1

Canadian Mathematical Olympiad, Problem #1

Determine the value of

.

Copyright (c) 2011 by Richard Hoshino.

This work is made available under the terms of the

Creative Commons Attribution-NonCommercial-NoDerivatives 2.5 license

2

Problem 1: Sum of Exponents

I stared blankly at the first problem, and shook my head in frustration.

I had no idea where to even start.

Olympiad problems are often characterized by beauty and elegance, where the statement of the

problem is clear and understandable despite the difficulties in finding a correct solution. This was not

such an example. Problem 1 was just ugly.

Over the years, I had developed the intuition of knowing which techniques and methods would most

likely lead to a solution. But not this time – not on this day, when I so desperately needed intuition.

I was quite familiar with the concept of exponents, yet Problem 1 was unlike anything I had previously

encountered. I was used to expressions like , where the exponent was a

positive integer, as well as fractional exponents, such as , which was just the square root.

But what was I supposed to do with the exponents in this question?

I stared at the first term in the expression of Problem 1, the ugly exponent . Was I actually

supposed to calculate the 2000th root of 9? In other words, was I supposed to determine the number

that equalled 9 when multiplied by itself 2000 times? This looked awful, and without a calculator, was

an impossible task. Surely this couldn’t be right.

There had to be an insight somewhere, that I knew. This was an Olympiad problem, and all Olympiad

problems have nice solutions that require imagination and creativity rather than a calculator.

I looked carefully at each of the terms in Problem 1. Each expression was of the form

. In the first

term, was equal to

; in the second term was equal to

; in the third term was equal to

; and so on, all the way up to the last term, where was equal to

. At least there was some

pattern to be found, although this pattern was completely useless to me at the moment.

In the entire expression, there was only one doable calculation, namely the term right in the middle. I

knew I could calculate

, using the fact that

. Since raising a quantity to the exponent

was equivalent to taking its square root, I reasoned that

.

But other than this, I had absolutely no idea what to do. Twirling my pen and closing my eyes, I

concentrated hard, hoping for some insightful spark.

One idea came to mind, that of a telescoping series. My mentor Mr. Collins introduced me to this

beautiful technique years ago at one of our Saturday afternoon sessions at Le Bistro. He illustrated the

concept with the following example:

Without using a calculator, determine the value of

.

3

Within a minute, I had obtained the correct answer by the standard technique of adding fractions by

finding the common denominator. In this case, the common denominator was 60, the smallest number

that evenly divided into each of 2, 6, 12, 20, and 30. The answer was:

Mr. Collins liked that approach, yet guided me towards a more elegant solution. He encouraged me to

see whether I could find any pattern among the denominators. I looked carefully, and after a few

minutes, I saw it:

2 = 1 × 2 6 = 2 × 3 12 = 3 × 4 20 = 4 × 5 30 = 5 × 6

He suggested I write

as the difference of two fractions. I quickly figured out that

. He then

asked whether there were any other terms in this expression that could also be written as the difference

of two fractions. I quickly noted that

and

. Once I saw the pattern, I discovered a

solution that was both beautiful and stunning.

can be re-written as

.

We then remove the parentheses to get

.

Since one negative fraction cancels out a positive fraction with the same value, all the terms in

the middle get “cancelled out”:

.

Like a giant telescope that collapses down to a small part at the top and a small part at the

bottom, this series collapses to the simple difference

, which equals

.

That day, Mr. Collins showed me many problems where the common denominator approach was too

laborious and lengthy, but the telescoping series approach would yield a simple solution by reconverting

the problem into a difference of two basic fractions. In recent years, I had seen other contest problems

involving telescoping series, and I was confident that it would work on this difficult Olympiad problem.

Yes, I got my insight and intuition!

I re-read Problem 1.

4

Determine the value of

.

I started with the general expression

, and tried to write it down as the difference of two functions.

I tried a bunch of different combinations to get the difference to work out to

, such as the

expression

which almost worked but not quite. I tried a bunch of other combinations for

another ten minutes using every algebraic trick I knew to set up a telescoping series. All of a sudden, I

realized the futility of my approach. I slapped my hand against my forehead. The denominator doesn’t

factor nicely, and so this approach cannot possibly work.

Damn it!

I looked at the clock.

9:19.

I started to panic. If I can’t solve this one, how was I supposed to solve the other four?

Come on Bethany, this is Problem #1! Focus!

I thought about the gentle soothing words of Mr. Collins, at one of the very first sessions we had

together. Simplify the problem by breaking it into smaller and easier parts, in order to find a pattern.

Okay, I’ll do that. I took a few deep breaths and found my heart rate slowly returning to normal.

I didn’t want to deal with the horrible expression given in the problem, a complicated sum of nearly two

thousand fractions. I had seen enough math contest problems to know that the number 2000 was a

smokescreen, and that it really had nothing to do with the question. By making the number big, the

problem looked a lot more intimidating than it actually was. For example, in the question that Mr.

Collins posed to me that day at Le Bistro, once I realized that the series telescoped, it didn’t matter

whether there were five fractions or five thousand fractions. In the former the answer was

,

and in the latter the answer would be

. The final answer was different, but at its heart,

it was the exact same problem.

I was sure the same was true with this Olympiad problem. Especially being the first problem, I knew

there had to be some clever insight that would lead to a short and elegant solution. Remembering the

advice of Mr. Collins, I decided to simplify the problem in order to discover a pattern, which would then

allow me to solve the actual problem.

I changed the denominator from 2000 to 4, to have just a few terms to play with. So now, instead of the

exponents ranging from

to

, I only had to consider

, and

.

5

Instead of adding 1999 ugly terms as in the actual problem, I only had three terms in the simplified

problem. By changing the expression into something easier, I could see whether any interesting

patterns emerged, which I could then apply to solve the actual problem.

So my simplified problem was to determine the value of

.

This looked much more reasonable. The middle expression was easy – I had already figured this one out

ten minutes earlier.

.

As I pondered how to calculate the values of

and

, a few ideas occurred to me. I scribbled

some calculations on my notepad, and added up the two fractions. Once I looked at what I had done, I

gasped.

Oh weird, why is it that?

I ran through the calculations one more time, double-checking that I hadn’t made any mistakes. All the

terms on the numerator aligned perfectly with all the terms on the denominator.

Wow.

I knew how to solve the problem. The solution was just staring me in the face. All I needed to do was

re-apply the same technique to the actual problem with 1999 expressions, and I was done. I knew it

would work out the same way, as I had discovered the key insight. I had determined the heart of the

problem.

Oh my god, this works for sure!

I laughed and shook my head at the irony of it all, finding an old friend who had arrived unexpectedly

into my life when I was a young girl in Grade 5 with no self-esteem and no sense of my own identity or

potential. My old friend, who first showed me the beauty of mathematics, and planted the initial seed

that brought me here today, competing for a spot on Canada’s team to the International Math Olympiad.

My old friend, forgotten for years, reappearing during the most important three hours of my life.

My old friend. The staircase.

6

The Staircase

“Two-hundred ten,” I blurted out.

Every head in the classroom turned towards me. Michael dropped his pen. Vanessa spilled her soft

drink. Several others stared at me in shock. Our Grade 5 teacher Mrs. Ridley just stood there with her

back leaning against the chalkboard and her jaw dropped, and for several uncomfortable seconds that

seemed like eternity, I could hear the sound of my own heart thumping.

One person broke the silence. Of course, it was Gillian.

“Did she get the right answer?”

Mrs. Ridley walked over to her desk and looked down at her teaching notes. She looked up in surprise.

“Yes, she did.”

Michael sat to my immediate left, and I noticed him clapping. “Wow, Gillian didn’t get the answer first.”

A few people joined in the applause, which prompted Gillian to turn around from her seat right in the

middle of the second row and glare coldly at Michael, who was sitting directly behind her.

“Bethany,” said Mrs. Ridley. “I can’t believe you figured that out so quickly. How did you do it?”

I shrugged and looked down at the floor, avoiding eye contact.

“Let me try that again, Bethany,” pressed Mrs. Ridley. “You got the answer, and I’m stunned that you

calculated it so quickly. You couldn’t have just added up the numbers. Could you present your solution?”

Could I present my solution? In front of the entire class? No way.

I slouched back into my chair, and stared at my notepad with the picture of the two joining staircases.

Mrs. Ridley knew how public speaking terrified me. She also knew that every time I spoke, my deep

baritone voice reduced half the class to giggles. What was Mrs. Ridley doing to me?

“I’m sorry,” she apologized. “I shouldn’t have asked you to share in front of the entire class. But how

did you add up the numbers from 1 to 20 so quickly? Could I walk over and see your sheet of paper?”

Without looking up, I shook my head furiously from left to right, and slumped my shoulders even further.

Even though my head was down, I could tell that everyone was staring at me. People started whispering.

All of a sudden, I heard a high-pitched cackle from Vanessa, who was sitting right in front of me. As

always, I’m sure it was her best friend Gillian who leaned over and whispered something cruel. I could

feel the sweat on my shoulders and back, and saw my hands trembling.

7

Also, if I showed Mrs. Ridley my piece of paper, I’d get in trouble for doodling in class, and not paying full

attention to my teacher with her mindless drills that drained the life out of me. I was only drawing

staircases because I was bored – in all shapes, sizes, and directions.

When Mrs. Ridley posed her daily “mental math” question like she always did at the halfway mark of the

class, it was a freak coincidence that my doodling provided the spark needed to answer her question of

calculating the sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20.

Or maybe the creative spark came from all those jigsaw puzzles I’d been doing since childhood, and

realizing how the two staircases would perfectly fit together. Either way, it was a complete fluke.

Mrs. Ridley’s mental math questions made me anxious, and hated how Gillian turned it into her personal

competition. Although I quickly understood mathematical concepts such as fractions and long division, I

just wasn’t fast at doing calculations. I could do them – they were easy – it just took time. Gillian was

excellent at both concepts and calculations, and always got the mental math question first.

Well… until today.

Gillian was Mrs. Ridley’s favourite student, and had been the teacher’s pet in every class since senior

kindergarten. While her teachers never saw Gillian’s cruel side, I certainly did, as I had been the target

of numerous verbal attacks in the playground. Gillian and Vanessa were inseparable, and formed a tight

clique with Alice and Amy, the two Chinese twins who sat directly in front of them in the first row.

Whenever a teacher wasn’t around, the four of them made fun of the way I looked and walked, calling

me names like Bigfoot and Big Ugly Bethany. I was so much bigger and taller, yet they were the ones

bullying me. I hated all of them.

“Bethany,” Mrs. Ridley whispered. “Instead of sharing your solution in front of the class, perhaps you

could write it down for me, and I’ll present it to the class? Would you do this for me?”

I looked at the clock. It was still thirty minutes to the end of class. Mrs. Ridley was now only three feet

away from me. There was no way I’d get out of this now. I wanted to fly away somewhere, someplace

safe where I could just be alone, or be hanging out in the park with Marie, my best friend – my only

friend. But I knew at this place, at this moment, there was no escape.

Trembling, I stared at Mrs. Ridley and nodded slightly. I picked up my pen and started writing.

Mrs. Ridley mercifully returned to the front of the classroom and started explaining another tedious and

repetitive drill while I focussed on writing my solution. I wondered whether I should talk about the

staircase. If I explained the staircase, would the truth come out, that I drew the picture only because I

was bored and was just scribbling on my page? Could I demonstrate my solution without the two

staircases? No, I couldn’t. But more importantly, I didn’t want to.

I started breathing heavily and focussed my thoughts. I took out a fresh sheet of paper, and a small

ruler from my pencil case. Despite all the noise around me, I was able to focus completely on the task in

8

front of me. When I had finished, I slowly lifted my hand, holding up a sheet of paper. Mrs. Ridley saw

me, gently walked over and took my solution.

With the class working on some other question, she saw that they were all occupied, and began reading

what I had written:

Answer = One staircase.

Two staircases joined together = One rectangle.

Two staircases = 21 × 20

One staircase = 21 × 10

Answer = 210

I didn’t take a breath as I stared at Mrs. Ridley reading my solution. Her eyes kept moving up and down

the page. Finally she glanced up with a confused look on her face.

“Sorry, I don’t get it.”

I felt deflated. I had presented a beautiful solution and my teacher didn’t get it. My shoulders sagged

down and formed a pout with my lower lip.

Mrs. Ridley bent down and leaned closer. “Could you explain it to me?”

She and I whispered back and forth, pointing to the diagram, until it all clicked in her mind.

“Oh wow,” gulped Mrs. Ridley. “I understand it. Wow.”

Mrs. Ridley got the attention of the class, and walked up to the front of the classroom, placing my sheet

of paper on top of the fancy projector before turning the machine on. After thirty seconds, the

projector warmed up and my staircase diagram filled the screen at the front of the classroom.

“Class, listen carefully,” said Mrs. Ridley. “In the mental math question, I asked you to calculate the sum

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20. I want to show you

Bethany’s remarkable solution. Look at the staircase on the left. There’s one square in the top row, two

21

20

9

squares in the second row, three squares in the third row, all the way down to 20 squares in the last row.

So the answer to the mental math question is equal to the number of squares in that one staircase.”

Mrs. Ridley paused. “Do you all understand Bethany’s picture?”

A bunch of heads nodded. I held my breath.

“Now here is Bethany’s clever solution. Take the same staircase and flip it,” said Mrs. Ridley, pointing to

the second staircase I drew. She then pointed to the rectangle on the right side of the page.

“And if you take those two staircases and stitch them together, you get a rectangle that’s 21 squares

wide and 20 squares high.”

“Oh, I get it!” exclaimed Michael. “It’s like a jigsaw puzzle. You take two staircase-shaped pieces and

join them together – and you get a rectangle.”

“That’s exactly right, Michael. Now, who can finish off the solution?”

Gillian raised her hand. With her chin lifted up, she spoke in her annoying and stuck-up high-pitched

tone.

“The answer is the number of squares in one staircase. Twice the answer is the number of squares in

two staircases, which is the same as the number of squares in that big rectangle. The number of squares

in that big rectangle is 21 × 20, width times height. So the number of squares in one staircase is just half

of that, which is 21 × 10. So the answer is 210.”

“Thank you, Gillian. That’s excellent. Did everyone understand that?”

Michael shook his head to indicate that he hadn’t. Mrs. Ridley called on Gillian again to explain why the

correct answer had to be half the number of squares in that big 21 x 20 rectangle, and why this worked

out to 21 x 10. This time, Michael nodded in understanding.

“Well done, Gillian,” said Mrs. Ridley. She turned to me. “And well done, Bethany.”

A few people turned towards me, and politely clapped. I turned red, although I felt a surge of pride.

Mrs. Ridley glanced towards the middle of the classroom, beaming at Gillian and me.

“Hey, we should call you the Bethany and Gillian team. The B & G Team.” My heart sank.

Gillian raised her hand. “The G should come first. The G & B Team.”

“Whatever,” replied Mrs. Ridley, dismissing the comment. She turned to write on the chalkboard.

After a few seconds, Gillian snickered and looked at Vanessa. She pointed to herself with her index

finger and then pointed her thumb directly at me.

“G & B. Good and Bad.”

10

Upon hearing Vanessa’s cackle, Mrs. Ridley turned back to face us. “What was that?”

“Oh nothing,” replied Vanessa. “I was just laughing about something that happened yesterday.”

Mrs. Ridley shook her head, not believing Vanessa. After a slight hesitation, she picked up her chalk and

continued writing on the board.

Gillian looked at Vanessa again. She pointed to herself and then stuck her finger towards me.

“G & B. Girl and Boy.”

This time Vanessa’s laugh could be heard by the entire classroom. Mrs. Ridley glared hard at Vanessa.

“I’m sorry, Mrs. Ridley,” replied Vanessa. “It won’t happen again.”

I glanced at the clock. The bell would ring in ten minutes. I look at the clock in desperation.

Hurry up, clock! Move faster!

I saw Gillian take out a clean sheet of paper to start drawing. She took some crayons out of her pencil

case and was rushing to complete something. I was dreading what it could be. Out of the corner of my

eye, I could see her broad smile. I leaned forward to see what Gillian was drawing.

A feeling of nausea came over me. It was a picture of two people. The girl on the left had a big G in the

middle of her shirt. She had long flowing black hair, tanned skin, and a perfectly-shaped face with high

cheekbones. The girl on the right had a big B in the middle of her shirt. She was really tall, with broad

shoulders, pale sticky white skin, a pointed noise, frazzled brown hair, and huge feet. The girl on the left

looked like a ballerina, while the girl on the right looked like a weightlifter or an Italian opera singer.

I didn’t need to be reminded which one was me. I felt a tear coming out of my right eye. I blinked and

felt tears coming out of my other eye.

Gillian scribbled some words below the two pictures and showed her masterpiece to Vanessa and the

twins sitting in front of them. They burst out in uncontrollable laughter.

As Gillian turned the page towards Vanessa, I saw the words written below the two girls. I didn’t

recognize the word below Gillian’s portrait. Under my portrait was a five-letter word that I most

certainly did recognize. The word BITCH.

Mrs. Ridley angrily walked over and grabbed the picture, and turned bright red upon seeing the two

images. Mercifully, the bell rang. As people began to exit the classroom, I was still processing the

seven-letter word that Gillian had written under her own portrait. GORJOUS. What on earth was that?

“Gillian, Vanessa, Alice, Amy, Bethany. You’re not dismissed.” I had never seen Mrs. Ridley this angry.

11

After everyone had left, there were just six of us in the classroom. Mrs. Ridley, me, and Gillian’s clique.

My heart was beating rapidly. The twins looked mortified knowing that they were in big trouble, while

Gillian and Vanessa looked indifferent. I quickly wiped the tears away from my eyes, and sniffled.

“Gillian,” snapped Mrs. Ridley, displaying the picture. “Did you draw this?”

“No,” she lied.

“Gillian, I’ll ask you one more time. Did you draw this?”

“Yeah,” she whispered. “But it was just a joke. I didn’t mean it.”

Just then, I figured out Gillian’s seven-letter word. I couldn’t help myself. I started to giggle.

“Bethany,” said Mrs. Ridley. “What Gillian did was completely unacceptable. It was deeply hurtful, and

it was absolutely wrong.”

She turned to Gillian. “What do you have to say?”

Gillian turned to me. “Sorry.”

Mrs. Ridley glared at her. “That was the most insincere apology I’ve ever heard. Try again.”

“Sorry,” said Gillian, a bit louder yet still unconvincing.

Mrs. Ridley sighed and looked at me, with a pained expression in her eyes. “Do you accept her apology?”

“Yeah,” I replied, and grabbed my books and binder to walk out the door. Just as I was about to exit the

classroom, a feeling came over me. A strange conviction, something I had never felt before.

I grabbed a piece of chalk from the side board and wrote down one word in big block letters.

GORGEOUS

I stood up tall and glared down at Gillian Lowell, the ultimate teacher’s pet who was the top student in

our Grade 5 class. I ignored her clique and locked my eyes on the spiteful princess who had bullied me

since senior kindergarten, and tapped the chalkboard right by the classroom door.

“I might not be gorgeous. But I can spell it.”

I strolled out the door, head held high. I turned right to enter the washroom and walked into a stall,

closing the door behind me. As I sat on the toilet, a huge grin came across my face.

Take that, BITCH.

12

Epiphany

“Happy Birthday darling!”

“Thanks Mom,” I replied, walking into the house and shutting the door behind me.

It was June 15th. My twelfth birthday.

On the weekend, Marie would be coming over and we’d have a small birthday celebration at our home.

But it would be a bittersweet occasion as Marie and her family were preparing to move to

Newfoundland at the end of July. Last week, I learned the heartbreaking news that Marie’s father got

transferred from his military post in Cape Breton, and would be starting a new job in St. John’s. I had

been depressed all week, knowing that my one friend would soon be gone.

Why couldn’t Gillian’s dad get transferred instead?

“So how was school?”

“Fine.”

School was never fine, but that’s what I always said to Mom so that she wouldn’t annoy me with her

questions. I didn’t want to talk to Mom about all the details of my personal life – it wasn’t her business.

Besides, she never shared with me the details of her life.

Thankfully, Grade 6 would finish in a couple of weeks and I wouldn’t have to see Gillian and Vanessa and

any of those other people at school until September, when we’d all start Grade 7 at Pinecrest Junior

High. The teachers selected Gillian as the class valedictorian. Were they blind? How could they have

picked her?

The noise in the living room was loud, as usual. “Can I turn off the TV?”

“No problem! It’s your birthday, so you can do whatever you want!” That was Mom, always cheery and

happy, like she was hiding something. It annoyed me that whenever she was preparing dinner in the

kitchen, she had to have the television on. It was so loud, and it was impossible to concentrate and read

my books, especially in our small bungalow house.

There was nothing good on TV anyway. It was all junk all the time. As usual, she had the CBC national

news on. It was always depressing stuff about something. This month alone, I overheard reports on the

execution of a crazy terrorist bomber in Oklahoma City, a tropical storm in Texas, a suicide bomber in

the Middle East, and yet another murder in Toronto.

I looked for the remote control. “Mom, where’s the remote?”

Mom didn’t look up from the pot she was stirring. “Should be in its usual place.”

The living room was always neat and tidy, since Mom was obsessed about keeping the house clean. But

the remote control was not where Mom always left it, on the right side of the main recliner.

13

“Can’t find it.”

“Then can you turn it off? Power button’s just under the TV. Dinner’s almost ready.”

I scowled and walked over to the TV. Just as I was about to hit the power button, I saw a clip of six

teenagers wearing matching red and white shirts writing what looked like math equations on a big

chalkboard. They were all smiling and laughing, and the camera focussed on the short freckled girl in

the middle, with her straight auburn hair falling just beneath her shoulders.

A deep male voice resonated from the television set.

“Are you a good problem-solver? Think you can match wits with the best in the country? When we

come back, meet Rachel Mullen and the rest of Canada’s team to the International Math Olympiad.”

I stood there in silence as a commercial came on. Math Olympiad?

“Ready for dinner, honey?”

I continued to stare at the screen.

“Bethany?”

“Uh, not ready. Two minutes – just wanna see something first.”

Mom turned off the stove, lifted the pot of stew, and set it on the kitchen table. Confused yet intrigued

by my sudden interest in the CBC evening news, she walked over and stood behind the worn-down Lazy-

Boy recliner and motioned for me to take her seat. I nodded and sat down, finding the remote control

sandwiched between the covers of the recliner.

“What are we watching?” she asked.

The commercials were still rolling. “I’m not sure.”

The screen returned to CBC news, and the anchor’s face appeared. I turned up the volume.

“When Rachel Mullen was growing up in Brandon, Manitoba, she was an over-enthusiastic child

interested in everything. Rachel enjoyed puzzles, and in particular, she loved mathematics. Although

some of her teachers encouraged her to pursue other interests, she stuck with math. And today, she is

one of our country’s brightest young minds, representing Canada at the biggest stage of them all, the

International Mathematical Olympiad next month in Washington D.C. Here is her story.”

The screen then turned to Rachel, writing some complicated math equations on the blackboard, looking

comfortable in a red and white polo shirt with the Maple Leaf stitched on its front. The next shot saw

Rachel sitting down opposite a middle-aged male reporter with the Canadian flag in the background.

With a confident look on her attractive dimpled face, Rachel smiled at the man.

14

“I was a hyperactive kid who had a hard time concentrating on tasks, always moving from one activity to

another. I was restless, constantly distracted, and got bored of things easily. I drove my parents crazy.

But they were so patient with me, and so supportive of my dreams and ambitions – even though some

of my dreams only lasted a few weeks! Luckily, I met an amazing teacher in Grade 7 who noticed my

interest in puzzles, and she introduced me to math contests. She saw my potential even before I did,

but eventually I realized it too.”

“And what was that?” asked the reporter.

“That I could develop a passion for a subject and grow to love it. That I could channel my high energy

and learn how to focus, how to concentrate, how to apply myself to a task that would help me achieve

an ambitious goal that I didn’t know was possible. And as I did more math and performed well on

contests, I met these great people from all across Canada at various math camps. Coming from a small

town, it was so good for me to be exposed to this world. My best friends are my math friends, because

we understand each other. Even though we all grew up in different cities and provinces, we have so

much in common. We’re interested in the same things. They’re cool. And they’ll be my closest friends

for the rest of my life.”

“What do you like about math, Rachel?”

“It’s the beauty, the patterns. Math isn’t about memorizing formulas and rules – it’s all about problem-

solving and critical thinking. That’s the subject at its heart. I’m not the most intelligent person at my

school, but am definitely the most creative and imaginative. That’s how I developed as I trained for the

Math Olympiad.”

“Tell me about the Math Olympiad. Are you looking forward to it?”

“Definitely. I’ve been wanting to make the IMO team for the past three years, and worked really hard to

make it. When I started out, I didn’t think it was realistic, but I set the bar really high and just went for it.

A couple of times, I thought about quitting, but my parents were really supportive – they taught me that

I could have no higher joy than to apply myself to the thing that filled my life with the greatest passion. I

realized that my passion was math, and that my dream was to make the IMO team for Canada. So that’s

what happened. Fortunately for me, I did it. And I’m so thankful for that.”

The clip then switched to a tall distinguished-looking gentleman with wrinkles around his eyes and deep

crease lines across his forehead. His perfectly-parted thin blond hair appeared almost child-like, and

looked so strange on top of his head. As the man began to speak, a small caption appeared on the

bottom of the screen: “J. William Graham, Executive Director, Canadian Mathematical Society”.

“The six members of our Canadian IMO team were selected from among two hundred thousand

students from Grades 7 to 12 who participated this year in local, provincial, and national mathematics

competitions. They represent the very best of Canada, and will be excellent ambassadors for our

country at the Math Olympiad.”

“Can you tell me more about the Olympiad competition?” asked the reporter.

15

“Our six team members will pit their skills against the top math students from over 100 countries. They

will attempt to solve six problems over nine hours, a mathematical `hexathalon’ that requires

exceptional problem-solving skills, mathematical understanding, daring, and imagination – the types of

skills that we Canadians require if we are going to be at the forefront of innovation in the 21st century.”

“Excuse me, Dr. Graham. Did you say six problems over nine hours?”

“Yes. Preparing for a competition like this one requires years of training, just like our sports athletes at

the Summer and Winter Olympics. Just as our Canadian athletes amaze us with their physical prowess

and push the boundaries of athletic performance, our mathletes do the same thing with their

intellectual prowess and should inspire all Canadians with their excellence and creativity.”

The feature then turned to the six students as they performed various activities. Rachel was the lone

girl on the team; the other five were boys. But none of the boys looked Canadian. Four were East Asian,

so they were probably Chinese. One was brown-skinned, and I had no idea what country he was from.

The brown-skinned boy was briefly interviewed – I was surprised how good his English was, with no

accent at all.

One of the boys was drawing a cartoon, while another played an incredible solo on the piano. Another

shot saw four of the boys sitting around a table playing cards, followed by a short clip of the entire team

outside in the park playing Frisbee. The camera focused on Rachel, who caught the Frisbee, turned her

body and flung it to one of her teammates. A perfect throw. Rachel raised her arms in celebration.

Finally, we saw Rachel with the reporter, in front of the Canadian flag.

“Rachel, what advice would you give to a young person who might be watching this?”

She paused. “Find out what gets you excited and passionate. Do what you love, and if others criticize

you because you challenge their way of thinking, ignore them. And make sure you persevere. Some of

my teachers tried to turn me away from math because I was a girl, and felt I should pursue other

subjects. But I stuck with math because that’s what got me excited. That’s what inspired my passion.

That’s what built my self-esteem and confidence. That’s what added meaning and purpose to my life. I

found my voice, and because I did, I’m now a Math Olympian.”

The feature ended, and the anchor moved to introduce the next story. I sat there stunned.

With a tear in my eye, I had an epiphany of what direction my life could take over the next six years. It

was a moment of absolute clarity, a voice from deep within shouting into my head and heart. I

shuddered with a feeling of excitement I had never felt before in my life. Since coming up with the

“staircase” insight last year in Mrs. Ridley’s Grade 5 class, I had been struck by the beauty of math but

never had anyone draw that out. I was never challenged or stretched in math class, so I found the

subject both boring and mindless – but I knew that there was something more.

There had to be something more to math. There had to be something more for me.

16

I wondered if math could be my escape from Gillian, Vanessa, and all the other bullies and brats at

school. Like Rachel, I wanted to do something special with my life and be accepted for who I was by

people with similar interests and passions.

I was deeply moved by Rachel’s words – self-esteem, confidence, meaning, purpose, finding one’s voice –

and in that one moment, I knew what I wanted to become.

“Mom,” I whispered, “I’m going to be a Math Olympian.”

She snickered, assuming I was joking.

“Mom!” I shouted. “I’m going to be a Math Olympian.”

She walked in front of the recliner and faced me, seeing the intensity in my eyes. Her face froze, and for

a brief moment I saw a look of absolute horror come over her, as if I had said something that brought

her immense grief and pain.

She reached over and hugged me.

“Bethany, honey, let’s enjoy our dinner together. It’s your birthday today. I made your favourite cake!”

She grabbed the remote, turned off the TV, and led me towards the dining table.

Oblivious to Mom, I ate dinner in silence daydreaming about the future. I wasn’t thinking about the

chicken stew, or about Mom’s birthday present, or about the triple chocolate cheesecake with fresh

raspberries that Mom made every year for my birthday. My mind kept coming back to a single thought

that gave me shivers, and renewed me with a sense of joyful hope.

I’m going to be a Math Olympian.

17

Golden Rachel

I shuffled slowly from my bedroom to the dining table, and found Mom sipping her coffee and reading

the Saturday edition of the Globe and Mail newspaper.

“Morning, sleepy head.”

“Morning,” I replied, yawning. I looked closely at Mom. “Hey, are you wearing my T-shirt?”

“You no longer wear it,” she replied. “It fits me perfectly.”

“Come on, Mom. It’s embarrassing – you wearing my stuff.”

“It’s a great shirt for the house. Look, I’m not going to go outside wearing this.”

Shaking my head, I walked into the kitchen to prepare my cereal. I knew that Marie’s clothes, once

outgrown, were passed along to her sister. I cringed at the contrast: Marie’s hand-me-downs went to

her younger sister; my hand-me-downs went to my mother.

Mom’s voice interrupted my thoughts. “Honey, come here. You’ll want to read this.”

I put down my bowl and walked over to the dining table, where Mom had the paper open to Page A-10.

Seeing the headline, I immediately sat down and began reading.

Canadian adds up to Math Olympiad Gold

Erin Murphy, Science Reporter

Special to the Globe and Mail

When Rachel Mullen clears customs at Pearson International Airport this afternoon, she will

declare that she is in possession of a bright and shiny object obtained during her recent trip to

Washington D.C. – a gold medal from the International Mathematical Olympiad (IMO).

Ms. Mullen, a 17-year old from Brandon, Manitoba, represented Canada at this year’s IMO, the

world championship of mathematical problem-solving for high school students. Out of 83

countries, the Canadian team finished in 24th place, with one gold medal and three bronze

medals amongst its six team members.

“This year’s IMO was one of the most difficult in years. The problems were extremely

challenging yet all six students performed exceptionally well,” said J. William Graham, the

executive director of the Canadian Mathematical Society, the organization responsible for the

selection and training of Canada’s IMO team.

This year’s IMO contest was set by an international jury of mathematicians, one from each

country, and was written on July 9th and July 10th in Washington D.C. On each day of the contest,

three questions had to be solved within a time limit of four-and-a-half hours.

18

Gold medals are given to the forty students with the highest score. Ms. Mullen correctly solved

five of the six problems, placing in a tie for 10th overall. She is the first female to win an IMO gold

medal for Canada.

“I’m so thrilled to have done well,” said Ms. Mullen, who will head to the University of Waterloo

in September on a full scholarship. “The problems were so difficult this year, so all of my hard

work and training paid off. I’m still in shock that I won a gold medal. I will treasure this

experience for the rest of my life.”

I put down the newspaper article. I was so inspired that my body was shaking. Rachel had done it – the

best in Canada, one of the best in the world. Seeing that newspaper article felt like a confirmation of

how my life would enfold over the next six years.

“That’s going to be me.”

Oops, I didn’t mean to say that out loud.

“Bethany, you said that last month after we saw that TV clip. Are you serious about this?”

“I’m serious.”

I sensed her doubt and glared at her. “What, you don’t think I’m good enough?”

Mom sighed. “Please, honey. I’m sure you are good enough, but is this what you want? The Math

Olympiad? Remember all those other kids in that TV clip? Big glasses, looking awkward?”

“So what? That’s how they look. You always say it doesn’t matter how people look.”

“I’m sorry,” replied Mom. “You’re right. But think about this. If you go for this Math Olympiad, you’re

going to need to make sacrifices and put in a ton of work. You won’t have time to hang out with friends,

play on sports teams, join school clubs, and enjoy just being a teenager. Trust me, it’s not worth it.”

“How do you know it’s not worth it?”

“From personal experience. Look, are you prepared to do all the work you need to become that good?

To be a world champion like Rachel?”

“Yes!”

“And how will you do that?”

I was getting angry. “I’m going to work hard, learn math at school, and do puzzles in my spare time.”

“But even if you work really hard and get an A+ in every math test at school, that’s not the same thing as

being one of these Math Olympian kids. Remember the TV clip? All that stuff about creativity and

imagination? Where will you learn that?”

19

“I’ll learn what I need to learn. And if I don’t learn that in school, I’ll learn it in my spare time. Math

books, stuff from the internet. I can do it.”

“Bethany, these kids have worked years to get to that level. Hours and hours each day. They don’t have

any social life outside of math, and there’s so much pressure to perform. It’s a brutal experience. It’s

not worth it.”

She paused and looked at me with a pained expression. “Trust me, I know something about this.”

“I want to do this.”

“Remember when we were watching Wimbledon a few weeks ago?”

“You’re changing the subject.”

“No, I’m not. Remember Wimbledon? Think about all those players we were cheering for – Capriati,

Hingis, the Williams sisters, Agassi, Sampras. Do you know how they all got so good?”

I didn’t answer.

“Their parents enrolled them in private tennis academies when they were little kids. They had personal

coaches. They practiced all day every day; they hired tutors help them with their school work in the

evenings. None of them had a normal life. I want you to have a normal life.”

“But what if I don’t want a normal life?”

“Honey, I don’t want to discourage you, but just look at our circumstances. I’m raising you on my own.

I’m sure all these Math Olympian kids have parents who are math professors, who teach them all sorts

of hard math, and work with them for hours every night. Or their parents have tons of money and can

send their kids to fancy private schools where their teachers give them special coaching. I can’t do that.”

I remembered the part in the TV clip when Rachel spoke so lovingly of her mom and dad, who supported

her dreams and encouraged her every step of the way.

“Mom, why can’t you be supportive of my dream?”

“Because your dream isn’t realistic,” she responded. “If you put all of your eggs in one basket and that

basket breaks, what happens then? You get shattered. Your life gets completely and totally shattered…”

Mom’s voice trailed off. She looked away from me. A few seconds later, I heard sniffling.

I stared. “Mom, are you… crying?”

Mom didn’t answer. She walked away from the dining room table and sat on the recliner, dabbing her

eyes with her fingers. I had never seen Mom cry before.

I sat on the floor facing Mom, and looked up at her. Neither of us spoke for several minutes. I was

shocked seeing Mom burst into tears, and felt really guilty.

20

After a long pause, Mom wiped her eyes with another tissue and put her hand on my arm.

“When I was your age,” whispered Mom. “I had an ambitious goal similar to yours. That dream took

over my life. It robbed me of my childhood. It robbed me of everything. I don’t want you to have to go

through what I did.”

“Go through what, Mom?” I asked.

She was so private about her past – about her childhood, about her adult life, even about my father’s

identity. I knew Mom grew up in Cape Breton but she didn’t hang out with people who knew her well;

she had the same government job for as long as I could remember but had no idea what she did before

that; and as for my father, all I knew was that he left Mom right after she got pregnant with me. I had

no picture of him; I didn’t even know his name.

Mom looked into my eyes. I sensed that she was opening up and was ready to share with me. I moved

closer.

“I’m sorry, honey, I don’t want to talk right now.”

With that, she gave me a tight hug, and headed towards her bedroom at the end of the hallway, closing

the door behind her. Well, so much for Mom opening up.

I ate breakfast in silence, reflecting on the conversation with Mom. What was in her past that she was

so afraid to share? What was she so ashamed about?

All I could think of was something Marie’s parents told me the first time I met them. They said that

Mom was an amazing figure skater many years ago. I remember asking Mom about it later that night,

but she laughed it off, saying that figure skating was just a hobby for her, and that Marie’s parents were

exaggerating. However, from the look on Mom’s face, I could sense there was something more.

What was she hiding?

I wanted to know more about Mom’s past so that I could better understand myself. But Mom’s parents

had passed away shortly after I was born, and she was an only child. So I had no uncles or aunts or

cousins to turn to – it was just Mom and me.

After thirty minutes or so, I got bored, and so I went back to my room and turned on the computer.

Waiting for the computer to load, I cut out the newspaper article on Rachel and pinned it on my

bedroom wall.

I then went online and did a Google search on “International Mathematical Olympiad”. I was curious:

just how difficult were these Olympiad problems?

I clicked on the first link. After a few more clicks, I found the six IMO problems from the contest that

Rachel wrote. My jaw dropped. What language was this written in?

21

Problem 1: Let ∆ABC be an acute-angled triangle with circumcentre O. Let P on BC be the foot of

the altitude from A. Suppose that . Prove that .

Problem 2: Prove that

for positive real numbers a, b, and c.

Problem 3: Twenty-one girls and twenty-one boys took part in a mathematical contest. Each

contestant solved at most six problems. For each girl and each boy, at least one problem was

solved by both of them. Prove that there was a problem that was solved by at least three girls

and at least three boys.

Problem 4: let n be an odd integer greater than 1, and let k1, k2, ..., kn be given integers. For each

of the n! permutations a = (a1, a2, ..., an) of (1, 2, ..., n), let . Prove that there

are two permutations b and c, with b ≠ c, such that n! is a divisor of S(b) - S(c).

Problem 5: In a triangle ∆ABC, let AP bisect BAC, with P on BC, and let BQ bisect ABC, with Q

on CA. It is known that BAC = 60° and that AB + BP = AQ + QB. What are the possible angles of

triangle ABC?

Problem 6: Let a, b, c, d be integers with a > b > c > d > 0. If ac + bd = (b + d + a - c)(b + d - a + c),

prove that ab + cd is not prime.

What was a circumcentre? A permutation? What were these weird symbols, ∑ and ≥? The only problem

whose meaning I actually understood was the third one, which looked like a pretty straightforward logic

question. But the Math Olympiad was a lot harder than what I thought was mathematics, i.e., logic

puzzles and calculations. My heart began to sink.

I found the webpage displaying the scores for each of the 473 contestants, arranged by country. I

scrolled down to “C” for Canada and clicked on the name of my hero.

P1 P2 P3 P4 P5 P6 Total Result

Mullen, Rachel 7 7 0 7 7 7 35 Gold

From the contestants’ scores, I saw that each of the six problems was marked out of 7 points, making 42

a perfect score. I saw that Rachel had solved every problem except for the seemingly-straightforward

logic question. So maybe Problem 3 wasn’t so easy after all. I clicked on the webpage a few times and

saw that only 20 of the 473 contestants had correctly solved that problem and gotten the full seven

marks; the majority of the contestants, Rachel included, had received zero. And these were the best

math students in the world? I was curious to know how to solve the problem. Was it really that hard? I

found the page with the solutions to each problem and clicked on the link for Problem 3.

I gulped. There were a few English words, but the official solution to Problem 3 looked like it was

written in another language. What on earth was supposed to mean?

Whatever it was, it appeared in the first paragraph and everything that followed it was absolute

22

gibberish. More strange symbols, very few words, and something to do with the “Pigeonhole Principle”.

The solution ran for a couple of pages. My head started to hurt. I sank down on to the bed, in shock.

So much for wanting to be a Math Olympian. I’d been thinking and dreaming about the Math Olympiad

every day for the past month, ever since watching that TV clip.

As if I could ever get to that level.

Mom was right. The Math Olympiad was just for super-special kids, the natural prodigies with math

professors for parents, or the gifted kids whose parents could afford private schools and personal tutors.

The Math Olympiad was for the Rachel Mullens of this world, not for ordinary people like me.

As I lay on my back, staring blankly at the ceiling, Mom walked into my open bedroom.

“Bethany.”

I sat up.

Mom spoke slowly. “I’m sorry.”

I got off the bed, and wordlessly pointed to the computer screen.

Mom turned around and looked at the monitor, scrolling through the solution to Problem 3. She turned

back to face me, looking confused.

“That’s the Math Olympiad,” I whispered. “I’ll never be able to do that. Ever.”

23

The Coach

“Bethany, honey, let’s go out for lunch.”

“I’d rather stay here.”

“But it’s the last Saturday before school. It’s a gorgeous day outside. Let’s take advantage of it.”

“I’m busy.”

“Busy with what?”

“Reading.”

“Bethany, there’s a new bistro that just opened on Charlotte Street. I overheard two ladies talking

about it the last time I was out grocery shopping. Brazilian coffee, oatmeal cookies, home-made choc…”

“No thanks.”

“And let me finish. Chocolate and raspberry pancakes.”

“What?”

“You heard me. Chocolate and raspberry pancakes. Your favourite. With freshly-picked berries and

dark chocolate.”

“I’m coming,” I replied, going upstairs and stuffing the latest Harry Potter into my bag.

Mom wouldn’t go down to the waterfront just to eat. She loved shopping and I knew she’d look for

some shoes and clothes after our meal, especially as her favourite stores were all on Charlotte Street.

Luckily for me, the Sydney Waterfront Boardwalk was just two blocks away, and I could happily wait

there, consumed by Harry’s latest adventures at the Hogwarts School of Witchcraft and Wizardry until

Mom was ready to go home.

We stepped into the car and drove towards downtown. I looked towards Mom, who looked quite nice

wearing a light blue summer dress that went well with her slim figure. It was a sharp contrast to what I

was wearing: a plain black T-shirt and faded jeans.

After parking the car, we walked along Charlotte Street until we arrived at our destination, Le Bistro. As

we walked in, we were hit with the aroma of freshly-cooked food. The smell was amazing. I noticed

plenty of people, of all ages, sitting around circular tables and chatting away.

We lined up to order our food. I got my chocolate and raspberry pancakes, while Mom got a coffee and

a spinach salad with maple dressing. We took the last empty table, and chatted for a few minutes until

the server brought us our food.

24

The pancakes looked amazing. Just as we were about to bite into our meals, I heard a loud voice over

my shoulder.

“Lucy? Is that you, Lucy?”

Mom stood up and smiled at the old man standing right behind me. “Mr. Collins, what a surprise.”

The old man gave Mom a warm hug. “Lucy, it’s so nice to see you. It’s been years since we last saw

each other. How are you doing?”

“Great. And you?”

“Super, though I miss teaching. I’m sad that school is starting next week and I won’t be in the

classroom.”

I glanced up at the man, who was as skinny as a rake, and stared at the thick grey hair on top of his

unusual pear-shaped face.

Turning to me, he smiled. “Hello there. You must be Bethany.”

He stretched out his hand. I hesitated before shaking it.

How did the old man know my name?

“Nice to meet you,” replied the old man. “Mind if I sit down and join you for a few minutes?” He

answered his own question as he grabbed an empty chair from another table and sat down next to us,

placing his take-out coffee right next to my pancakes. It was more of an invitation rather than a request.

“Bethany, it’s such a pleasure to meet you. My name is Taylor Collins. I taught mathematics at Sydney

High School for thirty-nine years. Your mother was in my class, quite some many years ago. I had the

privilege of teaching thousands of bright students during my career.”

Mom looked at me sheepishly. “He wasn’t including me in that list”.

Mr. Collins grinned, flashing a toothy smile that was partially hidden under his thick grey moustache.

“Well, Lucy, you were a bit preoccupied with something else when you were in high school.”

Mom forced a smile, and returned to picking away at her salad.

“Bethany,” said Mr. Collins. “I hear you’re really strong at math.”

How did the old man know I was strong at math?

“Oh sorry, I should have mentioned,” said Mr. Collins, noticing my reaction. “Cape Breton is a small

place, and it turns out that Donna Ridley and I are close friends. She told me about your creativity and

writing skills, and she thinks the world of you.”

25

I blushed. He smiled and sipped his coffee. Mom beamed, a look of great pride stretching across her

face.

“Bethany, from what Mrs. Ridley has told me about you, I don’t think you’ll find Grade 7 math either

challenging or interesting. The subject is so beautiful, but the curriculum is so boring. Yes, I’ll admit it.

The curriculum is boring.”

Mr. Collins waved his hands around as he spoke. I was shocked by his enthusiasm and energy.

“Imagine an art class where you spend the entire year practicing brushstrokes and learning how colours

combine without actually drawing a single painting, or being inspired by how simple techniques could

produce an artistic masterpiece revealing elegance and beauty? In art class, you create art and get

inspired by beautiful art. But in math class, it’s a different story. It’s an absolute shame.”

“Mr. Collins,” interrupted Mom, “are you offering to teach Bethany’s Grade 7 math class at Pinecrest?

That would be so wonderful.”

“No,” he sighed. “Well, I’d love to, but you know, the province has this mandatory retirement rule so

there’s no way of getting around that.”

“That’s too bad,” replied Mom, “since Bethany really enjoys math.”

Mr. Collins looked at me. “Say, just out of curiosity, have you ever written a math contest?”

I shook my head.

“Would you like to?”

Still shattered from seeing those IMO problems on the internet a couple of weeks ago, and realizing I

could never get to that level, I reluctantly shook my head and looked down at my uneaten pancakes.

“No problem. But I think you have the potential to do well in math contests since the key to success is

insight and creativity rather than one’s ability to perform rapid calculations or recall lots of formulas.

From what I’ve heard about you, I think you’d do really well. There’s a really fun contest open to all

Grade 7 students across Canada, and Pinecrest students always participate. Maybe you’ll consider

participating yourself.”

“You should offer to become Bethany’s personal math coach,” Mom said, laughing.

“That would be my pleasure.”

Mom and I froze.

Did I hear that right?

“I was just kidding,” Mom said slowly, her smile turning into a slight frown.

26

“Well, I wasn’t kidding,” Mr. Collins replied, the big grin returning to his face. He sipped his coffee again,

and looked at Mom. “Lucy, you know how much I love teaching. It’s my whole life, and it’s what I know.

This retirement thing is hard for me, and come next week, I’ll just be sitting around looking for

something to do. Sure, we’ll travel with the family, but we can’t do that year-round. What am I going to

do five days a week? Sit around and watch endless re-runs of Oprah?”

“That’s really kind of you, Mr. Collins, but shouldn’t Bethany be doing other stuff with her time than just

math?”

“Of course, Lucy. There are many classes that Bethany will be taking at Pinecrest, and I’m sure there will

be activities and clubs and sports teams that she’ll participate in as well. But if Bethany enjoys math,

and wants to discover her potential, then I’d be happy to work with her.”

“But we don’t have a lot of money,” Mom interrupted. “Raising Bethany on my own. I can’t afford to

pay…”

“It’s not about the money. I’ll do this for free.”

“That’s awfully nice of you Mr. Collins, but won’t Bethany be busy? You know, doing homework in the

evening from Monday to Friday, and on weekends, hanging out with her friends?”

Why was Mom saying this? She knew I didn’t have any friends. Marie had moved to Newfoundland last

month.

“You’re right, Lucy,” said Mr. Collins. “And Bethany, I’m sure you’re excited about starting Grade 7 next

week. I’m sure you’ll meet lots of nice people and that you’ll make many new friends. I know most of

the teachers at that school, and some of them are really good. But I don’t think you’ll find math class all

that exciting or inspiring, and Pinecrest doesn’t offer a Math Club like the one I ran at Sydney High. If

you’re looking for enrichment or challenge, you’ll be on your own and that’s why I’m offering to help.

“You don’t have to decide now, but think about it. I’m completely serious about offering my time to

work with Bethany, but I won’t pressure either of you. Let me know when you’ve made up your mind.

I’m really happy to have met you today, Bethany. And Lucy, it was such a treat to see you again.”

With that Mr. Collins leaned over from his seat and gave Mom a hug. He then stood up and stretched

out his hand to me.

I looked into his eyes, smiled, and gave him a firm handshake.

After Mr. Collins grabbed his coffee and walked out of Le Bistro, I gave Mom a cold stare.

“Mom, why did you blow him off like that?”

“Honey, do you really want a math coach? Remember how discouraged you were a couple of weeks ago,

after seeing that Olympiad problem on the internet? I don’t want to see you get all excited about

something and then getting hurt. You don’t want that, do you?”

27

“No,” I mumbled.

“It’s not worth the risk, is it?”

“No,” I whispered. I picked up my fork and bit into my pancake. It was cold. I felt overwhelmed by

everything, and the cold yucky pancake made matters worse. Tears began to form.

Mom moved closer to me, placed her hands on top of mine, and squeezed tightly. She gave me a

moment to compose myself and gently let go of my hands to give me a tissue. I dried the tears from my

eyes and sniffled.

“Is he a good teacher?” I asked.

“He’s a great teacher,” Mom whispered gently, holding my hands again. “Mr. Collins was the best

teacher I had when I was in high school. He made learning fun.”

“I’ve never had a teacher who made learning fun.”

“Bethany, honey.”

Mom took a deep breath and looked into my eyes. She sighed. “Would you like to work with him?”

I nodded.

“You could ask to meet him at Pinecrest. Maybe you two could work together after school once a week.”

“Wouldn’t that be weird?”

“You’re right,” said Mom. “Everyone would wonder what’s going on. Maybe it’s not a good idea after

all.”

We sat in silence for a few minutes. I grabbed another tissue.

Mom spoke up. “I have an idea.”

“What’s that?”

Mom paused and took a deep breath. After a long silence, she spoke again.

“We can offer to meet him right here, at Le Bistro. Every Saturday afternoon. It would be Mr. Collins

with the two of us.”

I snorted. “But you don’t even like math.”

Mom looked up at me with a warm and gentle smile. “Yeah, but I really like the coffee here.”

28

Cross Training

We walked into Le Bistro and found Mr. Collins waiting at the big table right by the window that

overlooked the waterfront. He stood up from his seat and greeted us with a warm smile.

“Lucy, Bethany, thanks so much for joining me.”

“It’s our pleasure,” said Mom. “Thank you for taking the time to meet with us.”

“Not at all. I’ve been looking forward to this all week! The past five days have been really boring for me.

There’s only so much puttering I can do around the house.”

We smiled. As we lined up to place our orders, Mr. Collins asked about my first week at Pinecrest.

“School’s great. I’m really enjoying Pinecrest.”

That was a partial truth. Some of my teachers were wonderful, while some were awful. I had made two

friends, Bonnie and Breanna, and was really happy that both were in all of my classes this semester. But

then again, so were Gillian and her clique.

Mr. Collins nodded and turned to Mom. “And Lucy, how are you? Are you still working for the feds?”

“Yup – same as always. My job is good.”

That was a complete lie. I knew Mom hated her job at the Canada Revenue Agency. She was the

administrative assistant to a senior director, who craved power and yelled all the time. I often heard

Mom complaining about him, especially whenever she was talking on the phone. I once asked why

Mom couldn’t just quit her job and find one that she enjoyed more. She replied that the job paid well

and came with all these perks, and it was too much of a risk to leave the government.

Mom and Mr. Collins both ordered a medium coffee, while I settled for a small hot chocolate. He kindly

paid for all three drinks.

“Mr. Collins, thanks for this,” said Mom.

“My pleasure,” he replied. “By the way Lucy, please call me Taylor. When you call me Mr. Collins, you

make me feel like an old man!” He laughed.

As I sipped my hot chocolate, I looked at him strangely. But he was an old man.

Mr. Collins turned to me. “Bethany, I’m delighted to work with you. Your mother and I spoke on the

phone a few days ago. As I’m sure she mentioned to you, we’ll work together every Saturday afternoon

for an hour, here at Le Bistro. Your mother will sit with us but not participate in our sessions. Together,

we’ll learn the real heart of mathematics, a beautiful subject that revolves around deep patterns

connected together in unexpected ways.”

He was moving his hands again, waving them in all directions.

29

“We’ll discover how math develops creativity and problem-solving skills, and makes important

connections to everything we see in the world. I won’t be teaching you a bunch of formulas for you to

memorize, nor will you be doing tons of rote drills and repetitive calculations. Instead, you’ll discover

key concepts for yourself, and take ownership of the material we cover. In this light, think of me not as

your professor or your teacher, but as your guide. Does that make sense?”

I nodded, pretending to understand, but was confused.

Math without formulas, drills, or calculations?

“Oh, and I should mention one other thing. I will carefully select problems that are extremely rich in

content and context, that will enable us to visualize mathematics, discover unexpected patterns, and

develop the creativity that I talked about earlier. That’s the primary purpose of our study together, and

who knows, maybe through our sessions you’ll find that you’ve got a knack for math contests. But

training you to succeed on math contests is not the goal. My goal is to inspire you with a love for the

subject of mathematics and reveal how it connects to everything in your life. That’s what I want to do.

Are you with me?”

I nodded. I was touched that Mr. Collins believed in my ability and potential – I remembered from the

TV clip that Rachel had found a math teacher in Grade 7 who believed in her ability and potential.

Was my life unfolding the same way?

“Bethany, before we start, do you have any questions for me?”

I shook my head.

“You ready to do some math?”

I nodded. In eager anticipation, I picked up my pen.

Mr. Collins placed a stack of white index cards in the centre of the table. “All right, let’s get started.

Please take the first index card and spell out your name.”

What?

“Bethany, I promise you there’s a reason behind my request”, said Mr. Collins. “Please take the first

index card and spell the letters in your first name.”

I shook my head and quickly wrote down my name at the top of the first card. B-E-T-H-A-N-Y.

“Great. Now spell your name backwards.”

Still staring at my card, I looked at the letters I just wrote and added them in the space below in reverse

order: Y-N-A-H-T-E-B. I dropped my pen and looked right at him. What does this have to do with math?

30

“Okay, now I want you to put down your pen and your card, and spell your last name backwards. And

this time spell it out loud.”

I hesitated. This wasn’t so easy. I said my last name a few times, visualizing the spelling in my head.

MacDonald, MacDonald, MacDonald.

I quickly got the first two letters: “D-L.”

MacDonald, MacDonald. Then I got the next letter. “A.”

Repeating the process, I finally came to the end a couple of minutes later.

All done. Wow, that was really hard.

“Great work,” said Mr. Collins. “Now Bethany, I’d like to introduce to you a problem-solving technique

that you can apply to almost any situation in life. The technique is to simplify the problem by breaking it

into smaller parts. Let’s take our example with backwards spelling. You have a nine-letter last name,

and it’s a lot to go through every time you figure out the next letter. I could tell you were saying your

name in your head many times to figure out how to spell it – MacDonald, MacDonald, MacDonald.

“Instead of thinking of your last name as MacDonald, think of it as Mac – Don – Ald. Three blocks, three

letters. Mac – Don – Ald,” he repeated, gesturing with three fingers on his left hand. “Now reverse-spell

your last name again. But this time, do it block by block, rather than letter by letter.”

Mac – Don – Ald, I thought to myself while staring at my hand, one finger for each syllable.

Ald, Ald, Ald. “D-L-A.”

Mac – Don – Ald. Don. “N-O-D.”

Mac. “C-A-M.” Wow, I got that in just a few seconds.

“Great work, Bethany. Now spell Cape Breton Harbour, doing exactly what you just did.”

Cape – Bre – Ton – Har – Bour, I said silently, using my five fingers to mark each block. I shook my

fingers a few times.

“R-U-O-B – R-A-H – N-O-T – E-R-B – E-P-A-C.” All done, in less than fifteen seconds.

“Excellent. See how much simpler that is?” I nodded.

“Let’s try some more.” Over the next fifteen minutes, Mr. Collins gave me 50 words to reverse-spell. I

was really good at it, and enjoyed the challenge. But what was the point of all this? How was this

mathematics?

“Well done, Bethany. Now let’s move on to something else.”

31

Oh good. We still hadn’t done any math. Maybe that was a warm-up, like in gym class yesterday when

were ran around for five minutes before stretching our muscles and playing indoor soccer.

“Before we move on to the next thing, do you need a little break? Your head must be hurting from all of

that backwards spelling.”

“I’m fine.”

“Okay then. I’m going to spell out a word you might not have seen before”. He proceeded to write

down the word BOOKKEEPER on a fresh index card. “Do you know what this word means?”

After I shook my head, Mr. Collins turned to Mom. “Lucy, do you know?”

“Is it a librarian?” she asked.

“Not quite, but that’s a good guess. A bookkeeper is someone who records business transactions, so it’s

basically another word for accountant. So I suppose many of your colleagues at the Canada Revenue

Agency are bookkeepers.”

Mom nodded.

Mr. Collins turned back to me. “Bethany, there’s something unique and special about the spelling of this

word. What is it?”

I saw it immediately. “The double letters. O-O there, K-K there, E-E there,” I said, pointing.

“Excellent. This word is one of only three words in the English language that have three consecutive sets

of double letters. The other words are BOOKKEEPING, which is basically the same word, and

SWEETTOOTH, although I’ve seen that spelled as two separate words.

“Bethany, here’s a puzzle for you. I’m thinking of a well-known 7-letter word that has two consecutive

sets of double letters. Let me write down the third and fourth letters of this word for you. I’d like you

to figure out the word I’m thinking.”

With that, he took an index card from the stack and wrote down

__ __ L L __ __ __

Mr. Collins just sat there, silently. Was I supposed to figure this out on my own? I looked at him

quizzically, confused that we had now moved on to word puzzles. He just smiled and nodded.

I thought I came here for math coaching not English coaching. Still in a bit of a daze, I proceeded to

ponder Mr. Collins’ question.

I quickly saw one word that fit the pattern, BALLBOY. But that didn’t have two consecutive sets of

double letters. I racked my mind to find other seven-letter words that fit that pattern. After a couple

32

minutes of concentration, I came up with another. TALLEST. But that didn’t work either. I was spent. I

looked up at Mr. Collins and shrugged.

“I don’t know.”

“No problem, Bethany. Have you found any seven-letter words that fit the pattern?”

“BALLBOY and TALLEST,” I answered.

“Great, that’s excellent. There are a few other words that fit the pattern, like CELLARS and CALLING and

BULLIED and WILLING, and also my last name, COLLINS. But the last one doesn’t count since it’s not an

actual English word. Now you know you’re looking for a seven-letter word with two consecutive pairs of

double letters. We talked about a powerful problem-solving strategy earlier. What was that strategy?”

“Simplify the problem.”

“That’s right. How do we simplify the problem?”

“Break it into small parts.”

“Excellent. So let’s apply that strategy to this problem. How do we do that?”

“The double letters.”

“Right. You want to focus on the double letters. You’ve got one pair of double letters already, and you

know that there must be another pair of double letters somewhere else. So what’s the next step?”

“Find out where they can go.”

“Great. Where can these double letters go? What are the possible cases?”

I took three index cards from the stack and scribbled an “x” where the double letters could go, using a

separate card for each case.

x x L L _ _ _ _ _ L L x x _ _ _ L L _ x x

Mr. Collins smiled. “Good job, Bethany. But you made one small mistake. Remember the statement of

the problem? What are we looking for? A seven-letter word with what?”

“Two pairs of double letters.”

“Two consecutive pairs of double letters.”

“Oh yeah,” I mumbled. I removed the third index card. So we were down to two cases.

33

“Excellent. Now let’s simplify the problem further by breaking it down into even smaller parts. What

are the possibilities for these double letters? Look at your first index card, the one where the first two

letters are the same. Can the first two letters be a pair of consonants?”

I thought for a few seconds. No, we coudn’t have words beginning with BBLL, CCLL, DDLL, FFLL, and so

on. That was impossible. The first pair of letters had to be vowels.

I didn’t need any further instructions. Just below my marking on the first index card, I listed the five

possible options:

A A L L_ _ _ , E E L L_ _ _ , I I L L_ _ _ , O O L L_ _ _ , U U L L_ _ _

Clearly the word couldn’t begin with AALL, IILL, or UULL. I thought for a couple of moments and realized

that OOLL didn’t make any sense either. I racked my brain for words beginning with EELL. I smiled.

“Is it EELLIKE?”

“Is it what?”

“You know, EEL-LIKE, like an eel?”

Mr. Collins laughed. “That’s a great try, but no, that’s not a word.”

I turned over the first card. I was convinced that the correct seven-letter word couldn’t begin with two

consecutive pairs of double letters, since EELLIKE was the only possibility that fit the pattern. I picked up

the second index card, the one where I had marked the pattern _ _ L L x x _ .

I thought for a few seconds and realized that this pair of double letters couldn’t be consonants either

(e.g. no word would fit a pattern like _ _ L L B B _ or _ _ L L C C _ ). So this pair of letters had to be

vowels. I listed the five cases on the second index card, writing down each case so that I could clearly

visualize them.

_ _ L L A A _ _ _ L L E E _ _ _ L L I I _ _ _ L L O O _ _ _ L L U U _

The first and third cases didn’t look right. Same with the fifth case. I figured that the correct pattern

had to be _ _ L L E E _ or _ _ L L O O _. I stared hard. What could the word be?

An idea came to me. I figured that the answer would probably be two separate words stuck together to

form a single word, like BALL-BOY or MALL-RAT. That made the most sense. But what could the first

four letters be? There were plenty of options: BALL, CALL, MALL, TALL, HELL, FILL, TOLL, DULL, and so on.

For each four-letter option, I tried to join it to a three-letter word to form the seven-letter answer.

But what could the last three letters be? EEL was the only possibility that I could think of. What other

three-letter words began with EE or OO? BALL-EEL certainly wasn’t a word. And the double O’s

certainly didn’t make much sense. Of course BALL-OOF, BALL-OON, and BALL-OOR weren’t words.

34

And then I just saw it: not BALL-OON, two words stuck together, but one single seven-letter word.

“BALLOON!” I shouted. A few customers from Le Bistro stared at us. I put my head down, embarrassed.

But I was glowing. Mom reached over and gave me a high-five.

Mr. Collins smiled. “Outstanding work. How do you feel?”

I let out a deep breath. “Good. Tired, but good.”

“Want to do a few more?”

“Okay!”

Mr. Collins proceeded to give me a half-dozen more puzzles with missing letters. On a couple of

questions, I saw the answer right away. For others, the answer wasn’t obvious at all, and I needed to

simplify the problem first, by breaking it down into smaller parts and figuring out which letters had to be

vowels and which letters had to be consonants. I answered some questions in a few seconds, while the

tough questions took several minutes.

I glanced at the clock, and was surprised to see that it was already 1:58PM.

“Well, Bethany, it was a pleasure working with you. You’re extremely bright, and once you learn a

technique, you quickly figure out how to apply it. That’s outstanding. You must be feeling tired.”

“Yeah, I’m tired.”

“I’m sure you’re exhausted. But do you feel good?” I nodded.

Although I was drained mentally, I felt alive and exhilarated, a feeling I hadn’t experienced since the

“staircase” moment in Mrs. Ridley’s Grade 5 class nearly a year-and-a-half ago.

“Now before we close, I want to ask you an important question. How much mathematics do you think

we did today?”

That was a weird question. Of course, we didn’t do any math. We spent the hour doing backwards

spelling and word puzzles. It was really good exercise for my brain, but we didn’t do any math.

Just as I was about to answer, I saw a quirky smile on Mr. Collins’ face, almost like a glow telling me that

he was asking me a trick question. I thought about his question some more. Simplifying difficult

problems, breaking it down into smaller parts, considering separate cases. Hmmm.

“A little?” I responded, hesitantly. “Simplifying questions into smaller parts. That’s math, right?”

“Absolutely. Our context was word puzzles and backwards spelling, but we spent every minute of our

time doing mathematics. You’re feeling exhausted because you’ve had an amazing intellectual workout

for the past hour. The exercises we did today have improved your concentration and your deductive

reasoning skills, and has trained your mind to make complex associations. All of these skills are a huge

35

part of developing your ability to think mathematically, and building your potential as a mathematician.

Because at the end of the day, math is not about remembering the right formula to get the right answer,

but applying your creativity and insight. Speaking of which, have you heard of Google?”

“I use it every time I’m online.”

“Did you know that Google’s search engine algorithm is just a little bit of Grade 12 math? It wasn’t

created by thousands of veteran engineers and computer scientists working for years on improving the

technology – Google was created by a pair of 23 year-old graduate students in California, who had the

insight of figuring out how to properly rank the relevance of trillions of websites in a fraction of a second.

Their solution is an ingenious application of linear algebra, something you’ll learn in a few years. While

their solution is simple, the ability to come up with that solution required extraordinary creativity and

innovative thinking. But these are skills that students can develop. These are the skills I want to

develop in you.”

“Those Google guys must be rich.”

“Yes, they’re incredibly rich. Their insight turned them both into billionaires. It still astounds me that

you can rank trillions of websites in the correct order in less than a quarter of a second. But that’s what

you can do with Grade 12 linear algebra, and that’s how internet search engines work. By building your

skills in critical thinking and deductive reasoning, this will make you a more confident, creative problem-

solver, able to apply your mind to serve society – just like these Google guys did.”

“How can I build that?”

“Build what? Build a billion-dollar company, or build your problem-solving skills?”

I blushed. “The second one.”

“I’d recommend you spend some time each week on your own, cross-training your mind, the same way

sports athletes cross-train their body to enhance their physical performance: tennis players run to

develop their endurance; football players lift weights to build muscle; and figure skaters do yoga for

more flexibility and range of motion.”

He turned to Mom. “Isn’t that right, Lucy?”

“I don’t know,” Mom said tersely.

“But Lucy…”

“Like I said,” replied Mom. “I don’t know.”

I stared at Mom, wondering why she had tensed up all of a sudden. Was it because Mr. Collins had

mentioned figure skating? Come to think of it, one of my Pinecrest teachers came up to me in

September and excitedly told me that he used to watch Mom skate, and how “honoured” he felt to have

me in his class. When I mentioned this to Mom, she immediately changed the subject.

36

As I was processing this unusual exchange, Mr. Collins swallowed the last drops of his coffee, paused for

a moment, and exchanged a knowing look with Mom before turning back to me.

“Bethany, as I was saying, you need to cross-train your mind, just as athletes cross-train their body. As a

young mathlete, I’d recommend you do lots of mental gymnastics to develop your mind, like we did

today. Our local newspaper here in Cape Breton has a page of puzzles every day, which are just perfect

for you. Have you heard of puzzles like the Cryptoquote, the Jumble, the Sudoku, and the Kenken?”

“Just the Sudoku. That’s in Mom’s newspaper.”

“Well, I’d encourage you to get the local Cape Breton paper so that you can do the others too. Really

easy to learn, and great training for the mind. We can discuss them next week if you’d like. Would you

like to do that?”

I nodded.

“Bethany, it was a pleasure working with you this week. I’ve got to run, but I look forward to seeing you

next week.”

He reached over and gave me a firm handshake.

“Thank you, Mr. Collins.”

Mom stood up. “Taylor, thank you so much. Really appreciate all this.”

Mr. Collins smiled and gave her a hug, whispering something in her ear that I didn’t quite catch.

Just as Mr. Collins walked out the door, he turned to the owner of Le Bistro who winked back at him. A

few minutes later, as we were packing up to leave, the owner stopped by our table, carrying two large

plates of piping hot chocolate and raspberry pancakes.

“Young ladies, these are for you.”

We blankly stared at him. Mom was the first to respond. “But we didn’t order this.”

“Oh I know,” the owner replied. “It was Taylor’s idea. And it’s on the house.”

“You serious?” I replied. He answered my question by handing us some cutlery and giving us a big smile.

“What a surprise… thank you,” replied Mom.

I grabbed a fork and bit into the first pancake. Wow, that was so good.

The owner looked at us. “Hopefully you two will be regular customers here.”

I laughed. “Oh yeah. Definitely.”

37

Locker Problem

We arrived at Le Bistro and found Mr. Collins waiting in line for his coffee.

“Hi Bethany and Lucy – perfect timing. What can I get you?”

“The usual dark roast for me, one milk and no sugar. Thanks, Taylor. How about you, Bethany?”

“Hot chocolate please.”

“Sure thing”. We stood two feet away from the cash register and saw the cashier ring up a bill of $5.89

for the three drinks. Mr. Collins opened his wallet and took out a ten-dollar bill. Just as the cashier was

pressing the register to get the change, he stopped her.

“Hang on a tick. I’ve got some change. Here’s $11.14,” he said, handing over the bill and six coins.

The cashier looked at him strangely, but entered the amount in anyway. She was surprised that the

change came out to exactly $5.25, and wordlessly handed Mr. Collins a five-dollar bill and a quarter.

Turning to us, he winked. “Useful application of math, eh?”

“Do you always do that?” asked Mom, “you know, with getting change?”

“Yes,” he confessed, “it’s a bit obsessive, I must say. I hate having a heavy wallet, so I have a simple

system to make sure I never have more than eleven coins in my wallet at any one time. My wife thinks

it’s overboard, but I think it’s a practical application of math.”

“Taylor, you’ve always been really quirky,” said Mom with a smile. “Glad to see you’re still the same.”

“Yes, I can’t stand change,” said Mr. Collins with a big grin. We both groaned as we took our seats. His

frequent use of puns were amusing yet painful.

“So Bethany, what did you think of today’s Jumble?”

From my backpack, I took out the puzzle page from the Cape Breton Post, with my completed answers.

The Jumble showed a picture of a well-dressed man yawning with the caption: Why the school

superintendent was always so tired.

To the left of the picture were six words that needed to be unscrambled, with the circled letters

requiring a final unscrambling to form the correct answer. Over the past two months, I improved my

ability to complete the Jumble puzzles, and built a deeper vocabulary as I learned new words. The

answer to the clue was always a visual or verbal pun, and always made me laugh or groan. No wonder

Mr. Collins enjoyed these Jumble puzzles; that’s probably where he got all his pun-ishing jokes.

I handed my completed Jumble to Mr. Collins.

38

SNURB ○B ○U R ○N S

AAMMD M A ○D ○A M

OIAPT P A ○T ○I O

WEFRE ○F E W ○E ○R

OOOODV V ○O ○O D ○O O

EEEDSC S E ○C E ○D ○E

The circled letters were B U N D A T I F E R O O O C D E, which unscrambled to the answer of

Why the school superintendent was always so tired:

He was “○B ○O ○R ○E ○D ” ○O ○F ○E ○D ○U ○C ○A ○T ○I ○O ○N

When Mom saw the answer to the Jumble, she shook her head and rolled her eyes. Mr. Collins laughed

out loud in hearty appreciation. I smiled at both of them.

“Okay Bethany,” said Mr. Collins. “Shall we work on a problem?”

I nodded.

“Excellent. Well, over the past two months, we’ve been looking at properties of natural numbers, which

we’ve been calling positive integers. We’ve looked at the properties of prime numbers, learned a way to

determine the greatest common divisor and least common multiple of a sequence of integers using

prime factorization, and studied the relationship between fractions and decimals. You’ve even figured

out some divisibility tricks, developing simple rules to show whether an integer is divisible by numbers

such as 6, 7, 8, 9, 11, or 99. We’ve covered some of the material from your Grade 7 class, but we’ve

gone far beyond that and seen how various strands connect in unexpected ways. Today we’re going to

take it one step further.

“To set the context for the problem we’re going to analyze today, let me ask you if Pinecrest’s hallway is

still the same from when I was last there, with several hundred lockers on one side.”

“Yup, it’s still like that.”

“And are the lockers still numbered 1, 2, 3, and so on?”

“Yup.”

“And what locker number do you have?”

“100.”

“Oh how perfect. So Bethany, how many students are at the school this year?”

39

“I think there are 407. Something like that.”

“Great. So here’s our problem for today. Pretend that the 407 students at Pinecrest Junior High play a

strange game with all the lockers. Suppose all of the lockers are initially closed. The first student, the

one with Locker #1, goes down the hallway and opens each of the lockers. After she’s done, the second

student, the one with Locker #2, goes down the hallway and closes all the lockers that are multiples of 2.

After he’s done, the third student, the one with Locker #3, changes the state of all lockers that are

multiples of 3, closing the open lockers and opening the closed lockers. And this continues, with each

student going down the hallway one after the other, altering the lockers whose numbers are multiples

of their locker number. You’ve got Locker #100, so when it’s your turn, you would alter the 100th, 200th,

300th, and 400th locker. Does that make sense?”

“I think so. So when it’s my turn, if Locker #100 is open, I would close it?”

“That’s right. And if it’s closed, you would open it. You do the same thing for Locker #200, #300, and

#400. Is that clear?”

“Yup.”

“So the exercise continues until all 407 students have completed their tour down the hallway. The last

student, the one with Locker #407, would just change the state of Locker #407. When she’s done, the

game ends. Here’s my question. At the end, how many lockers are open, and which ones are they?”

Oh boy. There was no way I was going to write down 407 marks on my notepad and figure out one by

one which lockers were going to remain open. That would take several hours, at least. So I just stared

blankly for about a minute and then looked at my napkin. This was my “white flag”, the symbol of

surrender. I had no idea where to proceed. I waved the white napkin.

“No problem. Over the past two months, we’ve discussed a powerful problem-solving strategy which

we’ve used almost every week. What is it?”

“Break down a difficult problem into smaller parts.”

“That’s right. When faced with a hard problem we don’t know how to solve, we break it down into

smaller components that are easier to solve. For example, a number is divisible by 99 precisely when it’s

divisible by both 9 and 11. So in coming up with the rule of divisibility for 99, you split the task into two

easier problems, figuring out a rule for divisibility by 9, then a rule for divisibility by 11. That’s how you

solved it.”

“But I can’t break this down into a smaller problem. You’ve got 407 people. You can’t just pretend that

you’ve got less people to make the problem easier.”

“Fewer people, not less people. So why can’t you pretend that you’ve got fewer people in the problem?”

I stared at him. “Well, uh, that’s illegal. You said there were 407 people and 407 lockers. You can’t just

change it to something less… I mean, to something fewer.”

40

Mr. Collins grinned. “The great thing about problem-solving is that we can change whatever we feel like,

and remove any restriction that we think is placed upon us. When we do math, there’s no strategy

that’s illegal. We can break any rule we want to. As long as we remember to put it all back together at

the end, no one will know, and we’ll never get caught!”

“Huh?”

“Breaking down a difficult problem into easier components simplifies the problem and allows us to make

headway. We’re turning something hard into something easier. In this case, we can apply the same

problem-solving strategy that we’ve used many times since September. By pretending than we have

fewer than 407 people, we’re simplifying the problem just so that we can discover a pattern. Once we

figure out what that pattern is, we can see if it holds for the actual problem with 407 students. But

there’s no way to find that pattern unless you simplify it first, since the problem is too cumbersome and

time-consuming with 407 students.”

“Are we allowed to do that?”

“Definitely.”

“It seems wrong.”

Mr. Collins nodded. “It might seem wrong, but when we refuse to be constricted by restrictions that we

think exist but really don’t, we become capable of achieving much more.”

“So what do we do?”

“My suggestion is for you to simplify the locker problem by pretending that there are only a handful of

students at Pinecrest, rather than 407. By considering this small scenario, you’ll be able to figure out

what’s going on, enabling you to solve the actual problem. So how many students do you want to

consider in our simplified problem?”

“How about five?”

“I think that’s too small. You probably won’t be able to figure out a pattern with only five students.”

“How about twenty-five?”

“I think that would be perfect. My recommendation is for you to create a square grid, and fill in the

numbers from 1 to 25 on the rows and on the columns.”

I created the grid. Mr. Collins then took my sheet and marked the first three rows of my diagram.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 O O O O O O O O O O O O O O O O O O O O O O O O O

2 C C C C C C C C C C C C

3 C O C O C O C O

41

“Let me explain what I just did. The first row marks the activity of the first student, the second row

marks the activity of the second student, and so on. So the first student opens all the lockers; that’s why

I marked it with the letter O. The second student closes every other locker, and that’s why I’ve marked

all the even lockers with the letter C. And for the third student, for each multiple of 3, every O locker

turns into a C, and vice-versa”. Mr. Collins gave me back my pen. “Please fill out the rest.”

It took just a few minutes to fill out the rest of the grid, especially since every student after the 13th

person could only alter one locker. I highlighted the lockers that remained open at the end.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 O O O O O O O O O O O O O O O O O O O O O O O O O

2 C C C C C C C C C C C C

3 C O C O C O C O

4 O O C O O C

5 C O O C C

6 C O C O

7 C O O

8 C C C

9 O O

10 C O

11 C O

12 C O

13 C

14 C

15 C

16 O

17 C

18 C

19 C

20 C

21 C

22 C

23 C

24 C

25 O

END O C C O C C C C O C C C C C C O C C C C C C C C O

“Well done. Now which lockers remain open?”

“Locker #1, #4, #9, #16, and #25.”

“Do you notice anything interesting about these five numbers?”

“They’re all perfect squares. 1×1, 2×2, 3×3, 4×4, and 5×5. Cool.”

“Excellent. Now here’s my question, and I’d like you to think about this for a few minutes. Why does

this work? Why is it that after you complete the game for twenty-five students, the lockers that remain

42

open are precisely those five lockers whose numbers are perfect squares? Why is it just those lockers

that remain open, while the others get closed? Take all the time you need.”

I thought about it for a few minutes, looking at the rows to see if I could find any pattern. Not seeing

anything insightful, I then looked at the columns. Well, I noticed that Locker #1, #4, #9, #16, #25 were

each altered an odd number of times, thus producing the end result that these lockers remained open.

For example, Locker #9 was altered three times (O to C to O), and Locker #16 was altered five times (O

to C to O to C to O). But it wasn’t clear to me why it worked this way, and why the other lockers, the

non-perfect squares, had to be altered an even number of times. I waved my white napkin.

“So let’s look at one of the lockers. How about Locker #16. How many times was this locker touched?”

“Five times.”

“Which five people altered this locker?”

“Student #1, #2, #4, #8, #16,” I said, pointing to the corresponding squares in my diagram.

“Now let’s look at Locker #12. Which students touched this locker?”

“Student #1, #2, #3, #4, #6, #12,” I said, looking at my diagram once again.

“Do you see the pattern?”

I nodded. I saw the pattern. Instead of just focussing on how many times each locker was altered, I

should have dug deeper and tried to figure out which students altered each locker.

“Okay if you see the pattern, tell me which students touch Locker #20, without looking at your diagram.”

“Student #1, #2, #4, #5, #10, #20,” I said, listing all the divisors of the number 20. Each student, whose

locker number evenly divided into 20, had to alter Locker #20 when they went down the hallway.

“And how about Locker #17?”

“Since 17 is a prime number, it’s just Student #1 and #17. ” The numbers 1 and 17 were the only divisors

of 17.

“Excellent. You’ve figured out the pattern. Since 20 has six divisors, this locker remains closed at the

end, since it’s altered by an even number of students. But the number 16 has five divisors, so this locker

remains open at the end, since it’s altered by an odd number of students. Bethany, can you explain why

perfect squares must have an odd number of divisors and non-perfect squares must have an even

number of divisors?”

Hmmm. It just does, I thought. It made logical sense, I just couldn’t find a way to explain why it made

sense. I shrugged.

“Remember how you solved the staircase problem in Mrs. Ridley’s Grade 5 Math class?”

43

I nearly jumped from my seat. “How do you know about that?”

“As I mentioned, Cape Breton is a small place. Sydney is even smaller. And the math teacher

community in Sydney is even smaller than that. We all know each other. Well, anyway, Mrs. Ridley and

I met up shortly after you solved that problem in her class, and she was glowing and telling everyone

about the staircase solution and how clever Lucy MacDonald’s daughter was.”

Mom leaned over. “You never told me about that.”

I was quite flattered that Mrs. Ridley spoke so highly of me. But why did Mrs. Ridley refer to me as Lucy

MacDonald’s daughter? Did Mrs. Ridley also know about Mom’s figure skating?

“So the staircase,” continued Mr. Collins, re-focussing my attention. “Mrs. Ridley asked you to sum up

the integers from 1 to 20. How did you do that?”

I drew a sketch of the two staircases and explained how to stick them together to form a rectangle. I

explained how each row contained twenty-one squares.

“Excellent. I love that solution. Let’s unpack that further. Your brilliant solution works because you

paired each number.” He pointed to the first row, with the one square on the top row of my first

staircase next to the twenty squares from the bottom row of my inverted staircase.

“Notice how you’ve got one square here, and twenty squares there, which adds up to 21. Do you see

that?”

“Yup, I see it.”

He pointed to the second row. “Here, you’ve got 2 squares and 19 squares, which adds up to 21.”

“You’ve grouped the numbers into pairs, so that the terms always add up to 21. In the third row, you’ve

got 3 squares and 18 squares, which add up to 21.” I nodded.

Mr. Collins took out a new sheet of paper and started scribbling.

20 squares 1 square

44

“In your solution, you took two copies of the staircase to form a rectangle and showed that the answer

to the problem was one-half the number of unit squares in the big rectangle. There’s a slightly simpler

solution: just cut off the bottom half of the first staircase, rows 11 to 20, flip it, and join it to the top half

of the staircase, rows 1 to 10. This produces a 10×21 rectangle, which has 210 total squares. So the

answer has to be 210.”

I nodded, appreciating the simple elegance of his one-staircase solution.

“So getting back to the locker room problem. I want you to pair each number. For example, let’s look at

Locker #20. You said that this locker was altered by six students. What should the three pairs be?”

I wrote the six numbers down on a piece of paper, namely 1, 2, 4, 5, 10, and 20. Like the staircase

problem, I paired the numbers from the endpoints, working in. So my pairs were (1,20), (2,10), and (4,5).

“Excellent. What do you notice about each pair?”

“They multiply to 20.”

“Super. In your staircase solution, each pair adds to the same number, 21. In the locker room problem,

each pair multiplies to the same number, in this case, 20. Locker #20 remains closed at the end because

20 has an even number of divisors. We have an even number of divisors since we can group the divisors

into pairs. To express it in another context, think of a wedding party where all the guests are married

and they come with their spouse. Since each person came with their partner, there must be an even

number of guests at the wedding. Does that make sense?”

I nodded. I liked that analogy. Considering the wedding party context, 1 would be “married” to 20, and

similarly 2 and 10 would be a couple, as would 4 and 5. And that explained why Locker #20 was touched

by an even number of people, and therefore remained closed at the end.

“Now do the same with Locker #24. What are its pairs?”

I wrote down the eight divisors of 24 on a separate sheet in my notepad, namely 1, 2, 3, 4, 6, 8, 12, and

24. The four pairs were (1,24), (2,12), (3,8), and (4,6).

“Great. One more. What about Locker #16?”

I wrote down the five divisors of 16, namely 1, 2, 4, 8, and 16. I got two pairs (1,16) and (2,8). But then

one number was left over. The only way I could pair 4 was if I could pair it to itself, to produce (4,4). But

1 square

10 squares 11 squares

20 squares

45

that was illegal since there was only one Student #4. I remarked how 16 = 4×4, and the light clicked in

my head. So that was why perfect squares behaved differently.

“Now let’s bring it home, Bethany. It’s clear you’ve got it. So let’s answer the original question posed at

the very beginning. You have 407 students and 407 lockers at Pinecrest. How many lockers remain

open at the end, and which ones are they?”

I only needed a minute to come up with the correct answer. “All the lockers whose numbers are perfect

squares. So 1×1, 2×2, 3×3, 4×4, and so on. Since 20×20 is 400, and that’s just under 407, there are

twenty lockers that remain open at the end. The lockers that remain open are 1, 4, 9, 16, and so on, all

the way up to 400.”

“Way to go.”

Mom leaned over and gave me a high-five.

“Bethany, let’s recap what we did. You learned another powerful problem-solving strategy today, of

simplifying a difficult problem in order to find a pattern, which you could then apply to solve the original

problem. We did that by pretending there were only 25 students at Pinecrest instead of 407, which

enabled you to discover the pattern of the perfect squares. Once you recognized the pattern, you were

able to figure out why this had to be the answer. That was 99% of the work. Having done that, it was

trivial to revert back to the original problem of 407 students to answer the given problem.

“Simplification is a great strategy for real-world problem-solving, whether it’s building a small model of

an airplane before creating the real thing, or making just a couple of pancakes to figure out the correct

proportion of flour to baking soda before cooking a huge batch to feed an army. After all, better to

make a mistake with a small amount rather than a large amount. That reminds me…”

Mr. Collins looked over at Joe, the owner of Le Bistro. He nodded. Joe winked and went into the

kitchen.

“Bethany, we had a great session today. You did a great job solving the Locker Problem, explaining how

the solution had to be the perfect squares. Here’s my challenge for you. Sometime this week, please

write up a full solution to the locker problem with 407 students, providing a clear proof that the answer

must be all the lockers whose numbers are perfect squares. We’ll go over your proof next week.”

“So you want me to write up what I just explained?”

“Yes. You did a great job explaining it orally; what I want you to do is to justify it formally. You’re

already a strong writer so you’ll do well with this. I find the high school math curriculum extremely

frustrating because it doesn’t emphasize proof-writing. I argued for years with the textbook writers and

the school board but got nowhere with them. So students go off to university and they can get most of

the answers, but they can’t properly explain how they arrived at those answers. What good is that? So

these engineers and computer programmers and laboratory scientists can get the right answers, but

46

can’t clearly explain it to anyone who might be analyzing their reports or de-bugging their computer

code or reading their research publications.”

“So you’re training me to become an engineer or programmer or scientist?”

“No, I’m helping you develop your problem-solving skills, as well as your oral and written

communication skills, which will come in handy for whatever career you end up choosing for yourself.”

“And what is that?”

“Anything you want to be.”

“Anything? I can have any career I want, as long as I put my mind to it and work hard to achieve it?”

“I believe that with all my heart.”

“So I can be a world-famous soprano singer? Or the trapeze artist for Cirque de Soleil?”

Mr. Collins paused. He broke out into a wide grin.

“Okay. Almost anything.”

47

Spelling Bee

Gillian leaned into the microphone. “S-H-E-P-H-E-R-D.”

“That is correct,” said Miss Carvery, our vice-principal. The students and teachers in the auditorium

applauded as Gillian strolled back to her seat on stage. As she walked by me, she calmly tossed her hair

back, lifted her chin up, and gave me a cold stare. I looked away.

It was Vanessa’s turn. After hearing her word, Vanessa hesitated before carefully responding.

“C-E-M-E-T-A-R-Y.”

A loud bell went off. “I’m sorry, that is not correct. The correct spelling is C-E-M-E-T-E-R-Y.”

Vanessa shook her head in disbelief, sagged her shoulders, and formed a pout with her lower lip. As she

slowly walked to the other side of the stage, Michael stepped up to the front. He received his word

from Miss Carvery, thought for a few seconds, and then leaned into the microphone.

“D-E-D-U-C-T-A-B-L-E.”

The bell went off again. “I’m sorry, that is not correct. The correct spelling is D-E-D-U-C-T-I-B-L-E.”

It was now my turn. I took a deep breath, stood up from my seat, and walked up to the microphone at

the centre of the stage. Miss Carvery gave me my word; I smiled. I knew this one.

“N-E-C-E-S-S-A-R-Y.”

The audience applauded, as I sat back down right next to Gillian. Neither of us acknowledged the other.

“We have now completed the eighth round of the inaugural Pinecrest Junior High spelling bee,” said

Miss Carvery. “Congratulations to all of you who participated today. After starting with fifteen spellers,

we are now down to our two finalists, Gillian Lowell and Bethany MacDonald. One of these two Grade 7

girls will represent Pinecrest at the provincial spelling bee in Halifax next month. The winner of the

provincials will represent Nova Scotia at the CanSpell National Spelling Bee in Ottawa.”

She looked at us. “Are you two ready?”

I quickly glanced at Gillian, who nodded confidently. I, on the other hand, was a wreck. My hands were

wet and slimy, and I could feel the sweat on the back of my neck. My legs were shaking.

I turned around and saw my friends Bonnie and Breanna sitting on the other side of the stage, and felt

awful that both of them were knocked out so quickly. Breanna knew her word in the third round, but

accidentally said S when she meant to say C. She asked if she could start over but the rules didn’t allow

that, and so she was eliminated. That was so unfair.

Why couldn’t the Spelling Bee have multiple winners, rather than continuing until there were fourteen

losers?

48

As Gillian stepped up to the microphone, I began having second thoughts of signing up. I read lots of

books and was a strong speller; however, this Spelling Bee competition was 100% stress and 0% fun.

But as I stared at Gillian’s back, I realized that I didn’t want to lose… to her.

My thoughts were interrupted as Gillian began spelling her word. “K-E-R-O-S-E-N-E.”

As she sat down to applause, I walked up to the front. Feeling a bit queasy, I relaxed slightly when I

heard my word. “P-A-R-A-L-L-E-L.”

Gillian was next. An easy one for her. “E-E-R-I-E.”

It was my turn. I hesitated slightly before answering. “K-H-A-K-I.” I was pretty sure that was right but

not completely sure, and exhaled when Miss Carvery said that my spelling was correct.

Gillian paused for a few seconds before correctly answering her word. “U-K-U-L-E-L-E.”

It was Round 10. Miss Carvery said a word that I had heard before, but wasn’t sure how to spell.

I paused. Was it one R or two R’s? Was it spelled with an O or an E? Picturing the index cards during

my first session at Le Bistro, I visualized the four possible options for the correct spelling:

CORELATE CORRELATE COROLATE CORROLATE

I looked towards Miss Carvery. “Can you say the word again?”

She repeated the word twice. It still wasn’t clear whether the ending was RELATE or ROLATE.

“Can you use it in a sentence?”

She used it in a sentence, but that shed no insight on the proper spelling.

I had one final question. “Can I have the definition?”

Miss Carvery spoke into her microphone. “To place in, or bring into, mutual or reciprocal relation.”

I keyed in on her last word: relation. That meant the word had to end in RELATE, not ROLATE. I smiled,

certain that the correct spelling had to be one of the following:

CORELATE CORRELATE

But which one was correct? Was it one R or two R’s? The former, CO + RELATE, seemed to be too

obvious to be a word in Round 10 of the Spelling Bee. The latter seemed more likely, especially as I

knew words like CORRUPT, CORRECT, and CORRODE that had two R’s following the letters C-O.

I leaned into the microphone, and crossed my fingers. “C-O-R-R-E-L-A-T-E.”

“That is correct.”

49

I let out a huge sigh of relief and walked back to my seat. Gillian took my place at the front of the stage.

Miss Carvery gave Gillian her word. It sounded like sigh-key. It was not a word I had heard before.

Gillian didn’t hesitate. “P-S-Y-C-H-E.”

I wiped the sweat off my forehead, knowing that I got super lucky. I wouldn’t have spelled that word

correctly. I stepped up to the front, and smiled when I received my word for Round 11.

This was an easy word I first heard in Grade 1 when we were discussing different colours and shades in

art class. Just to be sure, I visualized the index card and listed the two ways the word could be spelled:

MAROON MARROON

The ending was definitely OON. But was it one R or two R’s? I pictured similar words that had two pairs

of consecutive double letters, like RACCOON and BUFFOON and BASSOON, and of course, the word

BALLOON from my first session with Mr. Collins. Yes, it definitely had two pairs of double letters.

I smiled. “M-A-R-R-O-O-N.”

The loud bell went off. My mouth opened wide, in shock.

What?

“I’m sorry that is not correct. Maroon is spelled M-A-R-O-O-N.”

Stunned, I trudged off to the losers’ side of the stage, to join Bonnie and Breanna. How did I miss that

word? How could I have spelled that wrong?

The audience clapped to congratulate my effort but I didn’t turn to face them. I slumped down in a

chair, and ignored the pat on the back from Bonnie. I covered my face with my hands and closed my

eyes, shaking my head at my carelessness.

How did I miss that word?

“Gillian, please come up to the front. This is the championship word. If you miss it, Bethany will return

and you two will start Round 12 together. If you spell it right, you will be the champion of the Pinecrest

Spelling Bee. Are you ready to receive your word?”

“Yes,” replied Gillian.

Based on her excited yelp, I knew Gillian had it. I didn’t bother opening my eyes as Gillian calmly spelled

the championship word.

“C-A-T-A-C-L-Y-S-M.”

“Congratulations, Gillian Lowell, you are the winner of Pinecrest’s first ever Spelling Bee!” The audience

stood up and cheered. I got up from my seat too, and joined half-heartedly in the applause.

50

As Gillian received a large trophy and posed for pictures with Miss Carvery, a few people came around

to where I was sitting, to congratulate me.

“Great effort, Bethany.”

“I was pulling for you. Gillian’s such a stuck-up bitch.”

“Sorry about the last word.”

None of that was of any comfort. I was relieved that the competition was over, but was still disgusted at

myself for missing such a simple word. A few people patted me on the back but I ignored them.

“Ready to go?” asked Breanna. I was ready to go eat ice cream with her and Bonnie. Before the Spelling

Bee began, the three of us had decided that no matter who won, we were going to spoil ourselves to an

Oreo Overload at Cold Stone Creamery. I was definitely ready to leave.

“Just wait,” I replied, changing my mind. “Hold on just a sec.”

As Gillian held up the trophy and celebrated with her clique, I thought I should do the honourable thing.

Gillian saw me walking towards her and turned to face me. Her smile turned into a frown. She put the

trophy down.

“What do you want?”

I held out my hand. “Congrats. Good job.”

She looked at my hand for a few seconds. She ignored it.

“Maroon?” she hooted, in a mocking tone. “Maroon? We learned that back in Grade 1. Are you really

that stupid?”

Gillian flipped her long hair back, and with her chin lifted high in the air, turned back towards her friends.

Vanessa cackled out loud and also turned her back to me. The Chinese twins had a sympathetic look on

their faces, but I knew their allegiances were with Gorgeous Gillian and not with Big Ugly Bethany.

With a clenched jaw, I walked back angrily towards Bonnie and Breanna. “Yeah, I’m ready.”

As the three of us filed out of the auditorium, I felt someone tap me on the shoulder. I turned back and

saw Miss Carvery looking up at me.

“Bethany, can I have a word with you?”

“Uh… yeah,” I replied, staring into the eyes of the vice-principal we all adored.

Bonnie and Breanna saw that Miss Carvery wanted my attention, and told me that they’d wait for me

outside.

51

“Girls, just head on to wherever you’re going. Bethany will meet you there.”

“Yes, Miss Carvery.”

As Bonnie and Breanna exited the school to make their way towards Cold Stone Creamery, Miss Carvery

led me towards the far end of the auditorium, away from the crowd of teachers and students huddled

together in the middle.

“Please have a seat, Bethany,” she said as pointed towards the seat next to her.

I sat down and looked into the eyes of my favourite teacher at the school. She was a beautiful woman

with light black skin and hair just above the shoulders, reminding me of Halle Berry, the actress from X-

Men. Bonnie’s mom told us that Miss Carvery had won a huge scholarship to some famous university in

England, and returned back home to Cape Breton to start teaching five years ago. I couldn’t believe that

Miss Carvery was already vice-principal and only 33 years old – she was even younger than Mom.

“Bethany, I’m so sorry about what happened at the end there.”

“Yeah,” I sighed. “Such an easy word – I couldn’t believe I spelled it with two R’s.”

“I wasn’t talking about that.”

I stared at her, with a puzzled look on my face.

“I admire how you handled yourself just now, that even though you were disappointed at finishing

second, you had the class to walk over and congratulate the winner. I heard what Gillian said to you,

and I was disappointed by her insulting and hurtful words. But I wanted to commend you for what you

did, and the integrity you displayed by calmly walking away. I applaud that.”

“Thank you,” I replied, beaming.

“Other than the last word, did you enjoy the Spelling Bee?”

I swallowed. “Uh, can I be honest with you?”

“I expect nothing less than complete and absolute honesty. Yes, you may tell me how you felt about

today.”

“I really hated it, Miss Carvery. All that pressure, all that tension. I felt really anxious the whole time,

and felt so bad hearing that annoying bell whenever people spelled the word wrong. It wasn’t fair to

see Breanna get eliminated because she said the wrong letter by mistake; she definitely knew that word.

And I didn’t like how competitive the whole thing was, and why we couldn’t have multiple winners –

why we had to keep going and going until there was only one person left. It wasn’t fun at all.”

I paused. Did I say too much?

“Is there anything else?” she asked. I shook my head.

52

“I agree with you, Bethany. The Spelling Bee is very competitive. In many ways, it simulates the real

world where the bar is set really high, and fifteen people are competing for a prize that only one person

can win – maybe it’s a prestigious entrance scholarship to a university, or maybe it’s a particular job.

But the Spelling Bee serves much deeper purposes.”

“Such as?”

“To celebrate excellence in academic achievement. To encourage young people to develop their skills in

spelling and language through friendly competition. To make people stronger…”

“But it’s not friendly competition. Everyone who didn’t win feels like a loser.”

“Do you feel like a loser, Bethany? You correctly spelled ten difficult words, without a single error, in an

extremely challenging high-pressure environment, with hundreds of people watching you. That’s not

the mark of a loser. That’s the mark of a champion.”

“But look at how it ended. I missed a word that I knew back in Grade 1. Gillian was right.”

“I think this experience has made you stronger, and you’ll bounce back. I know it. In fact, I’m sure of it.”

“How do you know?”

“Because every time I look at you, I’m reminded of someone I went to high school with. Someone I

looked up to very much. Can I tell you her story?”

“Sure.”

“There was a girl in my school who was incredibly ambitious. She had big goals and took huge risks. She

battled back from disappointments, competed tenaciously in the face of enormous pressure and

adversity, and won everybody’s respect and admiration, regardless of the final result. I see so much of

this girl’s courage and determination inside of you.”

“Thank you.”

“That girl’s name was Lucy MacDonald.”

My jaw dropped. I stared at Miss Carvery in shock.

“Mom?”

53

The O-Ring

“Bethany, how did you find the practice Gauss contest?”

“Pretty good,” I replied, handing Mr. Collins a sheet of paper. “Better than last week.”

Mr. Collins smiled as he looked over my answers to the twenty-five questions from a previous Grade 7

math contest, which I had worked on right after breakfast this morning. The contest had taken me

about 75 minutes to complete, exceeding the one-hour time limit, but this was for recreation rather

than competition.

The Grade 7 paper was called the Gauss contest, named after some famous German mathematician, and

was written by nearly twenty thousand students each year from all across Canada.

In January, Mr. Collins and I had begun a new routine, where we spent some time at the beginning of

each session reviewing the problems from an old Gauss contest. While I decided that I wouldn’t be

writing the actual Gauss contest with other Pinecrest students in May, I really enjoyed these practice

problems – they were a lot more interesting than the stuff we covered in Grade 7 Math. And because it

wasn’t a high-pressure contest environment, I could take as long as I wanted, rather than having to rush

through the problems. Mom often told me how I had the right perspective: doing math because it was

fun, rather than competing or comparing myself with Gillian Lowell or anybody else at Pinecrest.

“I left Question #20 blank.” I said to Mr. Collins. “But I think I got the rest.”

“Excellent work. Let’s go over the questions.”

I went through the first nineteen questions and explained to Mr. Collins how I had solved each one. He

followed the logic in my explanations, and agreed with all of my answers. We arrived at Question #20.

I looked at Mr. Collins and shrugged. “I wasn’t sure how to do this.”

“What problem-solving methods did you try?”

“Uh, I got stuck because there’s just too many words. It’s not clear how to keep track of everything.”

“No problem at all. Hang on one second. I’ll be right back.”

With that, Mr. Collins stood up, walked over to the counter at the front of Le Bistro, and returned a few

seconds later with ten packets of white sugar.

“What’s this?”

“Hold these packets of sugar in your hand, so that you have something concrete to hold and visualize. It

will be much easier to solve problems like this if you use props.”

“Isn’t that cheating?”

54

“Not at all. It’s creative problem-solving.”

I laughed, taking the ten packets of sugar in my hand. I re-read the question.

20. Anne, Beth and Chris have 10 candies to divide amongst themselves. Anne gets at least 3

candies, while Beth and Chris each get at least 2. If Chris gets at most 3, the number of candies

that Beth could get is

(A) 2 (B) 2 or 3 (C) 3 or 4 (D) 2, 3 or 5 (E) 2, 3, 4, or 5

Anne got at least three packets of sugar, while Beth and Chris got at least two. So I put three packets on

my left, two in the middle, and two on my right, keeping the remaining three in my hand. I wanted to

know how many total packets could be placed in the middle, the pile belonging to Beth. Of the three

remaining packets, how could I distribute them among the three individuals?

Suddenly, the insight came to me, that the information about Chris getting at most one extra packet was

completely irrelevant. Since Anne and Beth could be given additional packets with no restriction, I could

just distribute the remaining three packets between Anne and Beth. So Beth could receive zero, one,

two, or three additional packets of sugar, and Anne would take the rest. Just making sure, I wrote down

in my notebook the four possible scenarios for the final number of sugar packets for each individual:

Anne Beth Chris

6 2 2

5 3 2

4 4 2

3 5 2

Yes, that worked for sure. I looked at the five possible answers on the contest sheet, and saw that

option (E) corresponded to Beth getting 2, 3, 4, or 5 candies.

“The correct answer is (E),” I replied.

“Great job, Bethany. Now let’s move on to Question #21.” Within a minute, we finished that problem

and moved on to Question #22.

22. The total number of square and rectangles, of all sizes, that appear in this figure is

(A) 28 (B) 30 (C) 32 (D) 34 (E) 36

“So how did you do this one?” asked Mr. Collins.

“I broke the problem down into easier cases. I figured out the types of squares and rectangles that

could appear, you know, 1×1, 1×2, 2×2, 2×3, and so on. I counted each case, and then added it all up.”

“Excellent. You simplified the problem by figuring out the possible dimensions of the squares and

rectangles that could appear in the picture, and then found its total. Okay, what did you get?”

55

I handed him a sheet of scrap paper which marked all the different possibilities.

There were nine 1×1 squares, since there were 9 unit squares in the diagram.

As for 1×2 rectangles, there were 12 of them in the picture: six “horizontal” and six “vertical”.

Counting 1×3 rectangles, I found 6.

Counting 2×2 squares, I found 4.

Counting 2×3 rectangles, I found 2.

Finally, there was the 3×3 square. Of course, there was only 1 such square.

56

“So I found all the cases, and added them up. I got 9 + 12 + 6 + 4 + 2 + 1, which equals 34. So that’s the

answer.”

“Are you sure you considered every case?”

I looked at him in confusion. “Yes, I’m sure.”

Mr. Collins picked up a piece of paper and proceeded to draw a little table.

“Let’s look at the row and column dimensions of each possible rectangle that can appear in our big 3 by

3 square. Let’s summarize this information in this nice table. Note that in any rectangle, its row

dimension or column dimension can’t be more than 3, since the big square is only 3×3. Do you agree?”

I nodded. We couldn’t have a 4×5 rectangle inside a 3×3 square, nor could we have rectangles with

dimensions such as 2×4 or 5×1. Both the row and column dimension could be at most three.

“Bethany, I’d like you to put an X mark next to every scenario you’ve considered. For example, you

counted 12 rectangles that were either 1×2 (horizontal) or 2×1 (vertical). So mark both of those entries

in my table with an X. Do the same with all the other cases you considered.”

Looking at my diagrams of shaded squares and rectangles, I put an X next to every case I had considered.

Oops. I had forgotten the 3×2 rectangles. It was easy to see that there were 2 such rectangles.

So the correct answer wasn’t 34, but 34+2 = 36. I grinned at Mr. Collins.

“The answer is 36.”

1 2 3

1

2

3

1 2 3

1 X X X

2 X X X

3 X X

Column Dimension

Row Dimension

Row Dimension

Column Dimension

57

“You got it.”

Mom leaned over and gave me a high-five.

“Let’s recap what we did,” said Mr. Collins. “To answer this difficult problem, you first simplified it by

breaking it down into smaller cases, which in this context, was to enumerate all possible rectangles of all

possible dimensions…”

“Enumerate?” I interrupted.

“Sorry. Enumerate is just a fancy word for count.”

“I see.”

“So what you did was break the problem into different cases and enumerated each of these cases

separately. You first looked at 1×1 squares, then you moved on to 1×2 rectangles, then 2×1 rectangles,

and so on. But then you forgot to consider the 3×2 case. You were convinced that you had found all the

possible cases, but once you summarized this information in my little table, you realized you had

forgotten to count the 3×2 rectangles. How were you able to see that you had forgotten a case?”

“The information was put in a table.”

“That’s right,” replied Mr. Collins. “We presented the information in a highly structured way. When

you’re breaking down a hard problem into smaller and easier cases, you need to figure out how to

structure your information so that you know for sure you haven’t forgotten anything. Having a clearly

defined structure is the difference between thinking you’ve considered every case, and knowing you’ve

considered every case. Say, that reminds me. Can I tell you a story?”

“Sure.”

“Have you ever heard of the Challenger space shuttle disaster, the one from 1986?”

I shook my head. That was before I was born.

“How about you, Lucy?” asked Mr. Collins, turning to Mom.

“Oh definitely. I remember that day clearly. I was watching it on TV. It was awful.”

“Yes, it was a terrible tragedy. Bethany, here’s the story. The American space shuttle Challenger

exploded a minute after takeoff, killing all seven astronauts on board. It turns out that on the day of the

launch, the temperature was just 2° C on the ground. NASA had never previously launched a shuttle

with the temperature below 10° C, but they took the chance for this particular space mission. They

made the decision to launch after checking each of the components of the space shuttle, figuring the

cold weather would not affect any of these components. But they didn’t check everything. They

overlooked something.

58

“The rocket boosters of a space shuttle are comprised of several parts, which are fitted together by

rubber bands called O-rings, to ensure air doesn’t escape. It’s basically a giant elastic, but of course,

much larger and stronger. In the cold weather, these O-rings didn’t seal properly, allowing pressurized

hot gas to escape the rocket booster and reach the external fuel tank. That created a deadly

combustion effect that led to the space shuttle’s explosion. What a tragedy: seven people lost their

lives aboard a billion-dollar space shuttle, and the part that malfunctioned was a rubber band that was

probably worth five dollars.”

“Did no one know about this?”

“Actually, people did know, and that’s what makes the disaster so awful. The NASA engineers knew that

the O-rings would work properly as long as the temperature remained above 10° C. But once the

temperature dropped to freezing, they predicted that the O-ring would lose resilience and pliability, and

pose a huge risk to the safety of the space shuttle. The engineers reported the problem to their higher-

ups, but they were overruled.”

“Overruled?” I asked.

“The mission commanders at NASA, the big guns making the decisions, insisted the engineers prove it

was not safe to launch rather than demonstrate the conditions were safe to launch. Notice the

difference between the two. The engineers couldn’t show that it was not safe to launch as they had

never tested the O-rings in such cold temperatures.

“During one of the public hearings in the aftermath of the disaster, a Noble-prize winning physicist took

a small O-ring, and placed it in some ice water for a few minutes. When he took out the O-ring from the

ice water, he showed how it lost elasticity and became brittle. The cold temperature that day made the

O-ring less resilient, creating space for the pressurized gas to escape the rocket booster. That’s what

caused the shuttle to explode.”

“But I don’t get it. Why did they launch if they knew something could go wrong?”

“Well, NASA was under a lot of pressure. The space shuttle launch was delayed a bunch of times that

previous week, due to bad weather. Everyone was waiting for the launch to occur, and people were

getting impatient due to the weather delays. So despite the objections and concerns from the engineers

and the other scientists, the mission control at NASA decided to lift off, with horrible consequences. It

was a sad case of public relations trumping scientific and engineering reality.

“The moral of the story is the importance of performing scientific analysis thoroughly. In the real world,

a single omission can lead to disaster. Even if you remember 99 out of 100 items, it’s not enough,

especially when the one item you forget is the five-dollar O-ring. One of the things I appreciate about

mathematics is how it shapes your mind to be rigorous, to ensure that analysis is complete and accurate.

“That’s one of the things that we will learn together as you master the art of writing mathematical

proofs. You will learn how to present your solutions so that your arguments are air-tight, where every

line follows logically from the previous line, with no holes in your reasoning. Of course, this is a great

59

skill to have when writing math contests, but has a more important application for your life beyond

school. There’s no career in which you won’t use logical reasoning or critical thinking. As far as I’m

concerned, there’s no better way to develop this skill than by doing mathematics. You’ll learn never to

forget the O-ring as you master a subject that’s never bo-ring.”

I laughed out loud, and stared in wonder at Mr. Collins, curious how anyone could have so much energy

and passion, let alone someone nearly twice Mom’s age.

“Let’s continue, Bethany. Of the first 22 questions on the practice Gauss contest, you’ve solved 20 of

them. Take me through the last three.”

We went through the final three problems, and found to my surprise that I got all three right. And all

three problems were so interesting.

“Wonderful work, Bethany. So how many questions did you correctly solve?”

“23 out of 25.”

“What’s that as a percentage?”

“Ninety-two percent,” I replied, multiplying 23 by 4 to answer his question.

“That’s right. That’s your best score so far. Do you know how many students across Canada got 92% on

that contest? Less than one percent. Bethany, look at you! You have become an outstanding mathlete.”

“I got lucky. Besides, it’s not the same thing. I had all the time I wanted this morning. Those people had

only 60 minutes, and had to write it under a lot of pressure. I bet I couldn’t do that.”

“Why do you say that?”

“Because I know myself. I don’t like pressure. I hate competition. Remember the Spelling Bee?”

“Oh Bethany, don’t be so hard on yourself. As you told me, you were eliminated in Round 11. So that

means you spelled 10 out of 11 words correctly.”

“But I lost to Gillian Lowell.”

“No, you didn’t. You came in second place.”

“Second place is the first loser. I remember hearing that from some Nike commercial.”

“Oh, Bethany, don’t think of it that way. That Nike commercial is terrible, and sends a horrible message,

especially to young people. I want to encourage you to think of it a different way: it was just you and

Miss Carvery that day. She gave you 11 words to spell and you correctly got 10 of them. That’s over

90%. And maybe next year, when you’re in Grade 8, you’ll get an even higher percentage right.”

“I won’t be doing the Spelling Bee next year.”

60

“That’s fine, Bethany. That’s your choice, and we obviously respect your decision. If the Spelling Bee

isn’t fun for you, then there’s no sense entering it next year. And it’s the same with this Gauss Contest.

If you find the problems interesting and want to do the contest just because you find math enjoyable,

then I’d encourage you to participate with the other Pinecrest students in May. But if it just leads to

stress and anxiety, and you find yourself competing against Gillian Lowell and feeling bad if she solves

more questions than you, then you’re right – then I think it’s better that you don’t participate.”

I nodded, and saw Mom doing the same thing.

“Bethany, let me change the subject for a second. I’m now 66 years old. I was never a star athlete but I

always enjoyed running. And about twenty years ago, I started running half-marathons, gradually

building up to full marathons.”

“Are you serious, Taylor?” said Mom. “I never knew that. No wonder you’re so skinny.”

Mr. Collins laughed. “There’s this big race in Halifax each May, called the Bluenose Marathon. I’ve done

it the past five years. And my goal each year is to simply beat my personal best time. I’m not interested

in how many people I can beat, or what place I finish in. It’s just me versus the clock. Last year I ran it in

4:02, forty-two kilometres on a cold windy day. I’m confident I can break four hours this year.”

“Four hours?” said Mom. “You can probably qualify for the Boston Marathon.”

“In fact, I qualified for Boston last year. But doing Boston was never my goal – it was just me stretching

myself to see if I could actually finish a marathon. And once I achieved that, I wondered how fast I could

actually go. My goal this year is four hours. That’s why I’m going to Halifax in May – not to compete

with others, but to push myself and see if I can do it. I’m motivated by my result, not by my ranking.”

“Do you enjoy running?” I asked.

“I love it. I feel so alive when I’m running, especially when I’m with my Sunday morning running group

along the Sydney Waterfront. Of course, they’re mostly younger and faster, but that doesn’t bother me.”

“Can I ask you a question?”

“Of course, Bethany.”

“I’m really enjoying working with you and doing all these practice Gauss questions. I got 23 out of 25

this time, and that’s my best score so far. So if I had the attitude that it was just me versus the

questions, that I couldn’t care less about how Gillian did, that I didn’t feel like I needed to prove

anything to anybody, that I could just try my best and be happy no matter what the final result – then

should I write the Gauss contest in May?”

“Only you can answer that question,” said Mr. Collins.

Mom turned to face Mr. Collins. “I think Bethany just did.”

61

Newcomb’s Law

Mr. Collins handed me an index card.

“What’s this?” I asked, taking the card in one hand while sipping my hot chocolate with the other.

“You tell me.”

I looked at the numbers on his card.

“They’re all the two-digit numbers,” I said.

“That’s right. They’re the set of positive integers with exactly two digits. And how many are there?”

“Ninety,” I replied, noting that the numbers on the card appeared in a 9 × 10 table.

“Now suppose you randomly select a number from this table. What’s the probability that the last digit is

a 1?”

We had covered probability in Grade 7 Math, completing an entire unit on the topic. So this was an easy

question for me.

Looking at the second column of the card, I saw that there were nine numbers with last digit 1, namely

the integers 11, 21, 31, 41, 51, 61, 71, 81, and 91. Since there were 90 numbers in all, the answer was

9/90, which reduced to 1/10.

“One-tenth,” I responded.

“Very good. Now what’s the probability that the first digit is 1?”

This was also straightforward. Looking at the first row of the card, I saw that there were ten numbers

with first digit 1, namely the integers 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19. So the answer was 10/90,

which reduced to 1/9. I looked confidently at Mr. Collins.

“One-ninth.”

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

62

“Excellent. Now suppose I gave you a card that listed all the three-digit integers. And say I got you to

pick a number from that card at random. What would be the probability that the first digit is 1?”

As the two-digit scenario had probability 10/90, I reasoned that the three-digit scenario would have

probability 100/900, which also equalled one-ninth.

“It’s the same.”

“Good. Now how about five-digit integers, or fifty-digit integers, or hundred-digit integers?”

“Exact same – it’s always one-ninth.”

“So what can you conclude?”

“That if you pick any integer at random, no matter how large it is, there’s a probability of one-ninth that

the first digit of that number is 1.”

Mr. Collins gave me another index card. “Let’s test your theory. Tell me what this is.”

It took me a while, but once I saw China and India, I quickly realized what the numbers represented.

“They’re population numbers.”

“That’s it, Bethany. China has 1.3 billion people, followed by India with 1.2 billion people, followed by

the United States with 312 million people, and so on. This is the list of the ten most populated countries.

I was surprised to see countries like Nigeria and Bangladesh appear; I had no idea they had so many

people! Anyway, let’s go back to what we were discussing earlier. If you were to pick a country at

random from this table, what’s the probability that its population has first digit 1?”

I looked at the numbers. Eight of the ten countries had their population beginning with the digit 1, with

the exception of the United States and Indonesia. So the probability was 8/10, or four-fifths.

“It’s eight out of ten – 80%.”

“But this is a lot more than 11%, or one-ninth. Bethany, didn’t you say that the expected probability was

one-ninth?”

“Yeah, but these are special numbers. It’s not the same as picking numbers randomly.”

“What do you mean by special?” asked Mr. Collins.

1) China 1,330,000,000 6) Pakistan 177,000,000

2) India 1,210,000,000 7) Nigeria 158,000,000

3) USA 312,000,000 8) Bangladesh 151,000,000

4) Indonesia 237,000,000 9) Russia 142,000,000

5) Brazil 190,000,000 10) Japan 127,000,000

63

“Well you just gave the populations for the top ten countries, so it was just luck that eight of them had

first digit 1. I bet the first digits are evenly spread out if you look at the populations for all countries.”

“So you’re saying that if we looked at population data for all the world’s countries, it would be equally

likely for the first digit to be 1 or the first digit to be 9? That the probability would be about one-ninth

for each of the nine possible first digits?”

“Yeah.”

Mr. Collins smiled and took out some papers from his clipboard.

“Well, let’s see, shall we?”

I stared at the stapled three-page handout, listing the population numbers for the top 200 countries

based on the most recent census, ranging from 1.3 billion for China (#1) to 56,000 for Greenland (#200).

Mom was stunned. “Taylor, how did you know she was going to say that?”

Mr. Collins smiled. “Just a lucky guess.”

He turned back to me. “Bethany, there are about 225 countries in the world, if you include small

dependent territories like the Falkland Islands and the Cayman Islands. I’ve taken the top two hundred,

just to make it easier. What I want you to do is make a table that lists the number of countries that have

populations beginning with the digits 1, 2, 3, and so on. Let me know when you’re done.”

I flipped through the handout, tallying up the first digits for each country’s population. Just before

handing my sheet to Mr. Collins, I also calculated the percentage, or proportion, of countries that had

that first digit. Since there were 200 countries in my list, the percentages were easy to calculate.

First Digit 1 2 3 4 5 6 7 8 9 TOTAL

# of Countries 58 28 26 18 17 17 10 17 9 200

Proportion 29% 14% 13% 9% 8.5% 8.5% 5% 8.5% 4.5% 100%

“Great work, Bethany. So what do you notice?”

“29% of countries have their population beginning with the digit 1, while only 4.5% of countries have

their population beginning with the digit 9.”

“So these statistics are not evenly spread out. It’s not one-ninth for each of the nine possible first digits.”

“But why? This makes absolutely no sense.”

“Yes, that’s exactly what I said when I first learned about this. Here’s the phenomenon. Whenever you

have a list of naturally-occurring numbers, like the lengths of the world’s longest rivers, the heights of

the world’s tallest buildings, the numbers that appear in your mother’s Globe and Mail newspaper, the

64

number of files stored on a person’s PC, and the house numbers of residents in Cape Breton, it is far

more likely for the numbers to have a low first digit than a high first digit.”

“I don’t believe you.”

Mr. Collins handed me another index card. “I didn’t believe it myself, until I investigated further. Here’s

the magic table that states the distribution of first digits in such naturally-occurring numbers.”

First Digit 1 2 3 4 5 6 7 8 9

Proportion 30.1% 17.6% 12.5% 9.7% 7.9% 6.7% 5.8% 5.1% 4.6%

“Notice how these numbers are really close to the ones you came up with for country population?”

I nodded, seeing the similarity between the two tables, but was still confused.

Mr. Collins continued. “Let me illustrate with another example that I found on the internet yesterday.

California has a whole bunch of cities: huge cities like Los Angeles with 3.9 million people and San Diego

with 1.3 million; medium-sized cities like Pasadena with 150,000 people; and tiny cities that you’ve

never heard of with just a few thousand people.”

He handed me another sheet of paper.

“If you determine the first digit of each city’s population, and tally the results, here’s what the

distribution graph looks like.”

“That’s crazy, eh? The actual statistics match up almost identically with what’s predicted by Newcomb’s

Law.”

“What’s Newcomb’s Law?”

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9

Nu

mb

er

of

Cit

ies

First Digit of City Population (California)

Predicted by Newcomb's Law Actual Statistics for California

65

“Oh sorry, that’s the magic table I just gave you: 30.1% for the first digit 1, and 17.6% for the first digit 2,

and so on.”

“But why do they call it Newcomb’s Law?”

“Oh, I’ll get to that in a moment. You know, Bethany, this first-digit phenomenon isn’t just for

population, it’s also for things like house numbers. If you ask 1000 people in Cape Breton for the first

digit of their house number, around 300 of them will say that the first digit is 1.”

“Why?”

“That’s something I would like you to think about, and as usual, I’d like you to write up something for me

next week. Here, let me give you a small hint. Think of various streets in Sydney, big ones like George

Street and Kings Road, as well as smaller streets like the one you live on. What’s the range of the house

numbers? Do they all go from 1 to 99, or 1 to 999, or do some streets cut off earlier? For example, I

think George Street goes from 1 to 2400, or something like that. So say you live on George Street. Is it

more likely that your house number starts with 1 or 9?”

I pondered his question. If I lived on George Street, and the first digit of my house number was 9, my

options would be limited to 9, 90 to 99, and 900 to 999. On the other hand, over a thousand houses on

George Street had first digit 1, namely everything from 1000 to 1999, in addition to 1, 10 to 19, and 100

to 199.

I lived on a street where the numbers went from 1 to 56. So that meant that there were eleven houses

that had 1 as its first digit (namely 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19), but only one with 9 as its first

digit (namely 9). I began to see why the first-digit distributions weren’t equal.

“Okay, I’ll write up something for next week trying to explain why the distributions aren’t equal. Hey

Mom, we can drive around the city a bit checking out the house numbers? Is that okay?”

Mom smiled and nodded.

A question still nagged me. “But how did they come up with those statistics? You know, 30.1% of

numbers have a first digit of 1 and 17.6% have a first digit of 2 and so on?” I asked, referring to the

magic table Mr. Collins referred to earlier.

“Great question. Take out your calculator and do the following: for each of the numbers you see in

Newcomb’s table, convert it to a decimal. For example, with the percentage 30.1%, change that to

0.301. Then calculate the value of 10 raised to the exponent 0.301.”

I followed his instructions and produced the following table:

First Digit 1 2 3 4 5 6 7 8 9

Proportion 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046

10 Proportion 2.000 1.500 1.333 1.250 1.200 1.167 1.143 1.125 1.111

66

“Notice anything interesting, Bethany?”

I saw it right away. “Those numbers in the bottom row. They’re just fractions.”

“That’s correct. Way back when, before we had calculators, people had to use something called

logarithm tables to do calculations like 123456789 × 654321. It was a thick book with thousands of

pages. Using logarithm tables was a tedious process but the calculations had to be precise – and was

particularly needed by astronomers for tasks like surveying and celestial navigation.

“One day, a mathematician named Simon Newcomb looked at these logarithm tables and noticed that

the earlier pages were more worn out than the later pages – in other words, there seemed to be more

naturally-occurring numbers with first digits 1 and 2 than first digits 8 and 9. He wondered why that was.

That was the insight. Then he used elegant mathematics to show that the first digit frequencies had this

beautiful property relating to powers of 10 and simple fractions.”

“That’s really cool.”

“And I haven’t told you the best part of the story. Guess where Simon Newcomb was from?”

“I don’t know,” I replied with a shrug.

“Guess.”

“Britain? America?”

“He’s from Nova Scotia! Simon Newcomb was born in Wallace, right next to Pugwash. It’s just a four-

hour drive from Wallace to Sydney.”

Mom looked at him incredulously. “He’s from Nova Scotia?”

“Yup,” said Mr. Collins with a big smile. “We Nova Scotians are quite famous for lots of things, not just

for fresh seafood, fiddle music, and great beer. We’re a lot more than the Trailer Park Boys and

Alexander Keith’s.”

“Who’s Alexander Keith?” I asked.

Mr. Collins looked a bit embarrassed.

“Umm, Alexander Keith was, uh, the former mayor of Halifax.”

Mom snickered. “And?”

“Uh, he made a special beverage that’s enjoyed by thousands of adults across Canada, including many of

my former students, even before they turn nineteen.”

67

I laughed. “Oh yes, the beer guy.”

“So, Bethany, changing the subject,” said Mr. Collins. “This Simon Newcomb guy discovered the first-

digit phenomenon back in 1881, and published his proof with the frequency distributions that involved

those logarithms I just mentioned. Fifty years later, an American mathematician named Frank Benford

discovered the same principle, and he published it too. For whatever reason, the first-digit discovery is

now known as Benford’s Law even though Newcomb discovered it first. I mean, what is that about? So

anyway, regardless of what others say, I’ll always call it Newcomb’s Law.”

Mr. Collins looked at the clock.

“Before we end today, I want to summarize what we discussed because it’s one of the most

fundamental uses and applications of mathematics.

“Simon Newcomb looked at these logarithm tables and noticed the pattern that the earlier pages were

more worn out than the later pages. Having found the pattern, he sought to understand and uncover its

hidden structure. To do this, he had to apply his problem-solving skills to find a logical explanation for

the first-digit phenomenon that 30% of the numbers start with 1, about 18% start with 2 and so on. The

mathematics was deep, and the end result was completely unexpected. That’s the process by which

mathematics is done in the real world. It’s not people sitting around memorizing formulas, but rather

employing a multitude of techniques to discover patterns to uncover truth and structure, seeing how

those patterns explain what we see in society, and propose fresh ideas based on rigorous evidence to

change our world.

“Newcomb couldn’t have predicted this, but his discovery has led to numerous important and practical

applications over a century later. My favourite is fraud detection. Your mom works at a place called the

Canada Revenue Agency. As you might know, the CRA is the federal government department that

administers all the tax laws, and processes the income tax reports of all businesses and individuals. Well,

if you look at a company’s tax returns, you’ll see thousands of different numbers that represent bits of

financial information: assets, profits, stock prices, interest, deductions, and so on. If you look at the first

digits of these numbers, what do you think happens?”

I saw where this was leading. “The first digits follow Newcomb’s Law: 30% have first digit 1, about 18%

have first digit 2, and so on.”

“Exactly. So a legitimate tax return will have far more entries like $1,805 and $105.36 than those

beginning with larger digits, such as $98.35 and $800. On the other hand, those who fudge the data and

try to make the numbers appear random tend to distribute the numbers equally, with the first digits

spread out evenly between one and nine. So using Newcomb’s Law, you can identify the criminals and

send them to jail. Isn’t that amazing?”

“But if you knew Newcomb’s Law, couldn’t you properly fudge the numbers and get away with it?”

68

“Ah, you’re a clever young lady,” said Mr. Collins with a laugh. “Yes, if you’re going to cheat, you may as

well do it using mathematics. Great, I just taught you how to cheat on your income taxes, and I did it in

front of a CRA employee. Oops.”

Turning to Mom he said, “Say Lucy, were you familiar with all this? I mean, the CRA uses Newcomb’s

Law as one method of risk assessment to detect tax fraud. Did you know that?”

“Oh no,” said Mom. “You know more about my department than you do.”

“Do you find this stuff interesting?”

“Yes, Taylor, I find this absolutely fascinating. I had no idea that anyone at the CRA did anything

interesting. Sorry, that’s just me being cynical; that’s what happens when you’ve got the exact same

dead-end job for a decade. By the way, how do you know about this?”

“Because a former student of mine has been working for the CRA in Ottawa, managing a research group

specializing in fraud detection. He does some fascinating work for the federal government, using math

and statistics for risk assessment, saving the Government of Canada millions of dollars each year. It’s a

wonderful application of mathematics, and so practical.”

“Who’s your former student?” asked Mom.

“It’s Ryan Dexter. Do you know him?”

“Angus Dexter’s son?”

“That’s him. In fact, old Angus has a grandchild who goes to Pinecrest. I think his name is Rodney. Is he

in your class, Bethany?”

“Yeah,” I replied. “There’s a Rodney Dexter in my class. Don’t really know him though. But I know he’s

writing the Gauss Contest next week.”

“Oh yes, of course,” said Mr. Collins, remembering. “You excited about the Gauss Contest?”

“Not excited… nervous.”

“I’m sure you’ll do great. We’ve been doing lots of practice questions over the past four months,

reviewing problems from previous contests. You’re definitely ready.”

“But that’s practice. Next week is the real thing. There’s more pressure; there are other people there.”

“Just ignore them, Bethany. It’s just you versus the clock. Same with me and my marathon next week.

It’s just me versus the clock. Try to solve as many questions as you can in the 60 minutes. Remind me,

what’s the best score you’ve gotten in a practice contest?”

“23 out of 25 questions.”

69

“That’s right. And you did that three times. You’ve gotten over ninety percent of the problems during

our practice sessions. You’ve definitely got it in you to get a score you’ll be proud of.”

“Thank you.”

“And here’s a small tip. If you’re going to be distracted by other people and worried about what they’re

doing, then just sit in the front row. This way, you won’t be distracted by anyone.”

“Thanks, Mr. Collins. And good luck with the marathon. Hope you break four hours.”

“Thank you. By the way, do you know about Gauss? You know, who he was, what he was famous for?”

“Nope.”

“Well, let me fill you in. Carl Gauss was almost surely the greatest mathematician since antiquity, having

made enormous contributions to dozens of fields including statistics, astronomy, geophysics,

electrostatics, not to mention creating new fields of mathematics which were centuries ahead of his

time. Quite simply, Gauss was amazing. I could go on and on about Gauss and his contributions, but

there’s one story about Gauss that he’s most famous for, and it’s a story you absolutely must know.

“When young Carl Gauss was in Grade 4, his teacher gave all of the students a mindless and tedious

addition problem. Of course, this was in the 1700s, and they had no access to calculators. So the

students performed the addition one by one, and it took them all a long time. And every single student

got the answer wrong, that is, every student except for young Gauss. It turns out that Gauss correctly

solved the problem, and he did so in mere seconds because he saw a clever insight that no one else did.

It was the first sign of his incredible talent in mathematics, an ability that he nurtured and developed to

become arguably the most famous and important mathematician in the history of the world.”

“What was the addition problem?”

Mr. Collins smiled. “It was to find the sum of the integers from 1 to 100. Gauss’ incredible insight was

to create a second sum, and write the numbers backwards from 100 down to 1.

S = 1 + 2 + 3 + 4 + … + 97 + 98 + 99 + 100

S = 100 + 99 + 98 + 97 + … + 4 + 3 + 2 + 1

He then added each column, showing that the left side was twice the desired sum, and that the right

side was 101 + 101 + 101 + … + 101, which worked out to 101 × 100. So the desired sum was just half of

101 × 100, which is just 101 × 50 = 5050. His classmates were amazed, and his teacher was absolutely

astounded by his creativity and imagination.”

Mr. Collins looked into my eyes. “I wonder if that story sounds familiar.”

I gulped, knowing full well what he was referring to.

Mr. Collins winked. “Good luck on the contest – you’ll do great.”

70

Gauss Contest

The announcement came over the PA system.

“This is a reminder for all Pinecrest students writing the Gauss math contest: please head to Room 105

right now.”

I briskly walked to Room 105 and took the middle seat in the first row, right below the clock. Others

strolled in moments later: Gillian and Vanessa; the twins Amy and Alice; and several boys including

Rodney Dexter, the one that Mr. Collins referred to. My friends weren’t there: Breanna had a bad cold,

and Bonnie wasn’t interested, saying one math class was more than enough for a Wednesday morning.

Since there were 30 desks in the classroom, and over half were unoccupied, we could sit anywhere we

wanted. I noticed that no one had joined me in the first row. That was a relief.

Miss Carvery had arranged a pizza lunch for those of us writing the math contest – we would be eating

together immediately after the end of the contest. I had no interest in staying for lunch, sitting around

Gillian and her friends; my plan was to take my pizza and sit next to Bonnie in the cafeteria.

I got out a pen and two pencils, as well as several sheets of paper, a ruler, compass, and calculator. Miss

Carvery handed me an answer sheet, where I first had to first bubble in the letters of my name. To the

right of that were 25 rows, corresponding to each of the questions on the contest, with five possible

answers for each, marked (A), (B), (C), (D), and (E).

Miss Carvery explained the rules of the contest. There were twenty-five problems, with ten questions in

Part A, ten questions in Part B, and five questions in Part C. Each correct answer was worth 5 points in

Part A, 6 in Part B, and 8 in Part C. So the highest possible score was 50 + 60 + 40 = 150 points.

When all of us had finished bubbling in our names, Miss Carvery asked whether any of us had questions.

When none of us did, she looked at the clock and said, “okay, it’s now 10:59. When the clock turns to

11:00, you can start. You’ll have exactly one hour. Good luck!”

I stared at the clock, ignoring the people whispering behind me. Don’t worry about Gillian, I told myself.

Forget about her. Just do your best. Answer as many questions as you can.

“Okay, you can begin!”

I opened the contest and took a deep breath. Here was Question 1.

1. When the numbers 8, 3, 5, 0, 1, are arranged from smallest to largest, the middle number is

(A) 5 (B) 8 (C) 3 (D) 0 (E) 1

No problem at all. Five seconds later, I bubbled in (C). I looked at Question 2.

2. The value of 0.9 + 0.99 is

(A) 0.999 (B) 1.89 (C) 1.08 (D) 1.98 (E) 0.89

71

Mr. Collins warned me not to make a careless mistake on the Part A questions, and to take a few extra

seconds to be certain the answer was correct. My first instinct was to bubble in the answer 0.999 but

then I looked at the numbers more carefully and saw that the correct answer was actually 1.89, since

ninety cents and ninety-nine cents added up to $1.89. I breathed a sigh of relief knowing that I had

avoided a potential trap, and bubbled in the correct answer, (B).

Less than ten minutes later, I was already at Question 10.

10. Two squares, each with an area of 25 cm2 , are placed side by side to form a rectangle. What is

the perimeter of this rectangle?

(A) 30 cm (B) 25 cm (C) 50 cm (D) 20 cm (E) 15 cm

I created the diagram on my scratch paper. A square with area 25 cm2 has side length

5 cm. So when the squares are placed side by side to form a rectangle, the width is 5 cm

and the length is 10 cm. So the perimeter is 5 + 10, which equals 15 cm. I bubbled in (E).

I was on a roll. I hit Question 13.

13. A palindrome is a positive integer whose digits are the same when read forwards or backwards.

For example, 2002 is a palindrome. What is the smallest number which can be added to 2002 to

produce a larger palindrome?

(A) 11 (B) 110 (C) 108 (D) 18 (E) 1001

I wasn’t sure how to solve the problem. As I thought how I could obtain the correct answer, I recalled

Mr. Collins’ advice that when it’s not clear how to solve the problem directly, start from the multiple-

choice answers and work backwards. This was just one of the many strategies that I called “cheating”

and he called “creative problem-solving”.

I read the question again – what is the smallest number which can be added to 2002 to produce a larger

palindrome? So I started with the smallest number among the five multiple-choice options, and worked

my way up. As soon as I found a palindrome, I would have my answer. Just to save time, I used my

calculator:

2002 + 11 = 2013. Not a palindrome.

2002 + 18 = 2020. Close, but not quite. A palindrome has to read the same forwards and backwards.

2002 + 108 = 2110. No.

2002 + 110 = 2112. Yes!

I found a palindrome! I smiled and bubbled in (B).

That felt so wrong, but I didn’t care. I grinned and moved on to the next question.

5

5

5

72

14. The first six letters of the alphabet are assigned values A=1, B=2, C=3, D=4, E=5, and F=6. The

value of a word equals the sum of the values of its letters. For example, the value of BEEF is

2+5+5+6 = 18. Which of the following words has the greatest value?

(A) BEEF (B) FADE (C) FEED (D) FACE (E) DEAF

I knew that I could solve this problem by mechanically finding the values of each of the five words and

seeing which one had the largest total. But that would take some time. I recalled another useful tidbit

from Mr. Collins, to take a few seconds right at the beginning of each question, and see whether I could

discover a piece of insight that would make the problem easier.

All of a sudden, I saw it: each of the five words had the letters F and E appearing, so these would

effectively “cancel” each other when determining which of the five words had the greatest sum! In

other words, I realized that finding the greatest value of these five words

(A) BEEF (B) FADE (C) FEED (D) FACE (E) DEAF

was equivalent to finding the greatest value of these five “words”:

(A) BE (B) AD (C) ED (D) AC (E) DA

And from here, it was easy to see that ED = 5+4 = 9 had the greatest value. I smiled and bubbled in (C).

Before I knew it, I was at Problem 20.

20. The word “stop” starts in the position shown in the diagram to the

right. It is then rotated 180 degrees clockwise about the origin, and

this result is then reflected in the x-axis. Which of the following

represents the final image?

(A) (B) (C) (D) (E)

Having poor spatial reasoning, I wasn’t sure how to solve this question. There were four letters to keep

track of in my head, in addition to one rotation and a flip. It was too much to hold in my mind. How

could I solve this problem?

I sat and pondered for at least one minute, concentrating. All of a sudden, an idea hit me.

Remembering Mr. Collins and his sugar packets at Le Bistro, I thought of something I could do.

I took out a fresh sheet of paper, drew two large perpendicular lines that filled the entire page,

signifying the horizontal x-axis and vertical y-axis, and wrote down the four letters s t o p in large block

s t o p

s t o p

p o t s

s t o p p o t s

73

letters with my pencil, making sure they were dark enough that I could see through them when I needed

to flip the page over.

I held the paper in my hand. First I rotated the sheet 180 degrees clockwise about the origin, which

gave me the following picture:

So the rotation was easy. For the reflection, I pinched the left and right side of the paper with my

thumbs lying on the same line as my horizontal x-axis, and just turned over my wrists. The paper flipped

over and I saw the following image by holding my piece of paper to the bright light above me:

Looking at the five multiple-choice options, I saw the answer I was looking for. I bubbled in (E).

I glanced at the clock, and saw that I had finished the first twenty problems in 28 minutes, giving me lots

of time to work on the Part C questions. I quickly checked my answer sheet. I had bubbled everything in

correctly – it would have been awful if I had skipped a bubble by accident, but thankfully I marked my

answers properly. I took a deep breath.

Over thirty minutes left. Lots of time for Part C!

It took me about fifteen minutes to solve the next four problems: Question 21 was a simple problem of

counting handshakes; Question 22 dealt with the surface area of a rectangular box; Question 23 was a

probability question involving coloured marbles; and Question 24 asked for the area of a trapezoid

shaded inside a particular triangle. I had no doubt I had correctly solved all four, as I was really careful

with my calculations.

I glanced at the clock. 11:44. Okay, sixteen minutes left for the last question!

25. Each of the integers 226 and 318 have digits whose product is 24. How many three-digit

positive integers have digits whose product is 24?

(A) 4 (B) 18 (C) 24 (D) 12 (E) 21

I closed my eyes and smiled. I knew I was capable of solving this problem, given how much time I had

spent working with Mr. Collins on integer factorization and divisibility.

s t o p

74

I knew all the ways we could multiply two integers to get to 24. The combinations were 1×24, 2×12, 3×8,

and 4×6. Now how could I adapt this to three integers?

I started listing the options. Taking out a fresh sheet of paper, I wrote all the ways three digits could

multiply to give 24.

2×3×4 2×2×6 1×4×6

That’s all I could come up with; I was sure that was it. I ignored options such as 1×1×24 and 1×2×12,

since 12 and 24 weren’t digits. So I had exhausted the possibilities, and simplified the problem down

into simpler cases in order to find a pattern. All I needed to do was determine how many 3-digit

integers could be made from each of the sets [2,3,4], [2,2,6], and [1,4,6], and I would be all done!

I knew that [2,3,4] could be made into six possible three-digit integers, namely 234, 243, 324, 342, 423,

and 432. I knew there had to be 6 = 3×2×1 possible rearrangements, since there were three choices for

the first digit, two choices for the second digit, and one choice for the final digit. This was an example of

a permutation, a topic that Mr. Collins and I covered a couple of months ago.

Similarly, [1,4,6] could be made into six possible three-digit integers by the exact same argument,

namely 146, 164, 416, 461, 614, and 641.

Finally, [2,2,6] could only be made into three possible three-digit integers since the digit 2 appeared

twice. The possible options were 226, 262, and 622.

So I was all done! The correct answer was 6 + 6 + 3 = 15, and I was all done with some time to spare! All

I had to do was fill in the right bubble and I could relax.

Oh no, where was 15? The number 15 wasn’t listed among the five possible multiple-choice options for

this problem. What was going on?

What happened? Did the contest organizers screw up? Where was 15?

Or did I make a mistake? Did I miss a case? Did I forget about the O-ring?

I started to panic. I was sure I didn’t forget a case. What else was there besides 2×3×4, 2×2×6, and

1×4×6? I was sure I got them all. Did I read the question wrong? I re-read Problem 25.

Each of the integers 226 and 318 have digits whose product is 24. How many three-digit positive

integers have digits whose product is 24?

No, I definitely read the problem right. So what happened? My mind started to race. Calm down

Bethany, calm down! I took a deep breath and just closed my eyes for twenty seconds.

When I opened my eyes, I re-read the question. Each of the integers 226 and 318…

And then I saw it. Right in the statement of the problem, the number 318 was written. I had forgotten

the case 3×1×8, another instance of three digits multiplying to 24. Phew!

75

From the missing case [1,3,8], I quickly determined the six additional three-digit numbers whose digits

multiplied to 24, namely 138, 183, 318, 381, 813, and 831. So my answer wasn’t 15, but 15+6.

Did 21 appear as one of the five possible options? Yes it did. I bubbled in (E). I was all done. I put my

hands to my face and felt my breath on my palms. I was exhausted. I couldn’t believe I did it.

Yes, I got them all!

I had a burning desire to glance over at Gillian, to see how she was doing. I turned to my left and saw

her in deep thought, rapidly scribbling something onto her sheet. I saw her pause, let out a silent curse,

and take out her eraser to start her calculation over again. I turned back towards the front and smiled,

knowing that I was done, Gillian was not, and that I still had time to verify all of my answers.

I knew I had answered everything correctly in Part A, and so I didn’t bother to re-check those ten

questions. I skimmed through the ten questions in Part B, making sure I didn’t make any careless

computation errors. After a few minutes of double-checking my answers, I realized everything was fine.

I definitely answered the first twenty questions correctly.

Five minutes left. I went through the five questions in Part C, double-checking and triple-checking that I

didn’t do anything incorrectly. Everything looked fine. I glanced up at the clock.

Just twenty seconds left.

I smiled confidently, and opened the contest to the first page, where all the Part A questions were.

And then my face froze when I saw Question 10 and the diagram that I had marked on the contest paper.

10. Two squares, each with an area of 25 cm2 , are placed side by side to form a rectangle. What is

the perimeter of this rectangle?

(A) 30 cm (B) 25 cm (C) 50 cm (D) 20 cm (E) 15 cm

The length was 10 and the width was 5. So the perimeter was not 10 + 5 = 15, but

10 + 5 + 10 + 5 = 30, since the perimeter represented the sum of the lengths of all

four sides of the rectangle. Oh no! Quickly grabbing my eraser, I found Question 10

on the answer sheet, erased my incorrect answer of (E), looked for 30 cm on the

contest sheet, found it, and bubbled in the correct answer of (A).

“Okay, stop writing!” shouted Miss Carvery. “Put your pencils down!”

I let out a deep breath.

“Please leave your answer sheet on the table and I will come around and collect them,” said Miss

Carvery. “Congratulations. The contest is done. Please stay in your seats, and the pizza will arrive in a

couple of minutes.”

5

5

5

P = 10 + 5 = 15

76

I started to hyperventilate. My mind was racing. I messed up Question 10 but was just barely able to fix

both. What else did I mess up? What other careless mistakes did I make? I didn’t bother to check my

answers to the Part A questions since I was sure I got them all, but did I? If I screwed up simple

Question 10, what else did I mess up?

My heart was beating rapidly, and I had trouble breathing. I had flashbacks to the Spelling Bee.

M-A-R-R-O-O-N.

Miss Carvery walked by my desk to pick up my answer sheet. She saw my face and grabbed my shoulder.

“Bethany, are you alright?”

I shook my head. I felt like throwing up.

“Do you need to use the washroom?”

I nodded. I got out from my seat, and ran to the girls’ washroom. I hurried into a stall, locked it, knelt

down next to the porcelain bowl, and lifted the toilet seat.

With a great heave, I barfed out my entire breakfast from this morning. There were big pieces of dried

toast and peanut butter everywhere. It was disgusting. I started to breathe a bit slower and had a

second round of vomiting. Yuck. I just sat down for a couple of minutes on the stone cold floor, and

leaned over to flush the toilet. My breathing calmed down.

I heard a knock on the stall. “Bethany, are you alright?”

It was Miss Carvery. I could recognize her voice anywhere.

“Yes,” I lied.

“No, Bethany, you’re not all right. Could you open the stall?”

I turned the lock and Miss Carvery came in. A couple of other teachers entered the washroom, and

seeing me sitting on the floor, were immediately concerned. I heard them speak.

“Does she need to go to the hospital?”

“What happened to Bethany? Did she overdo it in gym class?”

Miss Carvery turned to them. “She’ll be okay. She just needs some water, and to take it easy for the

rest of the day.”

After about five minutes, I was ready to go back to Room 105. Miss Carvery stayed with me the whole

time. She took me back to the contest room and handed me a bottle of water, as the students were

sitting around the tables in the back, and comparing their answers while eating pizza. Gillian noticed us

walk back into to the room, and pounced immediately.

77

“Bethany cheated, Miss Carvery.”

I nearly choked on my water. What?

“That’s a serious accusation,” said Miss Carvery. She looked coldly at Gillian. “What makes you believe

that Bethany cheated?”

She walked to my desk and held up a sheet of paper, the one with s t o p written in large block letters.

“I saw her take this page and hold it up to the light, seeing what the reflection would look like. That’s

how she must have solved one of the questions. That’s cheating.”

I was speechless.

“Gillian, take it easy. Bethany’s not feeling well.”

“But Miss Carvery, it’s not fair! I did all the questions properly; I didn’t cheat like Bethany.”

“Gillian, calm down. Don’t get yourself involved. I’ll take care of this.”

Miss Carvery turned to me and motioned towards the door. “Bethany, please come to my office. Let’s

chat there.”

Gillian smiled at me with a smug expression on her face, before walking back to join her clique.

I reluctantly followed Miss Carvery to her office, the one with VICE-PRINCIPAL written at the front. I was

both scared and angry. Just as Miss Carvery walked in the door, I noticed her passing by one of the

administrative staff workers and giving her a ten dollar bill.

“Rhonda, please go to the cafeteria and get me two tuna sandwiches, two cookies, and two small

cartons of milk. You can bring them to my office.”

She spoke with such authority that I could hear her every word. I walked in behind Miss Carvery, fearful

of being accused by my favourite teacher. I needed to set the record straight.

“I swear I didn’t cheat, Miss Carvery.”

“I believe you, dear. You’ve always been an honest person. Please, have a seat.”

“Yes, I did scan that page to the light, but there’s no rule to say I couldn’t do that. My coach Mr. Collins

told me that I could use props if I wanted to, if that would help me solve the problem faster. He said it

wasn’t cheating – he called it creative problem-solving.”

Miss Carvery laughed. “That sounds like exactly something Taylor Collins would say.”

“You know Mr. Collins?” I asked, surprised.

“I know him very well, my dear. I’m a graduate of Sydney High School. He was my teacher.”

78

“So you don’t think I cheated?”

“Of course not, dear! Gillian is just overreacting. Don’t worry about her.”

“She’s so mean.”

“Gillian can say mean things, but you don’t understand where she’s coming from. She’s got a lot of

pressure on her to be the top student in her class. You threaten her, Bethany. You’re the one person

that can challenge her. You’re the one person in her grade who’s just as bright and clever as she is. If

Gillian talks down to you, it’s not because she’s mean; it’s because she’s intimidated by you.”

“Gillian intimidated by me? You serious?”

“Yes, I’m serious. I know how you feel because I can relate to your situation. At board meetings, I get

disrespected by colleagues who resent that I’ve already become vice-principal. They talk down to me,

saying I was promoted only because of my gender or my skin colour. It’s completely ridiculous, but I’ve

learned to ignore them. Those old white men can think whatever they want to think. If they feel so

insecure around me that they have to resort to verbal abuse, then that’s their problem, not mine.”

I heard a knock on the door. It was Rhonda, who held a tray of food. Miss Carvery thanked Rhonda and

handed one of the sandwiches to me.

“After what happened, I thought you would prefer eating this, rather than greasy pizza.”

I smiled in appreciation, as I took off the plastic wrapping and took a big bite.

“People talk down to you?” I asked my mouth full of tuna and celery.

“You don’t know the half of it, Bethany. But I’m tough – just like your mother.”

I paused and swallowed my food before speaking.

“I’d been meaning to ask you about this, Miss Carvery. You know Mom?”

“Yes, I most certainly do. I looked up to your mother very much, since she was two grades above me at

Sydney High. We were all in awe of her. Like everyone else in Cape Breton, I was heartbroken about

what happened at the Olympic Trials. We all cried. We all thought she would make the team.”

“What team?” I asked, confused.

She stared at me. “You know, the women’s figure skating team. At the Winter Olympics.”

The Olympics. My jaw dropped.

Everything that Mom said, and didn’t say, suddenly made sense.

I finally understood why Mom didn’t want me to pursue the Math Olympiad, why she always changed

the subject when it came to figure skating, and why she was in so much pain to talk about her past.

79

Oh my god, how did I not realize this earlier?

“Bethany, are you alright? Your face is all blue.”

My mind flashed back to the conversations we had last summer, about the dangers of pursuing

unrealistic goals and the risks of putting all of our eggs in one basket. I remember Mom bursting into

tears as she talked about her shattered dream. I had no idea what that dream was… until now.

Oh my god, how could I have been so blind? Everyone knew Mom was a star figure skater.

“Bethany, talk to me. What’s wrong?”

I couldn’t answer. My face streaked with tears, felt the burden of Mom’s pain over all these years. As

she saw me getting excited about becoming a Math Olympian for Canada and working with Mr. Collins,

she must have been constantly reminded of the memories of her own childhood, and all the sacrifices

she was forced to make as she sought to become an Olympian herself.

I unfairly assumed that she was just a bad mother who tried to discourage my dreams, instead of a

loving mother who tried to protect me from pain.

Oh my god, how could I have been this selfish?

Miss Carvery sat down beside and held my hand. She handed me a tissue, and wrapped me in a big hug.

After a few minutes, I was able to compose myself.

“Bethany, what happened? Are you okay?”

I nodded, and took several slow deep breaths in an effort to calm down. I knew what I needed to do. I

dried my tears and looked up at Miss Carvery.

“Can you drive me to Mom’s work?”

“Uh, sure. When?”

“Right now. I need to tell her something important.”

“Now?”

“Yes. Right now.”

80

Canadian Mathematical Olympiad, Problem #1

Determine the value of

.

81

Solution to Problem #1

I looked up at the clock. 9:24.

As I closed my eyes, the memories came back in a flash, one after the other. The staircase in Mrs.

Ridley’s class, the chance visit to Le Bistro that brought Mr. Collins into my life, cross-training my mind,

learning how to simplify hard problems into easier components in order to detect patterns, developing

the skill of solving problems by uncovering hidden structures, and the amazing euphoria of achieving a

perfect score on my very first math contest. All of this happened years ago, before I even entered high

school. But the experiences were so memorable that I could still remember everything in vivid detail.

Most important, I’m so glad I was able to patch things up with Mom. It took a while, but things were

good now. Really good.

Many years later, I found myself in this cold room, writing the Canadian Mathematical Olympiad for a

spot on Canada’s IMO team. And for the first problem on the country’s most difficult math contest, all

of the tools and insights I needed were developed during my years at Pinecrest Junior High. I couldn’t

believe that. And the solution involved the staircase – that was so wonderful.

I re-read Problem 1.

Determine the value of

.

In my simplified problem, I wanted to determine the value of

.

I had already determined that middle term was equal to one-half.

.

I looked down at my notes for calculating the other two terms, applying various properties of exponents

that I could rapidly recall after years of practice.

The first term was

.

The last term was

Instead of determining the value of each term separately before computing its sum, I realized I could

just add the two expressions directly, revealing a fraction with the same numerator as denominator.

.

82

The first and last terms, when paired, added to one. When I saw that this total was one, my jaw

dropped in wonder, and I knew that I had found the heart of the problem. I instinctively knew that this

approach would work in the actual problem with 1999 terms, and used some algebra to show that when

we paired the terms from the two endpoints, its sum was always one. In other words, we had

Armed with this insight, there was a simple way to determine the sum, using the staircase picture. Just

to make sure I got all the details correct, I first applied it to my simplified problem with three terms. I

drew three rectangles, each with height 1. I made the width in the first rectangle

, the second

rectangle

, and the third rectangle is

. I attached the three rectangles together to form a

staircase. Since each rectangle had height of 1, the staircase had total area

.

Like I did back in Grade 5, I was able to represent the desired sum pictorially. But this time, instead of

counting the number of squares in the staircase, I wanted to find the total area. To do this, I flipped the

staircase diagram, and pasted it to the original figure.

So the two staircases becomes one simple rectangle, since the width of each “block” is guaranteed to be

1 by our pairing method. Therefore, the area of two-staircase figure is just 3 times 1, the height of the

rectangle multiplied by its width. It follows that the area of a single staircase is 3/2, namely half of 3.

Without doing any calculations, I knew that the answer to Problem 1 was 1999/2, by the exact same

argument. I started writing my solution, grinning the entire way. To ensure I would finish faster, I

decided to use “function notation” and “sigma notation” , a lazy way for mathematicians to use

fewer symbols without losing any meaning or rigour. I smiled when I was all done. Perfect solution.

I looked up at the clock. 9:41. One problem done, four more to go. Time for Question 2.

This step of the staircase has area

This step of the staircase has area

This step of the staircase has area

+ =

83

Problem Number: 1

Contestant Name: Bethany MacDonald

Define

.

We will prove that

.

Let

. We first establish that for any real value of x.

Since

, our

claim is established.

Now consider a “staircase” diagram with 1999 steps, each of height 1, where the kth step has length

Then the area of this staircase is

, which equals S by definition.

⁞

Invert a copy of the staircase and attach it to the original so that the kth step in the first staircase is

joined to the (2000 – k)th step in the second staircase, as shown in the diagram below.

⁞ ⁞

The area of this new figure is 2S, since it is just the piecing together of two staircases with area S.

By this construction, the kth step in the new figure has length

. From the above

identity with

, this value equals

. It follows that each of the 1999

“steps” in the new figure has area 1, since it is just a square with height 1 and length 1.

Therefore,

, from which we conclude

.

The kth

step of the staircase has area

(note: diagram not to scale!!)

+