Maths Magic CTYI Review Session David Goulding

27/03/09 Maths MagicDavid Goulding


• These slides were presented to the parents during the final class of the course

• Main aim is to give an overview of the different topics covered during the eight weeks

• CTYI is geared towards students demonstrating high academic achievement over a range of ages

• Maths Magic was run for students ranging in ages between 8 and 12

• While classes were planned in advance, it was sometimes necessary to change course during the class in order to keep interest levels high

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

Natural Numbers

Triangular Numbers


11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

H T

HH HT TH TT

HHH HHT HTH THH TTH THT HTT TTT

Odds of Getting an Exact Number of Heads or Tails

Rosencrantz - Heads???

Odds of getting a head or tail is 1/2


http://www.youtube.com/watch?v=NbInZ5oJ0bc

Rolling dice

Well there are six numbers on a die, 1-6, if the die is fair (that is not biased towards any one number), then the chances of getting any one of the numbers is 1/6

What about if I roll a pair of diceand look at the sum??

36

36

18

12


Odds of Winning Dave’s Lottery

Dave’s lotto is a simple game, there are six numbers (1-6) you must choose four of them: what are the chances of winning?

Let’s look at the combinations(1,2,3,4) (2,3,4,5)(1,2,3,5) (2,3,4,6)(1,2,3,6) (2,3,5,6)(1,2,4,5) (2,4,5,6)(1,2,4,6) (3,4,5,6)(1,2,5,6) (1,4,5,6)

(1,3,4,5)(1,3,4,6)(1,3,5,6)27/03/09 Maths Magic

There are 15 ways to choose 4 numbers from 6, we can see this from Pascal’s triangle

1 6 15 20 15 6 1The general formula for calculating combinations is in fact given by

Where n!=nx(n-1)x(n-2)x…x3x2x1Let’s look at example above, want to choose four numbers from 6, so that’s

)!(!

!

rnr

nCrn

1548

720

2*24

720

)1*2(*)1*2*3*4(

1*2*3*4*5*6

!2!*4

!64

6 C


Odds of Winning The Irish Lotto

In the Irish Lotto there are 45 numbers and we must choose the correct six numbers, using the same formula as before we have

This works out to be 8,145,060 different combinations(again trust me on this one!!)

Conclusion: Don’t play the Lotto

!39!*6

!456

45 C


Let’s suppose we have some sort of magic box, which we’ll call the squaring box, we put a number into the box and it give back the square of the number, for example

2 4

We then generalised this to the following function:

2)( ttf 27/03/09 Maths MagicDavid Goulding

This lead us to be able to draw our first graph with the students, so we drew a graph of “t-squared” – we looked at this then in relation to graphs of running times for the students


A perfect number is a number whose divisors (excluding itself) add up to the number!

Let’s look at my birthday : June 28th

Divisors of 6 are 1,2,3,6. Now 1+2+3=6 So 6 is perfect!!

Divisors of 28 are 1,2,4,7,14,28. Now 1+2+4+7+14=28 So 28 is perfect!!

The next two perfect numbers are 496 and 8128 – you can trust me on those ones!!

(unsolved) Are there any odd perfect numbers?27/03/09 Maths MagicDavid Goulding

A prime number is a number divisible only by itself and one.(We do not include one as a prime)

Let’s look at primes, well 2,3,5,7,11,13,17,19 are all primes…..is there a way of generating the primes, well yes we can use the famous Sieve of Eratosthenes

27/03/09 Maths Magic

Now the Sieve is all well and good but it is extremely time consuming to carry out….is there another way

Well there are tests for certain types of primes but there is no magic way of writing done all the prime numbers!!

How many primes are there?The answer is that there is an infinite number of primes, this was proven in the class as follows.

Suppose we had only a finite number of primes, say 2,3,5,7 and 11. Now we know that any number is either prime or composite (that is it is divisible by some number-in fact divisible by primes). What about the number (2x3x5x7x11) + 1 = 2311well this is not divisible by any of 2,3,5,7,11 therefore it is either prime or we have not included a prime in our list!! (Turns out this is prime as well) Maths MagicDavid Goulding

First introduction to Graph Theory!


One possible solution that the class came up with:


Why does this work? Well Euler said that if we have either 0 or 2 nodes of an odd degree then the problem will have a solution – Try find another possible solution, simply by adding additional paths and then counting the number of lines entering each node

4

4

5 3

So the Bridges of Konigsberg turned out to be the forerunner of modern day graph theory


Monty Hall Problem

The Monty Hall Problem is based on the game show “Let’s Make a Deal” which was a very popular tv show in the States in the 1960s and 1970s.

In the problem, there are three doors 1, 2 and 3, behind one of the doors is a car but behind the other two there are goats!!

Haven chosen a door, Monty Hallwould then open a door for the contestant to reveal a goat.Monty Hall would then offer the chance to swap doors. The problemwas then whether it was more beneficial to stick or swap

My favourite demonstration is by Professor Marcus du Satoy 27/03/09 Maths MagicDavid Goulding

http://www.youtube.com/watch?v=o_djTy3G0pg

Switching Loses

Switching Wins

Switching WinsSo 2 out of 3 times you win by switching, so probability of winning by switching is 2/3, while it is only 1/3 by staying


Summing Consecutive Numbers

When Gauss was fourteen, his teacher suffering from a heavy night at the opera the evening before decided that the best way to keep the students quiet was to get them to add all the numbers from 1 to 100. Unfortunately for him, Gauss was a bit too clever even by our standards this is what he did

1 + 2 + 3 + … + 98 + 99+100

100+99+98+ …+3 +2 +1

101+101+101+…+101+101+10127/03/09 Maths MagicDavid Goulding

Summing Consecutive Numbers

So we have that the sum is simply (100*101)/2 (since we counted from 1 to 100 twice!)

We can extend this to the generalised formula

Which in simply terms, means if we want to sum all the numbers from 1 up to a certain number, we simply take the number, add one and multiply by the original number and finally divide by 2…hey presto we have the sum

Incidentally it’s 5050 in the original problem

2

)1(1...321

1

nnnni

n

i


Number of Squares on a chess board?

Well exactly how many squares are there on a chessboard?

No, not 64!! You forgot about the squares of size 2,3,4, etc.

How many of size 1? 64Size 2? 49Size 3? 36…etc.

See the pattern?It’s the sum of the square numbers from 1 to 8

# of squares = 1+4+9+16+25+36+49+64 = 204

6

)12)(1()1(...321 22222

1

2

nnnnni

n

i27/03/09 Maths MagicDavid Goulding

Matrix Multiplication

27

611

23

01

14

32AB

14

32A

23

01B

Matrix Multiplication was introduced in order to show the students that sometimes the order of multiplication is important.

1114

32

14

32

23

01BA

Codes and Ciphers

In this class, the students were introduced to some simple codes and the corresponding ciphers

These included a simple numerical substitution, where A=1, B=2, C=3, etc.

The Caesar Cipher in which the letters are moved along the alphabet by a fixed number of positions for example A=H, B=I, C=J, etcSometimes these are easy to break…especially if somebody is a VAXTMXK Maths MagicDavid Goulding

Another example was the well known Pigpen cipher, shown to the left with coded and decoded message below

This class made use of Simon Singh’s Code Book


http://www.simonsingh.net/The_Code_Book.html

Final Problems

Parents you are all to enter into a deal with your kids, I am going to propose a problem to you all and you are going to try and solve it yourselves. If you fail to do so by tomorrow you must give 1c to your child, if you can not solve it on Monday you must give them 2c, 4c on Tuesday if you haven’t solved it and so on until 30 days have elapsed

Sounds like a pretty fair deal, don’t you agree?

Here’s the puzzle so:


Being an extremely cruel and vicious maths teacher, I have decided to take ten of the students and stand them in a line facing the wall, one behind the other. On each of their heads I will place a hat, either black or white, I don’t know how many of each colour that is up to me to choose when I see fit. Now the kids are not allowed to speak or communicate once they are placed in the line. When I say so the kids starting from the back must tell me what colour hat they are wearing, if they pick the wrong colour unfortunately they will die. However if they pick the right colour I will let them live (this time!).

Your puzzle is to come up with the best strategy for the kids in order to save as many as possible…and just so you know you can definitely save at least nine of the kids!!

And just for completion, if you don’t manage to solve the problem, in 30 days time you will owe your child

€10,737,418.23Best get working on that problem I guess!!


Maths Magic CTYI Review Session David Goulding

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