Treasure Hunt Secondary

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MathematicalTreasure Hunt

KEY STAGE 3GRADES 6-8


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Mathematical Treasure Hunt

IntroductionTe mathematical treasure hunt is a great activity or un and engaging mathemat-ics lessons: the pupils ollow a trail o clues and mathematical problems aroundthe school site; each clue contains a hint to where the next clue is hidden.

Tis document includes clues and questions intended or Key Stage 3 (UK) orgrades 68 (US).

Te treasure hunt works best when the class is divided into groups o about 5 chil-dren o different abilities. Working in a team, and in a competition, supports teamworking skills, and even children with difficulties in mathematics can participate.

Te questions are taken rom a wide range o different topics, and ofen not di-rectly related to the mathematics curriculum. Some o the problems lend them-selves to urther discussion aferwards; ofen there is an article on that topic in theMathigon World o Mathematics.

Te answer to each problem is an integer, and all the answers once decoded intoletters spell the location o the treasure.: the library.

Te Questions

Name Locations Solution Order of eams

A Cryptography 18 1 9 7 5 3

B Combinatorics 18 2 10 8 6 4

C Graph Teory 2 3 1 9 7 5

D Number Pyramid 9 4 2 10 8 6

E Pascals riangle 8 5 3 1 9 7

F Prime Numbers 25 6 4 2 10 8

G Probability 5 7 5 3 1 9

H Platonic Solids 12 8 6 4 2 10

I Geometry 1 9 7 5 3 1

J Sequences 20 10 8 6 4 2

PreparationFirst choose 10 locations in your school where to hide the the different questions(see previous table). Either use the prepared clues (pages 910) or come up withyour own clues (pages 1112) to lead to these questions. Print the clues once oreach team.

Make sure that the class is able to solve all the problems. Print the introductorysheets and questions (pages 38) once or every team and cut them in the middle.Print and cut the additional materials or various problems (pages 1314).

Put the questions, materials as well as the clues leading to the nextquestion intoan envelope, and hide the 10 envelopes around the school site. Keep the two intro-

ductory sheets or each team, as well as a different clue or each team the onesleading to their first problem.

At the beginning o the lesson, divide the class into a couple o teams and giveeach team the two introductory sheets, as well as their first clue. Te treasure ishidden in the library usually chocolate works well

able of Contents

Page 3 Introductory SheetsPages 48 ProblemsPages 910 Clues

Pages 1112 Customisable CluesPages 1314 Additional Materials

Copyright Notices

Te mathematical treasure hunt is part o the Mathigon Project and PhilippLegner, 2012. Graphics include images by the sxc.hu users ba1969, slafoandspekulator. o be used only or educational purposes.

Instructions for Teachers


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Mathematical Treasure Hunt

Professor Integer was one of the worlds most famous mathematicians,

who made discoveries that changed the world forever: from algorithms for

computers and internet to statistical calculations and quantum mechani-

cal predictions.

When he died, he had no relatives or close friends but a very large for-

tune. He believed that only the best mathematicians deserved to find his

treasure and created a trail of puzzles and problems.

Many of his diary pages, notes and letters are archived at the University

of Cantortown, and they all include clues and hints regarding the location

of the treasure.

This treasure hunt will require you to move around your school, find

the hidden clues and solve mathematical problems. Each question

will contain a clue about where the next problem will be hidden, but

every team solves the problems in a different order.

When you find an envelope, take one problem page and one clue. Try

to solve the problem, sometimes using additional materials in the en-

velope; then look for the next problem. You may not find the problems

in the correct order!

There are many other children in the school, so avoid any unneces-

sary noise. Dont leave your solutions behind for the next team to see,

and dont take more than one copy of each problem otherwise fol-

lowing teams might not be able to solve the problem.

You are now ready to receive the first clue and a copy of the last letter

written by Professor Integer.

Good luck!

INSTRUCTIONS

The Integer Files

Archive of the University of Cantortown Item 0

Item 0: Last letter of Prof. Integer Catalogue Nr. 0010

Dear Mathematicians,When you read this letter, I will be dead,and my treasure will be hidden in a very safelocation. Only the best mathematiciansdeserve to find it.In my notes and diaries, I have left 10problems which you need to solve. Theanswer to every problem is a single number,which you can write down here:

A B C D E F G H I J

Once you have solved all problems, turnthe numbers into letters (1-a, 2-b,3-c andso on) and bring the letters into the correctorder to spell the location of the treasure: _ _ _ _ _ _ _ _ _ _

Hurry, though, because otherteams may be onto it as well

Regards and good Luck!Prof. Integer


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The Integer Files


Item 2: Spiral Bound Notebook 1, piece of cardbord Catalogue Nr. 0556

Yesterday I was invited to a Birthday

party. There were 36 guests and every-

body shook hands with everybody else

exactly once.

Afterwards, I wondered how many hand

shakes there were in total. It clearly

is impractical to count them all one by

one; we need a clever mathematical idea

to fnd a simple equation...

ProblemB: Combinatorics

The Integer Files


Item 1: Lined Paper, Cards Catalogue Nr. 7644

ProblemA: Cryptography

I hink somebody has broken ino my sudy and solen imporan docu-mens and calculaions. I is a disaser ha I have los my noes, bu iis even worse ha he hie can read my discoveries and ideas.

In he uure, I need o decipher my noes, so ha only I can readhem. A very easy mehod was invened by Julius Caesar: you jus

shif ever leter along he alphabe, or example

a b c d e g h i j k l m n o p q r s u n w x y z

u n w x y z a b c d e g h i j k l m n o p q r s The word 'mahemaician' or example would be shifed o

fmaxfmbvbg'.

To decipher his code, one would have o ry all 24 possibiliies o shifhe leter, which could ake a very long ime. This should keep my noes

sae in he uure!

MAXTGLPXK BLXBZAmxxg

Note:Cryptographyis theareaof

mathematicsaboutfinding

andbreakingcodes. Itwas

especiallyimportantinwars:dur-

ingthesecondworldwar,theCambridgeMathematicianAlan

Turingsuccessfullybuiltoneofthe

firstcomputerstodecodethe

GermanEnigmacodingmachine.T

hiscouldhavewellbeenthe

singlemostimportantachievement

thatledtothealliedvictory.

Therearemanymuchmorecomplic

atedmethodstodecode

sentencestoday,someofwhich(we

think)areunbreakableand

withoutwhichinternetbankingw

ouldbeimpossible.Theyuse

primenumbersandmanyimporta

ntmathematicalresults.

Togetthekeynumber

forthisproblem,divide

thetotalnumberof

handshakesby35.


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The Integer Files


Item 3: Spiral Bound Notebook No 2 Catalogue Nr. 5478

The Integer Files


Item 4: Old piece of paper 1 Catalogue Nr. 1271

Problem D: Number Pyramid

Last night I wasthinkingabout alarge numberpyramid.Unfortunately I spilled my coffee, andIlostmany ofthe numbers only 6 remained legible.I wasthinkingabout itfor some time, andI think itis possible to reconstruct the wholepyramid usingonly those6numbers!

ProblemC: GraphTheory

LastweekIvisitedKnigsberg,acityinRussia.Knigsbergisdividedintoseveralparts byariver,andtheislandsareconnectedbybridges.

Manyyearsago, themathematicianLeonardEuleraskedwhetheritwouldbepossibletotourKnigsberg, sothatyou crosseverybridgeonce,butnotmorethanonce.

Hereareacoupleofothercitymaps.Inhow manymapsisitIMPOSSIBLEtofindatourthatcrosseseverybridgeexactlyonce?Youcanstart andfinish whereveryouwant.

82

47 55

20

6 11

The answer !


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This shape is called an Icosahedron. Allfaces are equilateral triangles, and itlooks the same from every direction.Therefore it is called a Platonic Solid,named after the Greek mathematicianPlato. Plato showed that there are onlyfive solids of this kind. He though thatthey corresponded to the four classical elementsfire, air, earth and fire, as well as the universe.

Here is a table showing all 5 platonic solids. Canyou find a pattern and fill in the gaps?

The Integer Files


Item 8: Old piece of paper 2, Icosahedron Catalogue Nr. 5512

The Integer Files


Item 7:Old notebook, playing cards Catalogue Nr. 4652

Iloveplayingcardspoker,blackjack

andmanyothergames.Andcard

games

areverymuchrelatedtoprobability!

HereisonequestionIwasthinkin

g

about:Anormaldeckofcardscontains

52cardsinfourdifferentsuits(

clubs

andspadesareblack,heartsanddia-

mondsarered.

Supposewechooseonecardatra

ndom

andputitback,thenchooseasecond

cardatrandomandputitback,and

thenchooseathirdcardatrando

m.

Problem H: Platonic Solids

Name Model Faces Vertices Edges

Tetrahedron 4 6

Cube 6 8 12

Octahedron 8 6

Dodecahedron 20 30

Icosahedron 20 30

Maybe Think about Faces + Vertces!

ProblemG:Probabil

ity

Whatistheprobability

(inpercent)that

allthreecardshavethesamecolour

(redorblack)?IftheprobabilityisX%,

iamlookingforthesquarerootofthe

numberX.

Playing Cards by Wikimedia Users Asimzb and Jfitch


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The Integer Files


Item 10: Spiral Bound Notebook No 2 Catalogue Nr. 9612

The Integer Files


Item 9: Two letters by Prof. Integer Catalogue Nr. 1972

A sequences is a list of numbers which follow a certain pat-tern. For example, the square numbers, the powers of two orthe prime numbers are all sequences.

A very famous sequence are the Fibonacci numbers.Starting with 1, 1, every following number is the sum of theprevious two numbers. The third number is 1+1=2, the fourthnumber is 1+2=3 and so on. We get

1, 1, 2, 3, 5, 8, 13, 21, 34, 55,

Discovered by the Italian mathematician Leonardo Fibo-nacci, these numbers appear in many places in nature: fromrabbit populations to sunflower seeds.

I love playing around with sequences. Here are a few ex-amples, you need to find the pattern and fill in the missingnumbers.

1, 3, 6, 10, , ,

1, 3, 9, 27, , ,

3, 6, 5, 10, 9, , , ,

The answer to this problem is the sum of the individualdigits of the numbers in the circles.

Problem J: SequencesI received these

two letters fromProf. Interger justa couple of daysbefore he died.

My dear friend,

Heres a fun problem: Canyou work out which propor-tion of this square is red?

My dear friend,

I realised that my previous problem was rather hard, sohere are some hints: Let us assume that the big squarehas length 1. First, we need to calculate the area of thebiggest circleand the area of the second biggest square.

Now notice that the original shape consists of a single

frame, which is repeated again and again just smaller.Thus the proportion red in the final shape is exactly thesame as the proportion red in the frame.

Can you work out the proportion red in this frame? Here are threepossibilities: p = 0.57 p = 0.49 p = 0.68

1 2 3key:

Problem I:Geometry


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Full of paper, books and files,

Pay the school office some smiles!

Bonjour, Hola, Goddag, Ni Hao,

And more if languages allow.

Find the riddle

that is given,

Where the bits and

bytes are livin.

With watercolour, crayons, pen,

The next puzzle is waiting then.

In breaktime its brawling,

in lessons is still,

On the playground the next riddle finding you will.

Where smoke and where fire

are common event,

The following mystery

I will present.

SchoolOffice

Language

sRoom

ComputerRoom

ArtandCraftsRoom

Playground

Chemi

stryLab


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No pupil may enter,

no child may come in,Where the next clue is hidden, so you can begin!

Up

an

ddown

an

dleftandrig

ht,

Thesta

ircases,

theydoex

cite.

The biggest room

that is in sight,

But try to knock

it is polite.

Hurry, less than

80 days,For you to reach the problem's place.

Mathematics

Room

StaffCom

monRoom

MusicRoom

GeographyR

oom

Hall/Aud

itorium

Starirc

ases

Where 10 divided 5 is 2,The next questions, it waits for you.

Trumpetfanfares

nodelay!

Andmusicsounds

willleadyourway.


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Pascals TrianglePrint several times for each group, cut out and add to problem E

Pa s c a l s

Tr i a n

g l e

1

1

1

1

2

1

1

3

3

1

1

4

6

4

1

1

5

10

10

5

1

1

6

15

20

15

6

1

1

7

21

35

35

21

7

1

1

8

28

56

70

56

28

8

1

1

9

36

84

126

12

6

84

36

9

1

1

10

45

120

210

25

2

210

120

45

10

1

1

11

55

16

5

33

0

462

46

2

33

0

16

5

55

11

1

1

12

66

220

49

5

79

2

92

4

79

2

48

5

220

66

12

1

1

13

78

286

71

5

128

7

17

16

17

16

128

7

71

5

286

78

13

1

1

14

91

36

4

1001

2002

300

3

34

32

300

3

2002

1001

36

4

91

14

1

1

15

10

5

455

13

65

300

3

500

5

64

35

64

35

599

5

300

3

13

65

455

10

5

15

1

1

16

120

560

1820

43

68

8008

11

44

0

128

70

1144

0

8008

43

68

1820

560

120

16

1

Pa s c a l s

Tr i a n

g l e

1

1

1

1

2

1

1

3

3

1

1

4

6

4

1

1

5

10

10

5

1

1

6

15

20

15

6

1

1

7

21

35

35

21

7

1

1

8

28

56

70

56

28

8

1

1

9

36

84

126

12

6

84

36

9

1

1

10

45

120

210

25

2

210

120

45

10

1

1

11

55

16

5

33

0

462

46

2

33

0

16

5

55

11

1

1

12

66

220

49

5

79

2

92

4

79

2

48

5

220

66

12

1

1

13

78

286

71

5

128

7

17

16

17

16

128

7

71

5

286

78

13

1

1

14

91

36

4

1001

2002

300

3

34

32

300

3

2002

1001

36

4

91

14

1

1

15

10

5

455

13

65

300

3

500

5

64

35

64

35

599

5

300

3

13

65

455

10

5

15

1

1

16

120

560

1820

43

68

8008

11

44

0

128

70

1144

0

8008

43

68

1820

560

120

16

1


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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 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 99 100

100 Number TablePrint several times for each group, cut out and add to problem F

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 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 99 100

Treasure Hunt Secondary

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