Math without Words

MATH WITHOUT WORDS Purely Visual Puzzles to Confound and Delight

James Tanton

©2008 James Tanton

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A FEW BRIEF INFORMAL WORDS(!!) MATH WITHOUT WORDS is a collection of 75 immediately accessible, but deeply mathematical, puzzles all designed to offer true joy in thinking mathematically in creative, innovative and surprising ways. Not a single word is written. These puzzles are universal: they transcend the barriers of language and culture, literally, and are thereby accessible to all people on this globe. Moreover, the puzzles themselves speak to a universal truth of mathematics. Many of these puzzles are hard and they are not intended to be solved in one sitting. (And sometimes the bulk of the challenge is figuring out what is being asked!) The idea is to sit with a puzzle or two over a number of days and wait for a moment of inspiration or insight to arrive. That is the true mathematical experience. Mathematicians call such flashes of understanding aha moments and recognize that they cannot be called upon within minutes, or even hours, or even days. (Mathematicians don’t believe that timed tests reflect the mathematical experience.) Moments of insight will come on their own accord and there is nothing to be done but let them take their time. There is never need for a rush in doing true mathematics! For this reason each puzzle should be mulled upon for quite some while. This book is designed to carry interest for many weeks, not just hours. Perhaps pull out a particular puzzle page and hang it as a poster on the wall. Let it become part of your subconscious. It’s surprising how fruitful indirect mulling can be! Some puzzles are relatively easy to answer, while others are quite deep. Some connect to current research problems in mathematics. Any relevant insight garnered with a particular puzzle should be deemed a success, even if it does not solve the challenge completely. (But for those that are looking for confirmation or guidance with any particular puzzle, brief answers appear at the back of the book.) Mathematics can be compelling and frustrating, enlightening and humbling, but most of all it should forever be joyful. So on that note … Enjoy!

- JST

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MATH WITHOUT WORDS: 1

4

5

MATH WITHOUT WORDS: 2

6

7

MATH WITHOUT WORDS: 3

8

9

10

11

MATH WITHOUT WORDS: 4

12

13

14

15

MATH WITHOUT WORDS: 5

16

17

MATH WITHOUT WORDS: 6

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21

MATH WITHOUT WORDS: 7

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MATH WITHOUT WORDS: 8

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25

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27

MATH WITHOUT WORDS: 9

28

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31

MATH WITHOUT WORDS: 10

32

33

MATH WITHOUT WORDS: 11

34

35

MATH WITHOUT WORDS: 12

36

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MATH WITHOUT WORDS: 13

38

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MATH WITHOUT WORDS: 14

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MATH WITHOUT WORDS: 15

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MATH WITHOUT WORDS: 16

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MATH WITHOUT WORDS: 17

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64

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66

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68

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MATH WITHOUT WORDS: 24

72

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MATH WITHOUT WORDS: 25

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MATH WITHOUT WORDS: 27

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MATH WITHOUT WORDS: 28

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MATH WITHOUT WORDS: 30

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MATH WITHOUT WORDS: 31

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MATH WITHOUT WORDS: 32

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MATH WITHOUT WORDS: 33

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MATH WITHOUT WORDS: 63

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MATH WITHOUT WORDS: 65

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MATH WITHOUT WORDS: 66

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MATH WITHOUT WORDS: 67

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MATH WITHOUT WORDS: 68

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MATH WITHOUT WORDS: 69

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MATH WITHOUT WORDS: 71

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MATH WITHOUT WORDS: 75

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OH SO BRIEF SOLUTIONS 1.

4.

2.

3.

2 5.

8.

9.

6.

7.

3 10.

13.

11.

12.

14.

4

15.

18.

19. 16.

20.

17.

5 21.

22.

24.

23.

6 26.

27.

25.

28.

7 29.

31.

32.

30.

__________

_________

_________

8 35.

33.

34.

36. ---------------

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37.

--------------38.

9 41.

42.

43.

44. 39.

45.

40.

10 46.

50.

47.

51. 48.

49. 52.

11 53.

57.

--------

------58.

54.

59.

55.

60.

61.

56.

12

63.

64.

62.

13 65.

69. 66. ??? 67.

70. 68.

14 71.

74.

72.

75.

73.