1. The Game of Contorted Fractions

2. 2 Rules of the Game Typical position has a number of real numbers in boxes. The typical legal move is to alter just one of these numbers. The number replacing a given one must have a strictly smaller denominator, or If the given number is already an integer, the new number must be an integer strictly smaller in absolute value.

3. 3 Rules of the Game The only legal move for player 1, Left, is to decrease the number The only legal move for player 2, Right, is to increase the number

4. 4 2525 Possible Left options: 1/3, ¼, 0, -2, etc. Possible Right options: ½, 2/3, ¾, 17 ¼, etc.

5. 5 2525 Left’s best option: 1/3 Right’s best option: ½

6. 6 Notice The game will come to an end when each fraction reaches zero, since we require that:  The number replacing a given one must have a strictly smaller denominator, or  If the given number is already an integer, the new number must be an integer strictly smaller in absolute value.

7. 7 2525 -2 3 3737

8. 8 0101 1010 1 1212 2121 1313 2323 3232 3131 1414 2525 3535 3434 4343 5353 5252 4141 1515 2727 3838 3737 4747 5858 5757 4545 5454 7575 8585 7474 7373 8383 7272 5151

9. 9 2525 3737

10. Combinatorial Game Theory

11. It’s All About Hackenbush

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13. 13 Blue-Red Hackenbush is an example of a game with 2 players, Left and Right. In Blue-Red Hackenbush, Left deletes any bLue edge, and Right deletes any Red edge. Whichever player cannot move, loses.

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26. Games

27. 27 Defined We are interested in games that:  Have just two players, often called Left and Right.  Have several, usually finitely many, positions, and often a particular starting position.

28. 28  There are clearly defined rules that specify the moves that either player can make from a given position to its options.  Have Left and Right moving alternately, in the game as a whole.

29. 29  In the normal play convention, has the first player unable to move losing.  Will always come to an end because some player will be unable to move.

30. 30  Allow both players to know what is going on; i.e. there is perfect or complete information.  Have no chance moves such as rolling dice or shuffling cards.

31. “Whose Game?” *Pun from Winning Ways

32. 32 Outcome Classes If Left starts If Right starts Left Wins Right Wins Left Wins Positive (L wins) Zero (2 nd wins) Right Wins Fuzzy (1 st wins) Negative (R wins)

33. 33 Fuzzy Games A fuzzy game is not > 0, not < 0, and not = 0 A fuzzy game is confused with 0

34. 34 Fuzzy Games -212 G

35. 35 Games and Numbers All numbers are games But not all games are numbers!!!

36. 36 Positive, Negative, and Zero Theoretic game value is determined using Surreal Numbers

37. 37 Surrealism in the Arts

38. 38 How to Count The old way of counting: 1 23 4

39. 39 How to Count The new way of counting: 0 12 3

40. 40 How to Count So we have 0, 1, 2, 3 = 4 bananas!!

41. Surreal Numbers

42. 42 Surreal Numbers: A Class The Surreal Numbers are a proper Class of numbers. Other proper Classes we already know:  - The Natural Numbers - The Integers  - The Rational Numbers  - The Real Numbers  - The Complex Numbers

43. 43 Construction { L | R }

44. 44 Construction {  |  }

45. 45 Construction { | }

46. 46 Construction { | } = 0

47. 47 The Zero Game of Hackenbush You go first, I insist!!!

48. 48 Construction { 0 | } = 1

49. 49 Construction { | 0 } = -1

50. 50 {0 | } = 1 { | 0} = -1 {-1 | 1} = 0

51. 51 Red-Blue Hackenbush Chains

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55. 55 a 2 b 0 c - ½

56. 56 d e f

57. 57 gih

58. 58 jk

59. 59 l m

60. 60 n o

61. So, what’s the value of that loooooong chain???

62. 62 Berlekamp’s Rule

63. 63 Berlekamp’s Rule

64. 64 Berlekamp’s Rule -4. 0 1 1 0 0 1 1

65. 65 Berlekamp’s Rule -4. 0 1 1 0 0 1 1.0110101 -(4+1414 + 1 8 +1 32 +1 ) 128

66. 66 Berlekamp’s Rule -4. 0 1 1 0 0 1 1.0110101 -(4+1414 + 1 8 +1 32 +1 ) 128 = - 4 53 128

67. 67 Berlekamp’s Rule Likewise, we can convert a given fractional number into a Hackenbush string Write the fractional number in binary  3 5/8 = 3.101 Replace the integer part with  LLL (since positive value)

68. 68 Berlekamp’s Rule Replace the point with  LR (since positive) Convert 1’s and 0’s thereafter into L’s and R’s, but omitting the final 1  1 0 1  L R

69. 69 Berlekamp’s Rule Result:  3 5/8 = 3. 1 0 1  L L L L R L R

70. 70 Berlekamp’s Rule This also works for real numbers that don’t terminate, except that there is no final 1 to be omitted  1/3 = 0. 0 1 0 1 0 1 0 1 0...  LR R L R L R L R L R...

71. Value of a Game of Contorted Fractions

72. 72 Use Continued Fractions We can use continued fractions, together with Berlekamp’s rule, to determine the game value of any fraction We convert the fraction into a continued fraction, and then read off the partial quotients a 1, a 2, …, as alternate numbers of 0’s and 1’s, except that the first 0 is replaced by a binary point.

73. 73 Example

74. 74 Example

75. 75 2525 -2 3 3737

76. 76 2525 -2 3 3737

77. 77 2525 -2 3 3737 3737 2525

78. 78 2525 -2 3 3737 Positive Game Value: Left Wins

79. 79 Ready to Play?? Here’s a link to a nice little applet that will let you play against the computer CONTORTED FRACTIONS