The Game of Contorted Fractions

1. The Game of Contorted Fractions

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

5. 25 Left’s best option: 1/3 Right’s best option: ½

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. 37 25 -23

8. 01 10 11 21 12 13 23 32 31 14 25 35 34 43 53 52 41 15 27 38 37 47 58 57 45 54 75 85 74 73 83 72 51

9. 37 25 -23

10. Combinatorial Game Theory

11. It’s All AboutHackenbush

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.

26. Games

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. 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. 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. 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. Outcome Classes

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

34. Fuzzy Games -2 -1 1 2 G

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

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

37. Surrealism in the Arts

38. How to Count • The old way of counting: 4 2 3 1

39. How to Count • The new way of counting: 3 1 2 0

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

41. Surreal Numbers

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. Construction { L | R }

44. Construction {  |  }

45. Construction { | }

46. Construction { | } = 0

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

48. Construction { 0 | } = 1

49. Construction { | 0 } = -1

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