1. Fibonacci Solitaire Cliff F. Smith June 8, 2010 Presented in partial fulfillment of the requirements for the MST in mathematics

2. Example Game

3. Example Game

4. Example Game

5. Example Game

6. Example Game

7. Example Game

8. Example Game

9. Example Game

10. Example Game

11. Example Game Removed Pairs

12. Example Game Removed Pairs

13. Example Game Removed Pairs

14. Example Game Removed Pairs

15. Example Game Removed Pairs

16. Example Game Removed Pairs

17. Example Game Removed Pairs

18. Example Game Removed Pairs

19. Example Game Removed Pairs

20. Example Game Removed Pairs

21. Example Game Removed Pairs

22. Game Over Removed Pairs

23. Game Over Removed Pairs We Lose 

24. Notation Removed Pairs

25. Notation

26. Notation 8 (6,5) (7,1) (4,3) 2

27. Notation (6,5) (7,1) (4,3) 2 8

28. Notation (6,5) (7,1) (4,3) 2 8 • Write pairs with the smaller card first

29. Notation (6,5) (7,1) (4,3) 2 8 • Write pairs with the smaller card first

30. Notation (5,6) (1,7) (3,4) 2 8 • Write pairs with the smaller card first

31. Notation (5,6) (1,7) (3,4) 2 8 • Write pairs with the smaller card first

32. Notation (5,6) (1,7) (3,4) 2 8 • Write pairs with the smaller card first • Organize numerically

33. Notation (5,6) (1,7) (3,4) 2 8 • Write pairs with the smaller card first • Organize numerically • (by singles and smaller card in pairs)

34. Notation (5,6) (1,7) (3,4) 2 8 • Write pairs with the smaller card first • Organize numerically • (by singles and smaller card in pairs)

35. Notation (1,7) 2 (3,4) 8 (5,6) • Write pairs with the smaller card first • Organize numerically • (by singles and smaller card in pairs)

36. Notation (1,7) 2 (3,4) 8 (5,6) • Write pairs with the smaller card first • Organize numerically • (by singles and smaller card in pairs)

37. Notation (1,7) 2 (3,4) (5,6) 8 • Write pairs with the smaller card first • Organize numerically • (by singles and smaller card in pairs)

38. Notation (1,7) 2 (3,4) (5,6) 8 • Write pairs with the smaller card first • Organize numerically • (by singles and smaller card in pairs)

39. Play!

40. Play! • Keep track of initial order of cards

41. Play! • Keep track of initial order of cards • Write final set of pairs and singles in correct notation

42. Some Questions

43. Some Questions • What is the probability of winning?

44. Some Questions • What is the probability of winning? • Hard to answer

45. Some Questions • What is the probability of winning? • Hard to answer • 2. Is the Fibonacci solitaire algorithm a function?

46. Some Questions • What is the probability of winning? • Hard to answer • Is the Fibonacci solitaire algorithm a function? • Yes

47. (1,7) 2 (3,4) (5,6) 8 More Notation

48. (1,7) 2 (3,4) (5,6) 8 More Notation FS(8, 5, 3, 4, 1, 7, 6, 2) = (1,7) 2 (3,4) (5,6) 8

49. The Function FS Orderings of a deck of cards. (permutations) Pairings (partial and total)

50. Some Questions • What is the probability of winning? • Hard to answer • Is the Fibonacci solitaire algorithm a function? • Yes • 3. Is the FS function injective?