A Generalized Ryuoh-Nim: A Variant of the classical game of Wythoff Nim

情報処理学会 第36回ゲーム情報学研究会にて．この発表にて，若手奨励賞を受賞．

Transcript

1. A Generalized Ryuoh - Nim: A Variant of the classical game of Wythoff Nim த԰
   ༔ࢿ†
   
   ٶࣉ
   ྑฏ†
   ෱Ҫ
   ণଇ*
   ށᅳ
   ༑و†
   † ؔ੢ֶӃߴ౳෦ * ฌݿڭҭେֶ
2. ݚڀͷഎܠɾ໨త ɾ૊Έ߹ΘͤήʔϜͷҰछͰ͋ΔੴऔΓήʔϜͷ
   
   
   
   มछΛݚڀ͍ͯͨ͠ɻ ɾ$PSOFS
   UIF
   2VFFO
   8ZUIP⒎`T
   (BNF
   ͱ͍͏
   
   
   
   ΋ͷΛ஌Γɺ͜ΕΛมܗͤͨ͞΋ͷͰɺ(SVOEZ਺
   ɹ͕៉ྷʹͳΔ΋ͷΛ୳ͨ͠ɻ
3. 1.ݚڀͷഎܠɾ໨త 2.Corner the “Ryuoh”(ཾԦ) problem 2.1 ήʔϜͷઆ໌ 2.2 Grundy਺ 2.3ͦͷଞಈ͖Λ෇Ճͨ͠৔߹ 3.Corner the “Ryuma”(ཾഅ) problem 4.Corner the “Ryuma”(ཾഅ) problem with pass 5.Grundy਺ͷ΋ͭۮحੑ શମͷྲྀΕ
4. Corner the “Ryuoh”(ཾԦ) problem w 
   ਓ͕ަޓʹνΣε൫্ͷཾԦ ੒ඈं Λಈ͔͢
   w ࠨ্୺ʹ࣋ͬͯߦͬͨϓϨΠϠͷউͪ
5. ੒Δ Corner the “Ryuoh”(ཾԦ) problem
9. ϓϨΠϠʔ̍
   ͷλʔϯ Corner the “Ryuoh”(ཾԦ) problem
10. ϓϨΠϠʔ̍
    ͷλʔϯ Corner the “Ryuoh”(ཾԦ) problem
11. ϓϨΠϠʔ
    ͷλʔϯ Corner the “Ryuoh”(ཾԦ) problem
12. ϓϨΠϠʔ
    ͷλʔϯ Corner the “Ryuoh”(ཾԦ) problem
13. ϓϨΠϠʔ
    ͷউͪʂʂ Corner the “Ryuoh”(ཾԦ) problem
14. x y
15. x y (x,y)=(2,3)
16. x y (x,y)=(2,3) (x,y)=(2,0) 2 3 2 0 x y
17. x y (x,y)=(2,3) (x,y)=(1,2) 2 3 1 2 x y
18. Ҿ͖෼͚ͳ͠ͷෆภήʔϜͳΒɺ
    ઌखඞউϙδγϣϯͱ
    ޙखඞউϙδγϣϯ͕͋Δɻ
19. ɾ࠷ऴہ໘Λ̌ͱ͢Δɻ
    ɾͦͷଞͷ(SVOEZ਺͸ߦ͖ઌʹͳ͍࠷খͷඇෛ੔਺ɻ (SVOEZ਺Λఆٛ͢Δ
20. x y
21. x y 0ͱఆٛ͢Δ 0
22. x y 0
23. x y 0 ߦ͖ઌʹͳ͍ ࠷খͷෛͰͳ͍੔਺
24. x y 0 ࠓճ͸̌Λআ͘ ࠷খͷෛͰͳ͍੔਺
25. x y 0 Grundy਺͸ 1ͱͳΔ 1
26. x y 0 1
27. x y 0 1 1
28. x y 0 1 1
29. x y 0 1 1 ߦ͖ઌʹͳ͍ ࠷খͷෛͰͳ͍੔਺
30. x y 0 1 1 0,1Λআ͘ ࠷খͷෛͰͳ͍੔਺
31. x y 0 1 1 2 Grundy਺͸ 2ͱͳΔ
32. x y 0 1
33. x y 0 1
34. x y 0 1 2
35. x y 0 1 2 2 1
36. x y 0 1 2 2 1 1,2Λআ͘ ࠷খͷෛͰͳ͍੔਺
37. x y 0 1 2 2 1 0
38. x y 0 1
39. x y 0 1
40. x y 0 1 2
41. x y 0 1 2 2 1
42. x y 0 1 2 2 1 0
43. x y 0
44. x y 0 1 1 2 2 0 2 0
45. x y 0 1 1 2 2 0 2 0 ߦ͖ઌʹͳ͍ ࠷খͷෛͰͳ͍੔਺
46. x y 0 1 1 2 2 0 2 0 1
47. G(x,y)=mex ({ G(u,v); (u,v)∈move(x,y)}) ࠷ऴہ໘ͷGrundy਺͸0
48. mex (A) ͸ ඇෛ੔਺ͷू߹ A ʹଘࡏ͠ͳ͍࠷খ ͷඇෛ੔਺ G((x,y))=mex ({ G(u,v); (u,v)∈move(x,y)})
49. mex ({0,1,2,3,5}) = G((x,y))=mex ({ G(u,v); (u,v)∈move(x,y)})
50. mex ({0,1,2,3,5}) = 4 G((x,y))=mex ({ G(u,v); (u,v)∈move(x,y)})
51. move ((x, y)) ͸ (x, y)
    ͔Β
    ̍खͰߦ͚Δશͯͷ৔߹ͷू߹ G((x,y))=mex ({ G(u,v); (u,v)∈move(x,y)})
52. move ( (2,1) )=((2,0),(1,1),(0,1),(1,0)) x y 0 1 2 1 2   ͔Β̍खͰಈ͚Δ
    શͯͷ৔߹ͷू߹
53. (x,y)ͷ̍खͰͷҠಈઌͷ Grundy਺ʹؚ·Εͳ͍ ࠷খͷෛͰͳ͍੔਺ G((x,y))=mex ({ G(u,v); (u,v)∈move(x,y)}) G(x,y)=G(0,0)=0
54. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 Grundy਺͕0ͷͱ͜Ζ͸ ޙखඞউϙδγϣϯ
55. 0 1 1 2 2 0 Grundy਺͕0ͳͷͰɺ ߦ͖ઌʹ0͸ͳ͍
56. 0 1 1 2 2 0 Grundy਺͕1ͳͷͰɺ ߦ͖ઌʹ0͕͋Δ 2 0 1
57. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 ઌख
    ͷλʔϯ
58. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 ޙख
    ͷλʔϯ
59. ઌख
    ͷλʔϯ 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1
60. ޙख
    ͷλʔϯ 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1
61. ޙखͷউͪ 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1
62. G((x, y)) = mod(x + y, 3) + 3(b x 3 c b y 3 c) mod(x+y,3)͸ɺ x+yΛ̏Ͱׂͬͨ༨Γͷ͜ͱɻ b x 3 c x 3 ͸ɺ ͷ੔਺෦෼ͷ͜ͱɻ ͸ɺഉଞత࿦ཧ࿨ͱಉ͡ɻ
    χϜ࿨
63. G((x, y)) = mod(x + y, 3) + 3(b x 3 c b y 3 c) mod(x+y,3)͸ɺ x+yΛ̏Ͱׂͬͨ༨Γͷ͜ͱɻ ͸ɺഉଞత࿦ཧ࿨ͱಉ͡ɻ
    χϜ࿨ b x 3 c x 3 ͸ɺ ͷ੔਺෦෼ͷ͜ͱɻ
64. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 19 20 18 2 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 20 18 19 3 3 4 5 0 1 2 9 10 11 6 7 8 15 16 17 12 13 14 21 22 23 4 4 5 3 1 2 0 10 11 9 7 8 6 16 17 15 13 14 12 22 23 21 5 5 3 4 2 0 1 11 9 10 8 6 7 17 15 16 14 12 13 23 21 22 6 6 7 8 9 10 11 0 1 2 3 4 5 18 19 20 21 22 23 12 13 14 7 7 8 6 10 11 9 1 2 0 4 5 3 19 20 18 22 23 21 13 14 12 8 8 6 7 11 9 10 2 0 1 5 3 4 20 18 19 23 21 22 14 12 13 9 9 10 11 6 7 8 3 4 5 0 1 2 21 22 23 18 19 20 15 16 17 10 10 11 9 7 8 6 4 5 3 1 2 0 22 23 21 19 20 18 16 17 15 11 11 9 10 8 6 7 5 3 4 2 0 1 23 21 22 20 18 19 17 15 16 12 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 13 13 14 12 16 17 15 19 20 18 22 23 21 1 2 0 4 5 3 7 8 6 14 14 12 13 17 15 16 20 18 19 23 21 22 2 0 1 5 3 4 8 6 7 15 15 16 17 12 13 14 21 22 23 18 19 20 3 4 5 0 1 2 9 10 11 16 16 17 15 13 14 12 22 23 21 19 20 18 4 5 3 1 2 0 10 11 9 17 17 15 16 14 12 13 23 21 22 20 18 19 5 3 4 2 0 1 11 9 10 18 18 19 20 21 22 23 12 13 14 15 16 17 6 7 8 9 10 11 0 1 2 19 19 20 18 22 23 21 13 14 12 16 17 15 7 8 6 10 11 9 1 2 0 20 20 18 19 23 21 22 14 12 13 17 15 16 8 6 7 11 9 10 2 0 1 G((x, y)) = mod(x + y, 3) + 3(b x 3 c b y 3 c)
65. G((x, y)) = mod(x + y, 3) + 3(b x 3 c b y 3 c) "" 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 1 1 2 0 4 5 3 7 8 6 2 2 0 1 5 3 4 8 6 7 3 3 4 5 0 1 2 9 10 11 4 4 5 3 1 2 0 10 11 9 5 5 3 4 2 0 1 11 9 10 6 6 7 8 9 10 11 0 1 2 7 7 8 6 10 11 9 1 2 0 8 8 6 7 11 9 10 2 0 1 9 9 10 11 6 7 8 3 4 5 10 10 11 9 7 8 6 4 5 3 11 11 9 10 8 6 7 5 3 4 12 13 14 15 16 17 18 19 20 "" 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 1 1 2 0 4 5 3 7 8 6 2 2 0 1 5 3 4 8 6 7 3 3 4 5 0 1 2 9 10 11 4 4 5 3 1 2 0 10 11 9 5 5 3 4 2 0 1 11 9 10 6 6 7 8 9 10 11 0 1 2 7 7 8 6 10 11 9 1 2 0 8 8 6 7 11 9 10 2 0 1 9 9 10 11 6 7 8 3 4 5 10 10 11 9 7 8 6 4 5 3 11 11 9 10 8 6 7 5 3 4 12 13 14 15 16 17 18 19 20
66. G((x, y)) = mod(x + y, 3) + 3(b x 3 c b y 3 c) "" 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 0 1 2 3 4 5 6 7 8 1 1 2 0 4 5 3 7 8 6 2 2 0 1 5 3 4 8 6 7 3 3 4 5 0 1 2 9 10 11 4 4 5 3 1 2 0 10 11 9 5 5 3 4 2 0 1 11 9 10 6 6 7 8 9 10 11 0 1 2 7 7 8 6 10 11 9 1 2 0 8 8 6 7 11 9 10 2 0 1 9 9 10 11 6 7 8 3 4 5 10 10 11 9 7 8 6 4 5 3 11 11 9 10 8 6 7 5 3 4 12 13 14 15 16 "" 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 0 1 2 3 4 5 6 7 8 1 1 2 0 4 5 3 7 8 6 2 2 0 1 5 3 4 8 6 7 3 3 4 5 0 1 2 9 10 11 4 4 5 3 1 2 0 10 11 9 5 5 3 4 2 0 1 11 9 10 6 6 7 8 9 10 11 0 1 2 7 7 8 6 10 11 9 1 2 0 8 8 6 7 11 9 10 2 0 1 9 9 10 11 6 7 8 3 4 5 10 10 11 9 7 8 6 4 5 3 11 11 9 10 8 6 7 5 3 4 12 13 14 15 16 "" 0 1 2 3 4 5 6 7 8 9 10 11 0 0 1 2 0 1 2 0 1 2 0 1 2 1 1 2 0 1 2 0 1 2 0 1 2 0 2 2 0 1 2 0 1 2 0 1 2 0 1 3 0 1 2 0 1 2 0 1 2 0 1 2 4 1 2 0 1 2 0 1 2 0 1 2 0 5 2 0 1 2 0 1 2 0 1 2 0 1 6 0 1 2 0 1 2 0 1 2 0 1 2 7 1 2 0 1 2 0 1 2 0 1 2 0 8 2 0 1 2 0 1 2 0 1 2 0 1 9 0 1 2 0 1 2 0 1 2 0 1 2 10 1 2 0 1 2 0 1 2 0 1 2 0 11 2 0 1 2 0 1 2 0 1 2 0 1 0 1 2 0 1 2 3 +3(2 3)
67. G((x, y)) = mod(x + y, 3) + 3(b x 3 c b y 3 c) "" 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 1 1 2 0 4 5 3 7 8 6 2 2 0 1 5 3 4 8 6 7 3 3 4 5 0 1 2 9 10 11 4 4 5 3 1 2 0 10 11 9 5 5 3 4 2 0 1 11 9 10 6 6 7 8 9 10 11 0 1 2 7 7 8 6 10 11 9 1 2 0 8 8 6 7 11 9 10 2 0 1 9 9 10 11 6 7 8 3 4 5 10 10 11 9 7 8 6 4 5 3 11 11 9 10 8 6 7 5 3 4 12 13 14 15 16 17 18 19 20 "" 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 1 1 2 0 4 5 3 7 8 6 2 2 0 1 5 3 4 8 6 7 3 3 4 5 0 1 2 9 10 11 4 4 5 3 1 2 0 10 11 9 5 5 3 4 2 0 1 11 9 10 6 6 7 8 9 10 11 0 1 2 7 7 8 6 10 11 9 1 2 0 8 8 6 7 11 9 10 2 0 1 9 9 10 11 6 7 8 3 4 5 10 10 11 9 7 8 6 4 5 3 11 11 9 10 8 6 7 5 3 4 12 13 14 15 16 17 18 19 20
68. දͷGrundy਺͔Β ΛҾ͘ͱ ... + y, 3) + 3(b x 3 c b y 3 c)
69. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 2 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 3 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 4 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 5 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 6 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 7 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 8 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 9 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 10 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 11 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 12 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 13 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 14 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 15 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 16 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 17 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 18 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 19 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 20 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1
72. Out[45]= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 2 3 0 5 6 7 4 9 10 11 8 13 14 15 12 17 18 19 16 21 2 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 18 19 16 17 22 3 3 0 1 2 7 4 5 6 11 8 9 10 15 12 13 14 19 16 17 18 23 4 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 20 21 22 23 16 5 5 6 7 4 1 2 3 0 13 14 15 12 9 10 11 8 21 22 23 20 17 6 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 22 23 20 21 18 7 7 4 5 6 3 0 1 2 15 12 13 14 11 8 9 10 23 20 21 22 19 8 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 24 25 26 27 28 9 9 10 11 8 13 14 15 12 1 2 3 0 5 6 7 4 25 26 27 24 29 10 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 26 27 24 25 30 11 11 8 9 10 15 12 13 14 3 0 1 2 7 4 5 6 27 24 25 26 31 12 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 28 29 30 31 24 13 13 14 15 12 9 10 11 8 5 6 7 4 1 2 3 0 29 30 31 28 25 14 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 30 31 28 29 26 15 15 12 13 14 11 8 9 10 7 4 5 6 3 0 1 2 31 28 29 30 27 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 17 17 18 19 16 21 22 23 20 25 26 27 24 29 30 31 28 1 2 3 0 5 18 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 2 3 0 1 6 19 19 16 17 18 23 20 21 22 27 24 25 26 31 28 29 30 3 0 1 2 7 20 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 4 5 6 7 0 G((x, y)) = mod(x + y, 4) + 4(b x 4 c b y 4 c)
73. දͷGrundy਺͔Β ΛҾ͘ͱ ... + y, 4) + 4(b x 4 c b y 4 c)
74. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 4 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 5 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 6 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 7 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 8 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 9 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 10 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 11 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 12 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 13 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 14 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 15 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 16 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 17 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 18 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 19 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 20 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0
75. G((x, y)) = mod(x + y, p) + p(b x p c b y p c) p-2 p-2
76. ੴऔΓήʔϜͷ৔߹ɺ ยํ͔ΒͳΒ͍ͭ͘Ͱ΋ͱͬͯ΋Α͍ɻ ྆ํ͔ΒͳΒɺ࠷େp-1·ͰͳΒͱͬͯ΋Α͍ɻ x y x-m y-n ( m+n ≤ p-1 )
77. ࣍ʹɺGrundy਺ͷද͔Βɺ ΛҾ͍ͨͱ͖ʹɺGrundy਺͕पظੑΛ࣋ͭ ৔߹Λߟ͑ͨɻ p(b x p c b y p c)
79. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 2 2 3 4 5 0 1 8 9 10 11 6 7 14 15 16 17 12 13 20 21 22 3 3 2 5 4 1 0 9 8 11 10 7 6 15 14 17 16 13 12 21 20 23 4 4 5 0 1 2 3 10 11 6 7 8 9 16 17 12 13 14 15 22 23 18 5 5 4 1 0 3 2 11 10 7 6 9 8 17 16 13 12 15 14 23 22 19 6 6 7 8 9 10 11 0 1 2 3 4 5 18 19 20 21 22 23 12 13 14 7 7 6 9 8 11 10 1 0 3 2 5 4 19 18 21 20 23 22 13 12 15 8 8 9 10 11 6 7 2 3 4 5 0 1 20 21 22 23 18 19 14 15 16 9 9 8 11 10 7 6 3 2 5 4 1 0 21 20 23 22 19 18 15 14 17 10 10 11 6 7 8 9 4 5 0 1 2 3 22 23 18 19 20 21 16 17 12 11 11 10 7 6 9 8 5 4 1 0 3 2 23 22 19 18 21 20 17 16 13 12 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 13 13 12 15 14 17 16 19 18 21 20 23 22 1 0 3 2 5 4 7 6 9 14 14 15 16 17 12 13 20 21 22 23 18 19 2 3 4 5 0 1 8 9 10 15 15 14 17 16 13 12 21 20 23 22 19 18 3 2 5 4 1 0 9 8 11 16 16 17 12 13 14 15 22 23 18 19 20 21 4 5 0 1 2 3 10 11 6 17 17 16 13 12 15 14 23 22 19 18 21 20 5 4 1 0 3 2 11 10 7 18 18 19 20 21 22 23 12 13 14 15 16 17 6 7 8 9 10 11 0 1 2 19 19 18 21 20 23 22 13 12 15 14 17 16 7 6 9 8 11 10 1 0 3 20 20 21 22 23 18 19 14 15 16 17 12 13 8 9 10 11 6 7 2 3 4
80. දͷGrundy਺͔Β ΛҾ͘ͱ ... 6(b x 6 c b y 6 c)
81. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 1 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 3 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 5 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 6 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 7 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 8 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 9 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 10 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 11 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 12 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 13 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 14 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 15 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 16 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 17 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 18 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 19 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 2 5 4 1 0 3 20 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4
82. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 1 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 2 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 3 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 4 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 5 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 6 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 7 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 8 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 9 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 10 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 11 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 12 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 13 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 14 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 15 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 G((x, y)) = mod(x + y, 6) + 6(b x 6 c b y 6 c)
83. 0 1 2 3 4 5 1 2 3 4 5 0 2 3 4 5 0 1 3 4 5 0 1 2 4 5 0 1 2 3 5 0 1 2 3 4 ̍पظΛऔΓग़͢ͱ͜ͷΑ͏ʹͳ͍ͬͯΔɻ G((x, y)) = mod(x + y, 6) + 6(b x 6 c b y 6 c)
84. 0 1 2 3 4 5 1 2 3 4 5 0 2 3 4 5 0 1 3 4 5 0 1 2 4 5 0 1 2 3 5 0 1 2 3 4
85. 0 1 2 3 4 5 1 0 3 2 5 4 2 3 4 5 0 1 3 2 5 4 1 0 4 5 0 1 2 3 5 4 1 0 3 2
86. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1
87. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 2 0 1 4 5 3 0 1 2 5 3 4 1 2 0
88. 0 1 2 3 4 5 1 0 3 2 5 4 2 3 4 5 0 1 3 2 5 4 1 0 4 5 0 1 2 3 5 4 1 0 3 2 0 1 2 3 4 5 1 2 3 4 5 0 2 3 4 5 0 1 3 4 5 0 1 2 4 5 0 1 2 3 5 0 1 2 3 4 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 2 0 1 4 5 3 0 1 2 5 3 4 1 2 0
90. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 2 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 18 19 16 17 22 3 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 19 18 17 16 23 4 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 20 21 22 23 16 5 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 21 20 23 22 17 6 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 22 23 20 21 18 7 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 23 22 21 20 19 8 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 24 25 26 27 28 9 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 25 24 27 26 29 10 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 26 27 24 25 30 11 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 27 26 25 24 31 12 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 28 29 30 31 24 13 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 29 28 31 30 25 14 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 30 31 28 29 26 15 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 31 30 29 28 27 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 17 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 1 0 3 2 5 18 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 2 3 0 1 6 19 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 3 2 1 0 7 20 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 4 5 6 7 0 G((x, y)) = (x y)
93. ۮ਺ݸ
94. ͜ͷΑ͏ʹࣼΊҰྻʹۮ਺ݸฒͿ΋ͷͷ (SVOEZ਺͸ҎԼͷࣜͰද͢͜ͱ͕Ͱ͖Δ G((x, y)) = x y
96. -1 ݸ Q͸ࣗવ਺ 2p+1
97. p͸ࣗવ਺ (x, y) (x 2p, y 2p) -1 ݸ 2p+1
98. 2p  mod(x, 2p+1) 2p  mod(y, 2p+1) x + y; even ͳΒɺ G((x, y)) = (x y) + 1 G((x, y)) = (x y) 1 ͳΒɺ x + y; odd ͔ͭ Λຬͨ͠ɺ G((x, y)) = (x y) ͦΕҎ֎ͷ৔߹͸ɺ
99. ੒Δ
103. 0 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 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 2 0 3 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 3 1 2 0 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 4 0 3 1 2 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5 1 2 0 3 5 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 6 0 3 1 2 4 7 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 7 1 2 0 3 5 6 4 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 8 0 3 1 2 4 7 5 6 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 9 1 2 0 3 5 6 4 7 9 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 10 0 3 1 2 4 7 5 6 8 11 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 11 1 2 0 3 5 6 4 7 9 10 8 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 12 0 3 1 2 4 7 5 6 8 11 9 10 12 13 12 13 12 13 12 13 12 13 12 13 12 13 13 1 2 0 3 5 6 4 7 9 10 8 11 13 14 15 14 15 14 15 14 15 14 15 14 15 14 14 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 12 13 12 13 12 13 12 13 12 13 12 15 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 14 15 14 15 14 15 14 15 14 15 16 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 17 16 17 16 17 16 17 16 17 17 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 19 18 19 18 19 18 19 18 18 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 16 17 16 17 16 17 16 19 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 18 19 18 19 18 19 20 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 21 20 21 20 21 21 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 23 22 23 22 22 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 23 21 20 21 20 23 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 20 23 22 23 24 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 23 21 22 24 25 25 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 20 23 25 26
104. Grundy਺ͷެࣜ
105. Corner the “Ryuma”(ཾഅ) problem with pass ਓͰճ͔͠ύε͕࢖͑ͳ͍
     ͕ۨ
     0,0) ʹߦ͘ͱύεͰ͖ͳ͍
106. x y
107. ϓϨΠϠʔ̍
     ͷλʔϯ Corner the “Ryuma”(ཾഅ) problem with pass
108. ϓϨΠϠʔ̍
     ͷλʔϯ Corner the “Ryuma”(ཾഅ) problem with pass
109. ϓϨΠϠʔ2 ͷλʔϯ Corner the “Ryuma”(ཾഅ) problem with pass
110. ϓϨΠϠʔ2 ͕ύεΛ࢖͏ Corner the “Ryuma”(ཾഅ) problem with pass
111. ϓϨΠϠʔ1 ͷλʔϯ Corner the “Ryuma”(ཾഅ) problem with pass
112. ϓϨΠϠʔ2 ͷλʔϯ Corner the “Ryuma”(ཾഅ) problem with pass
113. ϓϨΠϠʔ2 ͷউͪ Corner the “Ryuma”(ཾഅ) problem with pass
114. 0 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 0 0 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 1 0 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 1 0 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 3 0 3 1 4 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 4 1 2 0 3 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 0 3 1 2 4 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 1 2 0 3 5 6 7 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 7 0 3 1 2 4 7 5 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 8 1 2 0 3 5 6 4 7 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 0 3 1 2 4 7 5 6 8 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 1 2 0 3 5 6 4 7 9 10 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 11 0 3 1 2 4 7 5 6 8 11 9 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 12 1 2 0 3 5 6 4 7 9 10 8 11 13 12 13 12 13 12 13 12 13 12 13 12 13 12 13 0 3 1 2 4 7 5 6 8 11 9 10 12 15 14 15 14 15 14 15 14 15 14 15 14 15 14 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 13 12 13 12 13 12 13 12 13 12 13 15 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 15 14 15 14 15 14 15 14 15 14 16 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 16 17 16 17 16 17 16 17 16 17 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 18 19 18 19 18 19 18 19 18 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 17 16 17 16 17 16 17 19 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 19 18 19 18 19 18 20 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 20 21 20 21 20 21 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 23 22 23 22 23 22 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 20 21 20 21 23 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 23 21 22 23 22 24 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 20 23 25 24 25 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 23 21 22 24 27
115. Grundy਺ͷެࣜ y≤x ͷͱ͖ yʼxͷͱ͖
116. Corner the “Ryuoh”(ཾԦ) problem with pass 0 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 0 0 2 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15 18 17 20 19 22 21 24 23 26 25 28 27 30 29 1 2 1 3 0 6 4 8 7 5 9 12 10 14 13 11 15 18 16 20 19 17 21 24 22 26 25 23 27 30 28 32 2 1 3 0 2 4 5 6 9 8 7 10 11 12 15 14 13 16 17 18 21 20 19 22 23 24 27 26 25 28 29 30 3 4 0 2 1 5 3 7 11 6 12 8 9 10 17 16 18 14 13 15 23 22 24 20 19 21 29 28 30 26 25 27 4 3 6 4 5 0 1 2 10 12 8 7 13 15 9 18 11 17 19 14 16 24 20 23 21 22 28 30 26 25 27 31 5 6 4 5 3 1 2 0 12 9 11 13 8 7 10 15 17 19 14 16 18 21 23 25 20 27 24 22 29 31 26 28 6 5 8 6 7 2 0 1 3 4 13 14 15 9 11 12 10 20 21 19 17 16 18 26 27 25 22 29 23 24 32 33 7 8 7 9 11 10 12 3 0 1 2 4 5 6 16 17 14 13 15 22 24 18 25 19 28 20 21 27 31 29 23 26 8 7 5 8 6 12 9 4 1 2 0 11 3 13 19 10 20 22 23 21 14 15 16 17 18 29 31 24 32 34 33 25 9 10 9 7 12 8 11 13 2 0 1 3 4 5 6 20 19 21 24 25 15 23 14 18 16 17 30 32 22 33 31 34 10 9 12 10 8 7 13 14 4 11 3 0 1 2 18 5 6 23 20 26 22 25 15 16 17 19 32 21 24 35 34 36 11 12 10 11 9 13 8 15 5 3 4 1 2 0 7 6 21 24 22 23 25 14 17 27 26 16 18 19 20 32 36 35 12 11 14 12 10 15 7 9 6 13 5 2 0 1 3 4 8 25 26 24 27 28 29 30 31 18 17 16 21 19 20 22 13 14 13 15 17 9 10 11 16 19 6 18 7 3 0 1 2 4 5 8 12 26 27 28 25 30 20 31 33 22 21 23 14 13 11 14 16 18 15 12 17 10 20 5 6 4 1 2 0 7 3 9 26 8 28 29 30 31 19 33 34 21 24 37 15 16 15 13 18 11 17 10 14 20 19 6 21 8 2 0 1 3 4 5 7 9 12 31 32 28 23 34 35 36 22 38 16 15 18 16 14 17 19 20 13 22 21 23 24 25 4 7 3 0 1 2 6 5 8 9 10 11 12 35 36 37 38 39 17 18 16 17 13 19 14 21 15 23 24 20 22 26 5 3 4 1 2 0 8 7 6 10 11 9 33 36 12 38 35 40 18 17 20 18 15 14 16 19 22 21 25 26 23 24 8 9 5 2 0 1 3 4 10 7 6 12 11 13 37 39 40 41 19 20 19 21 23 16 18 17 24 14 15 22 25 27 12 26 7 6 8 3 0 1 2 4 5 10 9 37 11 13 41 42 20 19 17 20 22 24 21 16 18 15 23 25 14 28 26 8 9 5 7 4 1 2 0 6 3 13 34 10 38 12 11 43 21 22 21 19 24 20 23 18 25 16 14 15 17 29 27 28 12 8 6 10 2 0 1 3 4 5 7 9 13 11 37 44 22 21 24 22 20 23 25 26 19 17 18 16 27 30 28 29 31 9 10 7 4 6 3 0 1 2 8 5 14 15 12 11 23 24 22 23 19 21 20 27 28 18 16 17 26 31 25 30 32 10 11 6 5 3 4 1 2 0 13 7 8 9 14 15 24 23 26 24 21 22 27 25 20 29 17 19 16 18 30 31 28 11 9 12 10 13 5 2 0 1 3 4 6 7 8 45 25 26 25 27 29 28 24 22 21 31 30 32 18 17 20 19 23 12 33 11 9 34 7 8 13 3 0 1 2 4 5 6 26 25 23 26 28 30 22 29 27 24 32 21 19 16 31 33 34 35 36 13 37 10 9 5 7 4 1 2 0 6 3 12 27 28 27 25 30 26 29 23 31 32 22 24 20 21 33 34 35 36 12 37 11 38 13 14 8 6 2 0 1 3 4 5
117. Grundy਺ͷۮحੑ
118. ཾԦ(੒ඈं)ͷGrundy਺
119. 0 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 0 0 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 1 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 2 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 3 3 4 5 0 1 2 9 10 11 6 7 8 15 16 17 12 13 14 21 22 23 18 19 20 27 28 4 4 5 3 1 2 0 10 11 9 7 8 6 16 17 15 13 14 12 22 23 21 19 20 18 28 29 5 5 3 4 2 0 1 11 9 10 8 6 7 17 15 16 14 12 13 23 21 22 20 18 19 29 27 6 6 7 8 9 10 11 0 1 2 3 4 5 18 19 20 21 22 23 12 13 14 15 16 17 30 31 7 7 8 6 10 11 9 1 2 0 4 5 3 19 20 18 22 23 21 13 14 12 16 17 15 31 32 8 8 6 7 11 9 10 2 0 1 5 3 4 20 18 19 23 21 22 14 12 13 17 15 16 32 30 9 9 10 11 6 7 8 3 4 5 0 1 2 21 22 23 18 19 20 15 16 17 12 13 14 33 34 10 10 11 9 7 8 6 4 5 3 1 2 0 22 23 21 19 20 18 16 17 15 13 14 12 34 35 11 11 9 10 8 6 7 5 3 4 2 0 1 23 21 22 20 18 19 17 15 16 14 12 13 35 33 12 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 9 10 11 36 37 13 13 14 12 16 17 15 19 20 18 22 23 21 1 2 0 4 5 3 7 8 6 10 11 9 37 38 14 14 12 13 17 15 16 20 18 19 23 21 22 2 0 1 5 3 4 8 6 7 11 9 10 38 36 15 15 16 17 12 13 14 21 22 23 18 19 20 3 4 5 0 1 2 9 10 11 6 7 8 39 40 16 16 17 15 13 14 12 22 23 21 19 20 18 4 5 3 1 2 0 10 11 9 7 8 6 40 41 17 17 15 16 14 12 13 23 21 22 20 18 19 5 3 4 2 0 1 11 9 10 8 6 7 41 39 18 18 19 20 21 22 23 12 13 14 15 16 17 6 7 8 9 10 11 0 1 2 3 4 5 42 43 19 19 20 18 22 23 21 13 14 12 16 17 15 7 8 6 10 11 9 1 2 0 4 5 3 43 44 20 20 18 19 23 21 22 14 12 13 17 15 16 8 6 7 11 9 10 2 0 1 5 3 4 44 42 21 21 22 23 18 19 20 15 16 17 12 13 14 9 10 11 6 7 8 3 4 5 0 1 2 45 46 22 22 23 21 19 20 18 16 17 15 13 14 12 10 11 9 7 8 6 4 5 3 1 2 0 46 47 23 23 21 22 20 18 19 17 15 16 14 12 13 11 9 10 8 6 7 5 3 4 2 0 1 47 45 24 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 0 1 25 25 26 24 28 29 27 31 32 30 34 35 33 37 38 36 40 41 39 43 44 42 46 47 45 1 2
120. ཾഅ(੒֯ߦ)ͷGrundy਺
121. 0 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 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 2 0 3 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 3 1 2 0 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 4 0 3 1 2 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5 1 2 0 3 5 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 6 0 3 1 2 4 7 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 7 1 2 0 3 5 6 4 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 8 0 3 1 2 4 7 5 6 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 9 1 2 0 3 5 6 4 7 9 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 10 0 3 1 2 4 7 5 6 8 11 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 11 1 2 0 3 5 6 4 7 9 10 8 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 12 0 3 1 2 4 7 5 6 8 11 9 10 12 13 12 13 12 13 12 13 12 13 12 13 12 13 13 1 2 0 3 5 6 4 7 9 10 8 11 13 14 15 14 15 14 15 14 15 14 15 14 15 14 14 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 12 13 12 13 12 13 12 13 12 13 12 15 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 14 15 14 15 14 15 14 15 14 15 16 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 17 16 17 16 17 16 17 16 17 17 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 19 18 19 18 19 18 19 18 18 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 16 17 16 17 16 17 16 19 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 18 19 18 19 18 19 20 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 21 20 21 20 21 21 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 23 22 23 22 22 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 23 21 20 21 20 23 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 20 23 22 23 24 0 3 1 2 4 7 5 6 8 11 9 10 12 15 13 14 16 19 17 18 20 23 21 22 24 25 25 1 2 0 3 5 6 4 7 9 10 8 11 13 14 12 15 17 18 16 19 21 22 20 23 25 26
122. ۨʹΑͬͯGrundy਺ͷ ۮحੑ͕ҧ͏
124. ̎ͭͷήʔϜͷ࿨
125. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 x y x’ y’
126. 2 3 1 5 x y x’ y’
127. Grundy਺͸ɺ5⊕5=0ɹͳͷͰɺ ޙखඞউʹͳΔɻ 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 0 1 2 4 5 3 1 2 0 5 3 4 2 0 1 x y x’ y’
128. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8 Grundy਺͸ɺ5⊕4=1ɹͳͷͰɺ ޙखඞউͰ͸ͳ͍ɻ x’ y’ x y
129. ͭͷήʔϜͷ࿨
130. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8 Grundy਺͸ɺ6⊕4⊕2=0ɹͳͷͰɺ ޙखඞউʹͳΔɻ 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8
131. 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8 Grundy਺͸ɺ6⊕4⊕0=2ɹͳͷͰɺ ޙखඞউͰ͸ͳ͍ɻ 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8 0 1 2 3 4 5 1 2 0 4 5 3 2 0 1 5 3 4 3 4 5 6 2 0 4 5 3 2 7 6 5 3 4 0 6 8
132. ෳ਺ͷ൫໘ʹͳͬͨͱ͖ʹɺ
     ͦΕͧΕͷ൫໘ͷ(SVOEZ਺ͷχϜ࿨
     Λௐ΂Δ͜ͱͰޙखඞউ͔Ͳ͏͔͕෼͔Δ