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組合せゲーム理論への招待
- 1. 2018 4 14 ( ) 2018 4 14 1 / 27
- 2. ABUKU TOMOAKI 2 5 ( ) 2018 4 14 2 / 27
- 3. (combinatorial game) ( ) 2018 4 14 3 / 27
- 4. Impartial game Nim ( ) 2018 4 14 4 / 27
- 5. Nim Nim ( ) 2018 4 14 5 / 27
- 6. (Nim ) N0 m1, . . . , mn ∈ N0 2 Nim m1 ⊕ · · · ⊕ mn 3 ⊕ 5 = (011)2 ⊕ (101)2 = (110)2 = 6 ( ) 2018 4 14 6 / 27
- 7. Nim Nim (Bouton 1901) (m1, . . . , mn) Nim m1 ⊕ · · · ⊕ mn ̸= 0 ⇐⇒ (m1, . . . , mn) m1 ⊕ · · · ⊕ mn = 0 ⇐⇒ (m1, . . . , mn) ( ) 2018 4 14 7 / 27
- 8. Nim Nim (7, 8, 10) Nim 7 ⊕ 8 ⊕ 10 = (0111)2 ⊕ (1000)2 ⊕ (1010)2 = (0101)2 = 5 ̸= 0 7 7 ⊕ 5 = 2 ( ) 2018 4 14 8 / 27
- 9. Grundy Grundy Grundy number, Grundy value) . Grundy ( minimum excluded number) N0 T N0 T mexT mexT = min(N0 T) mex{0, 1, 2, 4, 5, 7} = 3, mex{1, 2, 4, 5, 7} = 0, mex{∅} = 0 ( ) 2018 4 14 9 / 27
- 10. Grundy , G Grundy G(G) (Grundy ) G N0 E G {G′ 1, . . . , G′ n} G ∈ G G : G → N0 G(G) = 0 (G = E) mex{G(G′ 1), . . . , G(G′ n)} (G ̸= E) G(G) G Grundy ( ) 2018 4 14 10 / 27
- 11. Grundy (Spruge 1936,Grundy 1939) G G(G) ̸= 0⇐⇒ G G(G) = 0⇐⇒ G Grundy ( ) 2018 4 14 11 / 27
- 12. Grundy Nim Grundy M m Nim G(M) = m Nim Grundy Nim Mk m k G(Mk) = n (mod k + 1) M = 31, k = 3 G(313) = 3 (mod 3 + 1) ( ) 2018 4 14 12 / 27
- 13. Grundy Grundy G H (G, H) G H Grundy G(G, H) = G(G) ⊕ G(H). (n Nim Grundy ) (m1, . . . , mn) Nim G(m1, . . . , mn) = m1 ⊕ . . . ⊕ mn (∵ G(mi ) = mi ) ( ) 2018 4 14 13 / 27
- 14. Grundy 0 1 • 3 • 5 6 7 • 9 . . . • • • 3 4 5 6 7 8 9 . . . ( ) 2018 4 14 14 / 27
- 15. Grundy 0 ≤ m1 ≤ · · · ≤ mr Grundy G(m1, . . . , mr ) = ⎧ ⎪⎪⎨ ⎪⎪⎩ (m2 − m1 − 1)⊕ · · · ⊕ (mr − mr−1 − 1) (r ) m1 ⊕ (m3 − m2 − 1)⊕ · · · ⊕ (mr − mr−1 − 1) (r ) (2, 4, 8) G(2, 4, 8) = 2 ⊕ (8 − 4 − 1) = 1 ( ) 2018 4 14 15 / 27
- 16. Grundy Nim Nim Nim (m, n, r) −→ (m + k, n, r) −→ (m, n, r) Nim Nim . ( ) 2018 4 14 16 / 27
- 17. Grundy A B ( ) 2018 4 14 17 / 27
- 18. Grundy Grundy n i mi (m1, . . . , mn) G(m1, . . . , mn) = m1 ⊕ . . . ⊕ mn (5, 5, 1, 0, 1, 5, 1, 1, 3) G(5, 5, 1, 0, 1, 5, 1, 1, 3) = 5 ⊕ 5 ⊕ 1 ⊕ 0 ⊕ 1 ⊕ 5 ⊕ 1 ⊕ 1 ⊕ 3=6 ( ) 2018 4 14 18 / 27
- 19. 3 3 Nim 313 (2, 4, 8) (5, 5, 1, 0, 1, 5, 1, 1, 3) Grundy G(313) = 3,G(2, 4, 8) = 1,G(5, 5, 1, 0, 1, 5, 1, 1, 3) = 6 Grundy G(313) ⊕ G(2, 4, 8) ⊕ G(5, 5, 1, 0, 1, 5, 1, 1, 3) = 3 ⊕ 1 ⊕ 6 = 4 6 ⊕ 4 = 2 Grundy 2 5 → 1 ( ) 2018 4 14 19 / 27
- 20. Mis`ere game Grundy Grundy ( ) 2018 4 14 20 / 27
- 21. Loopy game Loopy game Stopper Loopy game ( ) 2018 4 14 21 / 27
- 22. Transfinite game Grundy ( ) 2018 4 14 22 / 27
- 23. Partizan game ( (Alice,Bob ) ( ) 2018 4 14 23 / 27
- 24. 2 3 10 10 ( 1 1 1 3 1 3 ( ) 2018 4 14 24 / 27
- 25. 4 ( ) 2018 4 14 25 / 27
- 26. ( ) 2018 4 14 26 / 27
- 27. , , , 2014. , , , 2011. J. H. Conway, On Numbers And Games (second edition) A. K. Peters, 2001 E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays Vol 1-4, A. K. Peters, 2001-2004 A. N. Siegel, Combinatorial Game theory, American Mathematical Society, 2013 ( ) 2018 4 14 27 / 27