[4] 1.1. x,y 2 x = n i=0 x i2 i,y = n i=0 y i2 i (x i, y i {0, 1}) x y x y = w i 2 i, (1.1) w i = x i + y i (mod 2) (a) (N -Position)
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- とき うるしはた
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1 () ,3, {3, 3, 2} {x, y, z} 3 {x, y, z} 3 x y z [1] x y z x y z x y z [1] Z 0 x i {0, 1} x i {x, y, z} (x, y, z Z 0 ) 1.1 {3, 3, 2} 1.2 {4, 4, 9} 1.3 {4, 3, 13} 1.4 {4, 3, 13} 1.1. {3, 3, 2} 1.2. {4, 4, 9} 1.3. {4, 3, 13} 1.4. {4, 3, 13} 1
2 [4] 1.1. x,y 2 x = n i=0 x i2 i,y = n i=0 y i2 i (x i, y i {0, 1}) x y x y = w i 2 i, (1.1) w i = x i + y i (mod 2) (a) (N -Position) i=0 (b) (P-Position) 1.3. G H 2 G + H G H G p move(p) 1.1. move (i) mex S (ii) G G(p) = mex{g(h) : h move(p)} mex 1.1. x Z 0 k = 1, 2,..., n y k Z 0 (i) (ii) (i) x = mex({y k, k = 1, 2,..., n}). (ii) k x y k u < x u Z 0 u = y k k. 1.5 (mex ) 1.2. S {1, 2, 3,..., m 1} S mex(s) m. 1.5 mex G H G G G H G H (i) G g g P-position G G (g) = 0 (ii) G + H {g, h} G G (g) G H (h) [4] 2
3 1.6. f 2 (i) t Z 0 f(t) Z 0 (ii) f u v u, v Z 0 f(u) f(v) 1.7. y, z y f(z) z i t(i) = min(y, f(i)) + 1 CB(f, y, z) y, z 1.2. CB(f, y, z) CB(f, 3, 13) f(t) = t CB(f, 4, 9) f(t) = t CB(f, 2, 9) f(t) = t CB(f, 8, 31) f(0) = f(1) = 0 t > 1 f(t) = 2 log2t f f CB(f, y, z) {y, z} 1.3. f(t) = t 2 CB(f, 2, 5) {2, 5} CB(f, 1, 3) {1, 3} CB(f, 0, 5) {0, 5} CB(f, 1, 5) {1, 5} f CB(f, y, z) {y, z} move f ({y, z}) 1.4 move (z ) 2 min(y, f(w)) 1.8. y, z Z 0 move f ({y, z}) = {{v, z} : v < y} {{min(y, f(w)), w} : w < z}, (v, w Z 0 ) f(t) = t {y, z} = {2, 5} 1 y {1, 5}, {0, 5} {1, 5}, {0, 5} move f ({2, 5}) {y, z} = {2, 5} 2 z y min(2, 4/2 ) = min(2, 2) = 2 {2, 4} move f ({2, 5}) {y, z} = {2, 5} 2 z y min(2, 3/2 ) = min(2, 1) = 1 {1, 3} move f ({2, 5}) {y, z} = {1, 3} {0, 5} {0, 5} / move f ({1, 3}) 3
4 CB(f, y, z) G({y, z}) = y z f 1.2, 1.3 CB(f, y, z) 1 1 x {x, y, z} P-position x y z = 0 CB(f, y, z) G({y, z}) = y z 2 G({y, z}) = y z 2.1 G({y, z}) = y z h h (a) z, z Z 0 i z 2 i = z 2 i (2.1) z) 2 i 1 = ) z (2.2) 2i (a) (b) (b) k = 0, 1,..., n z k, y k {0, 1} z k 2 k ) = y k 2 k k = 0,..., i 1 z k {0, 1} k = 0,..., i 2 y k {0, 1} i i 1 z k 2 k + z k2 k ) = i 2 y k 2 k + y k2 k (a) 2.1. k z) = z 2k z) (2.2) u Z 0 i (2.1) (2.3) (2.4) z 2k z 2k = u < u + 1 (2.3) 2i 1 2i 1 z 2k u2i i 1 1 < (u + 1)2 i 1 z 2k z 2k(u2 i i 1 1) + 2k 1 < 2k(u + 1)2 i 1 z. (2.4) z 2 i k(u + 1) 1 z < k(u + 1) 2i 2 i z z < k(u + 1) 2i 2 i. (2.1) 4
5 2.2. 0) = 1) = 0 z 2 z Z 0 z) = 2 log2z 1 z) (2.2) i 2 i 1 < 2 log2z 1 2 i 1. (2.5) 2 log 2z 1 z = 2 n+m, z = 2 n +m n, n Z 0, 0 m, m < 1 (2.5) (2.6) (2.7), (2.8) 2 n i < 2 n i. (2.6) n i 0 (2.7) z 2 i = 2n+m i < 2 n i 2 n +m i = z 2 i. (2.1) h i. n z k 2 k ) = n + 1 n. (2.8) z k 2 k ) z k 2 k ) y k 2 k (2.9) y k 2 k. u k 2 k. u k {0, 1}, (2.9), u k 2 k u k 2 k y k 2 k. y k 2 k. (2.10) k = 0, 1, 2,..., i 1 z k = 0 (2.10), 2.1, 2.1 k = 0,..., i 2 y k = i 1 z k 2 k ) = z k 2 k + z k2 k ) i 2 u k 2 k + y k2 k u k 2 k y k 2 k k = 0, 1,..., n p k, q k Z 0 (i), (ii) (i) i p k 2 k ) < q k 2 k (2.11) 5
6 (ii). (i) n p k 2 k ) = 0, 1, 2,..., i 2 r k, p k 2 k ) < p k 2 k ) p k 2 k ) q k 2 k. q k 2 k q k 2 k r k 2 k r k {0, 1} 2.1, k = p k 2 k + i 1 q k 2 k > (2.12) (2.13) j Z 0 i 1 j n, k = j + 1, j + 2,..., n (2.12) (2.13), (ii) (i) n+1 q k 2k = 2.4 (i) 0 2 k ) = i 2 r k 2 k + r k2 k r k 2 k + i 2 r k 2k., (2.11) q n > r n. (2.12) q k = r k q j > r j (2.13) = q k 2 k > r k 2 k = p k 2 k ). p k 2 k ) < p k 2 k ) < r k 2 k + i 2 r k 2 k q k 2 k + 2 i 1 (2.14) q k 2 k + 2 i 1. q k 2 k + 2 i 1 p n+1 = 0., (2.14), n+1 p k 2 k ) n+1 p k 2 k ) < n+1 q k 2k = p k 2 k ) < n+1 q k 2k q k 2 k + 2 i x y z 0 y z) (2.15) 1 (1) u < x u Z 0 u y z = 0. (2) v < y v Z 0 x v z = 0. (3) w < z, y w) w Z 0 x y w = 0. (4) v < y, w < z, v = w ) v, w Z 0 x v w = 0. 6
7 . x = x k 2 k, y = s y k 2 k, z = z k 2 k n = 0 n 1 x n s 1 + y n s 1 + z n s 1 0 (mod 2) (2.16) i = n, n 1,..., n s x i + y i + z i = 0 (mod 2) Case (i) x n s 1 = 1 u = n i=1 u i2 i i = n, n 1,..., n s u i = x i u n s 1 = 0 < x n s 1 i = n s 2, n s 3,..., 0 u i = y i + z i (mod 2) u y z = 0 u < x (1). Case (ii) y n s 1 = 1 (i), v < y v Z 0 x v z = 0 (2). Case (iii) z n s 1 = 1 (2.17) i = n, n 1,..., n s w i = z i (2.18) i = n s 1,..., 0 w i = x i + y i (mod 2) (2.16) (2.17) w n s 1 = x n s 1 + y n s 1 = 0 (mod 2) w n s 1 = 0 < 1 = z n s 1. (2.19) 2 Subcase Subcase (iii.1) y w) (3) Subcase (iii.2) y > w) (2.20) (2.15) y k 2 k n z k 2 k ) 2.3 (2.18) y k 2 k z k 2 k ) = w k 2 k ) w). (2.21) k=n s 1 k=n s k=n s (2.21) (2.20) y k 2 k w k 2 k ) = w) (2.22) j (2.21) (2.23), k=n j y k 2 k > w k 2 k ) = w) (2.23) n j 1 < n s 1. (2.24) (2.22) 2.4 (ii), y k 2 k w k 2 k ) w k 2 k ). (2.25) k=n j k=n j+1 k=n j (2.23), y k 2 k > w k 2 k ). (2.26) (2.25) (2.26) k=n j 7
8 y k 2 k w k 2 k ) < y k 2 k (2.27) k=n j k=n j i = n, n 1, n 2,..., 0 v i w i v w i = n, n 1,..., n j w i = w i v i = y i (2.28) v n j 1 = 0 < 1 = y n j 1 (2.29) w n j 1 = x n j 1 + v n j 1 v n j 1 = 0, y n j 1 = 1 (2.27) v k 2 k w k2 k ) < v k 2 k + 2 n j 1. (2.30) (2.30) 2.5 v k 2 k w k 2k ) k=n j < v k 2 k + 2 n j 1 (2.31) t = n j 1, n j 2,..., 2, 1, 0 (2.32) t 1 v k 2 k w k2 k ) < v k 2 k + 2 t. (2.32) (2.31) t = n j 1 (2.32) t n j 1 t (2.32) (2.33) (2.36) (2.35) (2.38) v k 2 k + 2 t 1 w k2 k ) < v k 2 k + 2 t (2.33), v t 1 = 1 w t 1 = x t 1 + v t 1 (mod 2) v t 1 = 1, (2.33), v k 2 k w k2 k ) < v k 2 k + 2 t 1 (2.34) v t 1 2 t t 1 = 2 t 2.5 (2.34), v k 2 k w k2 k ) < v k 2 k + 2 t 1. (2.35) v k 2 k + 2 t 1 > w k2 k ) (2.36) v t 1 = 0 w t 1 = x t 1 + v t 1 (mod 2) v t 1 = 0, (2.36) (2.32) v k 2 k w k2 k ) < v k 2 k + 2 t 1 (2.37) 8
9 (2.37) 2.5, v k 2 k w k2 k ) < v k 2 k + 2 t 1 (2.38) (2.32) (2.35) (2.38) (2.35) (2.38), v k 2 k w k2 k ) < v k 2 k v k 2 k = n w k 2k ) (2.19), (2.24), (2.29) (2.28), v < y w < z (4) 2.7. x y z = 0 y z) (i) u < x u Z 0 u y z 0. (ii) v < y v Z 0 x v z 0. (iii) w < z w Z 0 x y w 0. (iv) v < y, w < z v = w) v, w Z 0 x v w 0.. (i),(ii),(iii) 1.1 () (iv) v < y, w < z w Z 0 v = w) i = n, n 1, n 2,..., j w i = z i w j 1 < z j 1 (2.39) y z), n z k 2 k ) y k 2 k 2.3 z k 2 k ) y k 2 k (2.40) k=j k=j 1 v = w), v < y (2.39) n z k 2 k ) = n w k 2 k ) w) = v = (2.40) (2.41) k=j k=j v k 2 k < y k 2 k (2.41) y k 2 k v k 2 k < y k 2 k k=j 1 k = n, n 1, n 2,..., j 1 v k = y k (2.42) x y z = 0 x j 1 + y j 1 + z j 1 = 0 (mod 2) (2.43) (2.39) (2.42) (2.43) x j 1 +v j 1 +w j 1 0 (mod 2) x v w , 2.2 CB(f, y, z) 1 1 x CB(f, y, z) {x, y, z}
10 2.1. {4, 4, 9} 2.2. {4, 3, 13} f(t) = t 2 f(t) = t 4 move h ({x, y, z}) {x, y, z} () 2.2. x, y, z Z 0, move h ({x, y, z}) = {{u, y, z} : u < x} {{x, v, z} : v < y} {{x, min(y, w)), w} : w < z}( u, v, w Z 0 ) 2.3. A k = {{x, y, z} : x, y, z Z 0, y z), x y z = 0}, B k = {{x, y, z} : x, y, z Z 0, y z), x y z 0} A k B k (i), (ii) (i) A k B k (ii) B k A k move h ({x, y, z}) {x, y, z} (i) (ii) A k B k A k B k CB(h, y, z) 1 P-positions N -positions. {x, y, z} A k 2.8 {p, q, r} B k 2.8 A k A k {0, 0, 0} A k A k (P-Position) {x, y, z} B k 2.8 {p, q, r} A k 2.8 B k A k {0, 0, 0} B k (N -position) 2.2. h 2.1 CB(h, y, z) y z. 2.1, 1 CB(h, y, z) {x, y, z} x y z = 0 P-position 1.1 (i) (ii) 1 CB(h, y, z) 0 1 CB(h, y, z) 1 x CB(h, y, z) x = y z CB(h, y, z) G({y, z}) = y z 2.1 CB(h, y, z) G({y, z}) = y z 10
11 2.2 y z 2.4. f (a) Z 0 Z 0 (a) G({y, z}) CB(f, y, z) G({y, z}) = y z. f 2.4 (a) 2.1 (a) 2.9. y = f(z), y f(z + 1), y < y y, z, y Z 0, G({y, z + 1}) < G({y, z + 1}). y = f(z) w z w f(w) y < y move f ({y, z + 1}) = {{v, z + 1} : v < y } {{min(y, f(w)), w} : w < z + 1} = {{v, z + 1} : v < y } {{f(w), w} : w < z + 1} = {{v, z + 1} : y v < y } {{v, z + 1} : v < y} {{f(w), w} : w < z + 1} = {{v, z + 1} : y v < y } {{v, z + 1} : v < y} {{min(y, f(w)), w} : w < z + 1} = {{v, z + 1} : y v < y } move f ({y, z + 1})( v, w Z 0 ). G({y, z + 1}) = mex({g({v, z + 1}) : y v < y } {G({a, b}) : {a, b} move f ({y, z + 1})} G({y, z + 1}). (2.44) {y, z + 1} move f ({y, z + 1}) G({y, z + 1}) G({y, z + 1}). (2.44) G({y, z + 1}) < G({y, z + 1}) y f(z) y, z Z 0 {G({min(y, f(w)), w}) : w < z} = {y w : w < z}. w Z 0 w < z, n = log 2 max(y, z) + 1 (2.45) (2.45) y w < y (z + 2 n ) = G({y, z + 2 n }) {a, b} move f ({y, z + 2 n }), G({a, b}) = y w a, b Z 0 move f (2.46) (2.47) w Z 0 w < z + 2 n G({min(y, f(w )), w }) = y w. (2.46) y (z + 2 n ) = G({y, z + 2 n }) = y w. (2.47) y Z 0 y < y (2.47) (2.45) (2.46) w z (2.48) f(w ) f(z) y. G({min(y, f(w )), w }) = G({y, w }) = y w. (2.49) (2.46), (2.49) 11
12 y w = y w (2.50) w < w (2.50) (2.48) w < z (2.46) {y w : w < z} {G({min(y, f(w)), w}) : w < z} {y w : w < z} {G({min(y, f(w)), w}) : w < z} {G({min(y, f(w)), w}) : w < z} = {y w : w < z} d, e, i Z 0, d i {0, 1}, e < 2 i 0 < d i 2 i + e a = d 2 i+1 + d i 2 i + e 1 c Z 0 c 2 i+1 f(a) < c 2 i i f(a + 1) < c 2 i i. 0 t < 2 i f(a) = c 2 i+1 + t (2.51) ( ) Case (i) d i = 1 f(a + 1) c 2 i i (2.52) G({c 2 i i, a + 1}) = (c 2 i i ) (d 2 i+1 + d i 2 i + e) = (c d)2 i+1 + e < (c d)2 i+1 + d i 2 i + (t e) = G({c 2 i+1 + t, a + 1}). (2.53) (2.51), (2.52) 2.9 G({c 2 i+1 + t, a + 1}) < G({c 2 i i, a + 1}) (2.53) Case (ii) d i = 0 G({c 2 i i, a + 1}) = (c 2 i i ) (d 2 i+1 + e) = (c d)2 i i + e > (c d)2 i i. d i 2 i + e > 0, d i = 0 e > 0 (c d)2 i i {G({p, q}) : {p, q} move f ({c 2 i i, a + 1})}. (2.54) {G({p, q}) : {p, q} move f ({c 2 i i, a + 1})} = {G({v, d 2 i+1 + e}) : v = 0, 1, 2,..., c 2 i i 1} {G({min(c 2 i i, f(w)), w}) : w = 0, 1, 2,..., d 2 i+1 + e 1}. (2.55) a = d 2 i+1 + e 1 w a, f(w) f(a) = c 2 i+1 + t (2.55) = {G({v, d 2 i+1 + e}) : v = 0, 1, 2,..., c 2 i i 1} {G({min(c 2 i+1 + t, f(w)), w}) : w = 0, 1, 2,..., d 2 i+1 + e 1}. (2.56) 12
13 c 2 i+1 + t = f(a) f(a + 1) = f(d 2 i+1 + e) 2.10 {G({min(c 2 i+1 + t, f(w)), w}) : w = 0, 1, 2,..., d 2 i+1 + e 1} = {(c 2 i+1 + t) w : w = 0, 1, 2,..., d 2 i+1 + e 1} 2.4 (2.56) = {v (d 2 i+1 + e) : v = 0, 1, 2,..., c 2 i i 1} {(c 2 i+1 + t) w : w = 0, 1, 2,..., d 2 i+1 + e 1} (i), (ii), (iii), (iv) = {(c 2 i+1 + k) (d 2 i+1 + e) : k = 0, 1, 2,..., 2 i 1} (2.57) {k (d 2 i+1 + e) : k = 0, 1, 2,..., c 2 i+1 1} (2.58) {(c 2 i+1 + t) (d 2 i+1 + k) : k = 0, 1, 2,..., e 1} (2.59) {(c 2 i+1 + t) k : k = 0, 1, 2,..., d 2 i+1 1}. (2.60) (i) (2.57) (c d)2 i+1 + (k e) (c d)2 i i k, e < 2 i (ii) (2.58) 2 i+1 c d (c d)2 i i (iii) (2.59) (c d)2 i+1 + (t k) (c d)2 i i k e 1 < 2 i t < 2 i (iv) (2.60) 2 i+1 c d (c d)2 i i (i), (ii), (iii), (iv) (2.54) (2.52) 2.3. f 2.4 f 2.1. f 2.1 z < z z, z Z 0 j (2.61) k = 0, 1, 2,..., n z k, z k Z 0 z = z 2 j = z 2 j (2.61) f(z) 2 j 1 < ) f(z. (2.62) 2j 1 z k 2 k z = k=j z k 2 k + j 1 z k 2k (2.62) i j 1 k = 0, 1, 2,..., n y k, y k Z 0 y i = 0 < 1 = y i, c = +1 f( f( z k 2 k ) = +1 j 1 z k 2 k + z k2 k ) = k=j y k 2 k (i+1). c 2 i+1 = i 1 y k 2 k + y i 2 i + y k 2 k (2.63) i 1 y k 2 k + y i 2 i + y k2 k (2.64) y k 2 k 13
14 (2.63) (2.64) i 1 f( z k 2 k ) = c 2 i i + y k 2 k (2.65) j 1 i 1 f( z k 2 k + z k2 k ) = c 2 i i + y k2 k (2.66) k=j a = max({z : f(z) < c 2 i i }) (2.67) b = min({z : f(z) c 2 i i }) (2.68) (2.67) (2.68) b = a + 1 d = +1 z k 2 k (i+1)., d 2 i+1 = < c 2 i i (2.67) i + 1 j, (d + 1) 2 i+1 = +1 z k 2 k (2.65) f(d 2 i+1 ) f( n z k 2 k ) +1 a z k 2 k d 2 i+1 (2.69) z k 2 k + 2 i+1 j 1 > z k 2 k + z k2 k (2.70) k=j. (2.66) (2.68) (2.70) (2.71) (2.69) j 1 z k 2 k + z k2 k b. (2.71) k=j (d + 1) 2 i+1 > b = a + 1 (2.72) a + 1 > d 2 i+1 (2.73) (2.72) (2.73) (d + 1) 2 i+1 > b = a + 1 > d 2 i+1 e < 2 i 0 < d i 2 i + e d i e a + 1 = d 2 i+1 + d i 2 i + e (2.74) (2.65) (2.69), f(a) f( z k 2 k ) c 2 i+1 (2.75) (2.67) f(a) < c 2 i i (2.76) f(a + 1) c 2 i i (2.77) (2.74) (2.75), (2.76) (2.77) 2.11 f G({y, z}) = y z CB(f, y, z) 14
15 [1] S. Nakamura and R. Miyadera, Impartial Chocolate Bar Games,Vol.15, Integers [2] A.C.Robin, A poisoned chocolate problem, Problem corner, The Mathematical Gazette Vol. 73, No. 466 (Dec., 1989), pp [3] D.Zeilberger, Three-Rowed CHOMP, Adv. Applied Math Vol. 26 (2001), pp [4] M. H. Albert, R. J. Nowakowski and D. Wolfe, Lessons In Play, A K Peters, p-139. [5] A.N.Siegel, Combinatorial Game Theory (Graduate Studies in Mathematics), American Mathematical Society (2013). 15
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