Identifier
- St000682: Binary words ⟶ ℤ
Values
0 => 0
1 => 0
00 => 0
01 => 1
10 => 0
11 => 0
000 => 0
001 => 2
010 => 1
011 => 2
100 => 0
101 => 1
110 => 0
111 => 0
0000 => 0
0001 => 3
0010 => 2
0011 => 0
0100 => 1
0101 => 1
0110 => 2
0111 => 3
1000 => 0
1001 => 2
1010 => 1
1011 => 2
1100 => 0
1101 => 1
1110 => 0
1111 => 0
00000 => 0
00001 => 4
00010 => 3
00011 => 6
00100 => 2
00101 => 5
00110 => 0
00111 => 6
01000 => 1
01001 => 4
01010 => 1
01011 => 5
01100 => 2
01101 => 4
01110 => 3
01111 => 4
10000 => 0
10001 => 3
10010 => 2
10011 => 0
10100 => 1
10101 => 1
10110 => 2
10111 => 3
11000 => 0
11001 => 2
11010 => 1
11011 => 2
11100 => 0
11101 => 1
11110 => 0
11111 => 0
000000 => 0
000001 => 5
000010 => 4
000011 => 0
000100 => 3
000101 => 5
000110 => 6
000111 => 1
001000 => 2
001001 => 6
001010 => 5
001011 => 0
001100 => 0
001101 => 7
001110 => 6
001111 => 0
010000 => 1
010001 => 3
010010 => 4
010011 => 7
010100 => 1
010101 => 0
010110 => 5
010111 => 5
011000 => 2
011001 => 1
011010 => 4
011011 => 6
011100 => 3
011101 => 3
011110 => 4
011111 => 5
100000 => 0
100001 => 4
100010 => 3
100011 => 6
100100 => 2
100101 => 5
100110 => 0
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Description
The Grundy value of Welter's game on a binary word.
Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
References
[1] Nowakowski, R. J. $…$, Welter's Game, Sylver Coinage, Dots-and-Boxes,$\,…$ MathSciNet:1095544
Code
@cached_function
def statistic(w):
"""Return the Grundy value of the binary word w for Welter's game.
sage: statistic((0,)*3 + (1,) + (0,)*15 + (1,))
15
sage: statistic((0,)*7 + (1,) + (0,)*15 + (1,))
15
sage: statistic((0,)*5 + (1,) + (0,)*7 + (1,))
7
sage: statistic((0,)*11 + (1,) + (0,)*5 + (1,))
25
sage: statistic((0,0,1))
2
sage: statistic(tuple(1 if is_prime(i) else 0 for i in range(24)))
28
"""
def children(w):
for i in range(len(w)):
if w[i] == 1:
for j in range(i):
if w[j] == 0:
yield w[:j] + (1,) + w[j+1:i] + (0,) + w[i+1:]
l = [statistic(tuple(v)) for v in children(tuple(w))]
i = 0
while i in l:
i += 1
return i
Created
Jan 06, 2017 at 20:23 by Martin Rubey
Updated
Jan 06, 2017 at 20:23 by Martin Rubey
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