Identifier
- St000792: Binary words ⟶ ℤ
Values
=>
0=>0
1=>1
00=>0
01=>2
10=>1
11=>3
000=>0
001=>1
010=>2
011=>3
100=>1
101=>0
110=>3
111=>2
0000=>0
0001=>4
0010=>1
0011=>5
0100=>2
0101=>6
0110=>3
0111=>7
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1001=>5
1010=>0
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1100=>3
1101=>7
1110=>2
1111=>6
00000=>0
00001=>1
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1011111110000000=>15
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1111110001000010=>6
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1111111000000010=>5
111111110000000010=>13
101111111100000000=>13
111111101000000010=>4
101111111010000000=>14
111111100100000010=>7
111111011000000010=>13
101111111001000000=>11
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11111111100000000010=>12
10111111111000000000=>12
11111111010000000010=>15
10111111110100000000=>9
1111111111000000000010=>14
1011111111110000000000=>8
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Description
The Grundy value for the game of ruler on a binary word.
Two players alternately may switch any consecutive sequence of numbers that ends with a 1. The player facing the word which has only 0's looses.
Two players alternately may switch any consecutive sequence of numbers that ends with a 1. The player facing the word which has only 0's looses.
References
[1] Lugo, M. Sprague-Grundy sequence for the ruler game MathOverflow:39165
Code
def statistic(w):
two_adic = Zp(2)
res = 0
for i, e in enumerate(w):
if e == 1:
res = res^^(2^ZZ((two_adic(i+1).ordp())))
return res
# CodeLanguage: Sage, naive implementation
@cached_function
def statistic_alt(w):
"""Return the Grundy value of the binary word w for the ruler game.
"""
def children(w):
for i in range(len(w)):
if w[i] == 1:
for j in range(i+1):
yield w[:j] + tuple(1-w[k] for k in range(j,i+1)) + w[i+1:]
l = [statistic_alt(tuple(v)) for v in children(tuple(w))]
i = 0
while i in l:
i += 1
return i
Created
May 01, 2017 at 07:36 by Martin Rubey
Updated
May 02, 2018 at 23:04 by Martin Rubey
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