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Identifier

St000753: Binary words ⟶ ℤ

Values

0 => 0

1 => 1

00 => 0

01 => 1

10 => 1

11 => 2

000 => 0

001 => 1

010 => 1

011 => 2

100 => 1

101 => 0

110 => 2

111 => 3

0000 => 0

0001 => 1

0010 => 1

0011 => 2

0100 => 1

0101 => 0

0110 => 2

0111 => 3

1000 => 1

1001 => 0

1010 => 0

1011 => 3

1100 => 2

1101 => 3

1110 => 3

1111 => 1

00000 => 0

00001 => 1

00010 => 1

00011 => 2

00100 => 1

00101 => 0

00110 => 2

00111 => 3

01000 => 1

01001 => 0

01010 => 0

01011 => 3

01100 => 2

01101 => 3

01110 => 3

01111 => 1

10000 => 1

10001 => 0

10010 => 0

10011 => 3

10100 => 0

10101 => 1

10110 => 3

10111 => 2

11000 => 2

11001 => 3

11010 => 3

11011 => 0

11100 => 3

11101 => 2

11110 => 1

11111 => 4

000000 => 0

000001 => 1

000010 => 1

000011 => 2

000100 => 1

000101 => 0

000110 => 2

000111 => 3

001000 => 1

001001 => 0

001010 => 0

001011 => 3

001100 => 2

001101 => 3

001110 => 3

001111 => 1

010000 => 1

010001 => 0

010010 => 0

010011 => 3

010100 => 0

010101 => 1

010110 => 3

010111 => 2

011000 => 2

011001 => 3

011010 => 3

011011 => 0

011100 => 3

011101 => 2

011110 => 1

011111 => 4

100000 => 1

100001 => 0

100010 => 0

100011 => 3

100100 => 0

100101 => 1

100110 => 3

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Description

The Grundy value for the game of Kayles on a binary word.
Two players alternately may remove either a single 1 or two adjacent 1's. The player facing the word which has only 0's looses.

References

[1] wikipedia:Kayles

Code

@cached_function
def statistic(w):
    """Return the Grundy value of the binary word w for Kayles's game.
    """
    def children(w):
        for i in range(len(w)):
            if w[i] == 1:
                yield w[:i] + (0,) + w[i+1:]
        for i in range(len(w)-1):
            if w[i] == w[i+1] == 1:
                yield w[:i] + (0, 0) + w[i+2:]

    l = [statistic(tuple(v)) for v in children(tuple(w))]
    i = 0
    while i in l:
        i += 1
    return i

Created

Apr 06, 2017 at 22:58 by Martin Rubey

Updated

Apr 06, 2017 at 22:58 by Martin Rubey