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Identifier
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 100111=>2 101000=>0 101001=>1 101010=>1 101011=>2 101100=>3 101101=>2 101110=>2 101111=>0 110000=>2 110001=>3 110010=>3 110011=>0 110100=>3 110101=>2 110110=>0 110111=>1 111000=>3 111001=>2 111010=>2 111011=>1 111100=>1 111101=>0 111110=>4 111111=>3 0000000=>0 0000001=>1 0000010=>1 0000011=>2 0000100=>1 0000101=>0 0000110=>2 0000111=>3 0001000=>1 0001001=>0 0001010=>0 0001011=>3 0001100=>2 0001101=>3 0001110=>3 0001111=>1 0010000=>1 0010001=>0 0010010=>0 0010011=>3 0010100=>0 0010101=>1 0010110=>3 0010111=>2 0011000=>2 0011001=>3 0011010=>3 0011011=>0 0011100=>3 0011101=>2 0011110=>1 0011111=>4 0100000=>1 0100001=>0 0100010=>0 0100011=>3 0100100=>0 0100101=>1 0100110=>3 0100111=>2 0101000=>0 0101001=>1 0101010=>1 0101011=>2 0101100=>3 0101101=>2 0101110=>2 0101111=>0 0110000=>2 0110001=>3 0110010=>3 0110011=>0 0110100=>3 0110101=>2 0110110=>0 0110111=>1 0111000=>3 0111001=>2 0111010=>2 0111011=>1 0111100=>1 0111101=>0 0111110=>4 0111111=>3 1000000=>1 1000001=>0 1000010=>0 1000011=>3 1000100=>0 1000101=>1 1000110=>3 1000111=>2 1001000=>0 1001001=>1 1001010=>1 1001011=>2 1001100=>3 1001101=>2 1001110=>2 1001111=>0 1010000=>0 1010001=>1 1010010=>1 1010011=>2 1010100=>1 1010101=>0 1010110=>2 1010111=>3 1011000=>3 1011001=>2 1011010=>2 1011011=>1 1011100=>2 1011101=>3 1011110=>0 1011111=>5 1100000=>2 1100001=>3 1100010=>3 1100011=>0 1100100=>3 1100101=>2 1100110=>0 1100111=>1 1101000=>3 1101001=>2 1101010=>2 1101011=>1 1101100=>0 1101101=>1 1101110=>1 1101111=>3 1110000=>3 1110001=>2 1110010=>2 1110011=>1 1110100=>2 1110101=>3 1110110=>1 1110111=>0 1111000=>1 1111001=>0 1111010=>0 1111011=>3 1111100=>4 1111101=>5 1111110=>3 1111111=>2 00000000=>0 00000001=>1 00000010=>1 00000011=>2 00000100=>1 00000101=>0 00000110=>2 00000111=>3 00001000=>1 00001001=>0 00001010=>0 00001011=>3 00001100=>2 00001101=>3 00001110=>3 00001111=>1 00010000=>1 00010001=>0 00010010=>0 00010011=>3 00010100=>0 00010101=>1 00010110=>3 00010111=>2 00011000=>2 00011001=>3 00011010=>3 00011011=>0 00011100=>3 00011101=>2 00011110=>1 00011111=>4 00100000=>1 00100001=>0 00100010=>0 00100011=>3 00100100=>0 00100101=>1 00100110=>3 00100111=>2 00101000=>0 00101001=>1 00101010=>1 00101011=>2 00101100=>3 00101101=>2 00101110=>2 00101111=>0 00110000=>2 00110001=>3 00110010=>3 00110011=>0 00110100=>3 00110101=>2 00110110=>0 00110111=>1 00111000=>3 00111001=>2 00111010=>2 00111011=>1 00111100=>1 00111101=>0 00111110=>4 00111111=>3 01000000=>1 01000001=>0 01000010=>0 01000011=>3 01000100=>0 01000101=>1 01000110=>3 01000111=>2 01001000=>0 01001001=>1 01001010=>1 01001011=>2 01001100=>3 01001101=>2 01001110=>2 01001111=>0 01010000=>0 01010001=>1 01010010=>1 01010011=>2 01010100=>1 01010101=>0 01010110=>2 01010111=>3 01011000=>3 01011001=>2 01011010=>2 01011011=>1 01011100=>2 01011101=>3 01011110=>0 01011111=>5 01100000=>2 01100001=>3 01100010=>3 01100011=>0 01100100=>3 01100101=>2 01100110=>0 01100111=>1 01101000=>3 01101001=>2 01101010=>2 01101011=>1 01101100=>0 01101101=>1 01101110=>1 01101111=>3 01110000=>3 01110001=>2 01110010=>2 01110011=>1 01110100=>2 01110101=>3 01110110=>1 01110111=>0 01111000=>1 01111001=>0 01111010=>0 01111011=>3 01111100=>4 01111101=>5 01111110=>3 01111111=>2 10000000=>1 10000001=>0 10000010=>0 10000011=>3 10000100=>0 10000101=>1 10000110=>3 10000111=>2 10001000=>0 10001001=>1 10001010=>1 10001011=>2 10001100=>3 10001101=>2 10001110=>2 10001111=>0 10010000=>0 10010001=>1 10010010=>1 10010011=>2 10010100=>1 10010101=>0 10010110=>2 10010111=>3 10011000=>3 10011001=>2 10011010=>2 10011011=>1 10011100=>2 10011101=>3 10011110=>0 10011111=>5 10100000=>0 10100001=>1 10100010=>1 10100011=>2 10100100=>1 10100101=>0 10100110=>2 10100111=>3 10101000=>1 10101001=>0 10101010=>0 10101011=>3 10101100=>2 10101101=>3 10101110=>3 10101111=>1 10110000=>3 10110001=>2 10110010=>2 10110011=>1 10110100=>2 10110101=>3 10110110=>1 10110111=>0 10111000=>2 10111001=>3 10111010=>3 10111011=>0 10111100=>0 10111101=>1 10111110=>5 10111111=>2 11000000=>2 11000001=>3 11000010=>3 11000011=>0 11000100=>3 11000101=>2 11000110=>0 11000111=>1 11001000=>3 11001001=>2 11001010=>2 11001011=>1 11001100=>0 11001101=>1 11001110=>1 11001111=>3 11010000=>3 11010001=>2 11010010=>2 11010011=>1 11010100=>2 11010101=>3 11010110=>1 11010111=>0 11011000=>0 11011001=>1 11011010=>1 11011011=>2 11011100=>1 11011101=>0 11011110=>3 11011111=>6 11100000=>3 11100001=>2 11100010=>2 11100011=>1 11100100=>2 11100101=>3 11100110=>1 11100111=>0 11101000=>2 11101001=>3 11101010=>3 11101011=>0 11101100=>1 11101101=>0 11101110=>0 11101111=>2 11110000=>1 11110001=>0 11110010=>0 11110011=>3 11110100=>0 11110101=>1 11110110=>3 11110111=>2 11111000=>4 11111001=>5 11111010=>5 11111011=>6 11111100=>3 11111101=>2 11111110=>2 11111111=>1 000000000=>0 000000001=>1 000000010=>1 000000011=>2 000000100=>1 000000101=>0 000000110=>2 000000111=>3 000001000=>1 000001001=>0 000001010=>0 000001011=>3 000001100=>2 000001101=>3 000001110=>3 000001111=>1 000010000=>1 000010001=>0 000010010=>0 000010011=>3 000010100=>0 000010101=>1 000010110=>3 000010111=>2 000011000=>2 000011001=>3 000011010=>3 000011011=>0 000011100=>3 000011101=>2 000011110=>1 000011111=>4 000100000=>1 000100001=>0 000100010=>0 000100011=>3 000100100=>0 000100101=>1 000100110=>3 000100111=>2 000101000=>0 000101001=>1 000101010=>1 000101011=>2 000101100=>3 000101101=>2 000101110=>2 000101111=>0 000110000=>2 000110001=>3 000110010=>3 000110011=>0 000110100=>3 000110101=>2 000110110=>0 000110111=>1 000111000=>3 000111001=>2 000111010=>2 000111011=>1 000111100=>1 000111101=>0 000111110=>4 000111111=>3 001000000=>1 001000001=>0 001000010=>0 001000011=>3 001000100=>0 001000101=>1 001000110=>3 001000111=>2 001001000=>0 001001001=>1 001001010=>1 001001011=>2 001001100=>3 001001101=>2 001001110=>2 001001111=>0 001010000=>0 001010001=>1 001010010=>1 001010011=>2 001010100=>1 001010101=>0 001010110=>2 001010111=>3 001011000=>3 001011001=>2 001011010=>2 001011011=>1 001011100=>2 001011101=>3 001011110=>0 001011111=>5 001100000=>2 001100001=>3 001100010=>3 001100011=>0 001100100=>3 001100101=>2 001100110=>0 001100111=>1 001101000=>3 001101001=>2 001101010=>2 001101011=>1 001101100=>0 001101101=>1 001101110=>1 001101111=>3 001110000=>3 001110001=>2 001110010=>2 001110011=>1 001110100=>2 001110101=>3 001110110=>1 001110111=>0 001111000=>1 001111001=>0 001111010=>0 001111011=>3 001111100=>4 001111101=>5 001111110=>3 001111111=>2 010000000=>1 010000001=>0 010000010=>0 010000011=>3 010000100=>0 010000101=>1 010000110=>3 010000111=>2 010001000=>0 010001001=>1 010001010=>1 010001011=>2 010001100=>3 010001101=>2 010001110=>2 010001111=>0 010010000=>0 010010001=>1 010010010=>1 010010011=>2 010010100=>1 010010101=>0 010010110=>2 010010111=>3 010011000=>3 010011001=>2 010011010=>2 010011011=>1 010011100=>2 010011101=>3 010011110=>0 010011111=>5 010100000=>0 010100001=>1 010100010=>1 010100011=>2 010100100=>1 010100101=>0 010100110=>2 010100111=>3 010101000=>1 010101001=>0 010101010=>0 010101011=>3 010101100=>2 010101101=>3 010101110=>3 010101111=>1 010110000=>3 010110001=>2 010110010=>2 010110011=>1 010110100=>2 010110101=>3 010110110=>1 010110111=>0 010111000=>2 010111001=>3 010111010=>3 010111011=>0 010111100=>0 010111101=>1 010111110=>5 010111111=>2 011000000=>2 011000001=>3 011000010=>3 011000011=>0 011000100=>3 011000101=>2 011000110=>0 011000111=>1 011001000=>3 011001001=>2 011001010=>2 011001011=>1 011001100=>0 011001101=>1 011001110=>1 011001111=>3 011010000=>3 011010001=>2 011010010=>2 011010011=>1 011010100=>2 011010101=>3 011010110=>1 011010111=>0 011011000=>0 011011001=>1 011011010=>1 011011011=>2 011011100=>1 011011101=>0 011011110=>3 011011111=>6 011100000=>3 011100001=>2 011100010=>2 011100011=>1 011100100=>2 011100101=>3 011100110=>1 011100111=>0 011101000=>2 011101001=>3 011101010=>3 011101011=>0 011101100=>1 011101101=>0 011101110=>0 011101111=>2 011110000=>1 011110001=>0 011110010=>0 011110011=>3 011110100=>0 011110101=>1 011110110=>3 011110111=>2 011111000=>4 011111001=>5 011111010=>5 011111011=>6 011111100=>3 011111101=>2 011111110=>2 011111111=>1 100000000=>1 100000001=>0 100000010=>0 100000011=>3 100000100=>0 100000101=>1 100000110=>3 100000111=>2 100001000=>0 100001001=>1 100001010=>1 100001011=>2 100001100=>3 100001101=>2 100001110=>2 100001111=>0 100010000=>0 100010001=>1 100010010=>1 100010011=>2 100010100=>1 100010101=>0 100010110=>2 100010111=>3 100011000=>3 100011001=>2 100011010=>2 100011011=>1 100011100=>2 100011101=>3 100011110=>0 100011111=>5 100100000=>0 100100001=>1 100100010=>1 100100011=>2 100100100=>1 100100101=>0 100100110=>2 100100111=>3 100101000=>1 100101001=>0 100101010=>0 100101011=>3 100101100=>2 100101101=>3 100101110=>3 100101111=>1 100110000=>3 100110001=>2 100110010=>2 100110011=>1 100110100=>2 100110101=>3 100110110=>1 100110111=>0 100111000=>2 100111001=>3 100111010=>3 100111011=>0 100111100=>0 100111101=>1 100111110=>5 100111111=>2 101000000=>0 101000001=>1 101000010=>1 101000011=>2 101000100=>1 101000101=>0 101000110=>2 101000111=>3 101001000=>1 101001001=>0 101001010=>0 101001011=>3 101001100=>2 101001101=>3 101001110=>3 101001111=>1 101010000=>1 101010001=>0 101010010=>0 101010011=>3 101010100=>0 101010101=>1 101010110=>3 101010111=>2 101011000=>2 101011001=>3 101011010=>3 101011011=>0 101011100=>3 101011101=>2 101011110=>1 101011111=>4 101100000=>3 101100001=>2 101100010=>2 101100011=>1 101100100=>2 101100101=>3 101100110=>1 101100111=>0 101101000=>2 101101001=>3 101101010=>3 101101011=>0 101101100=>1 101101101=>0 101101110=>0 101101111=>2 101110000=>2 101110001=>3 101110010=>3 101110011=>0 101110100=>3 101110101=>2 101110110=>0 101110111=>1 101111000=>0 101111001=>1 101111010=>1 101111011=>2 101111100=>5 101111101=>4 101111110=>2 101111111=>3 110000000=>2 110000001=>3 110000010=>3 110000011=>0 110000100=>3 110000101=>2 110000110=>0 110000111=>1 110001000=>3 110001001=>2 110001010=>2 110001011=>1 110001100=>0 110001101=>1 110001110=>1 110001111=>3 110010000=>3 110010001=>2 110010010=>2 110010011=>1 110010100=>2 110010101=>3 110010110=>1 110010111=>0 110011000=>0 110011001=>1 110011010=>1 110011011=>2 110011100=>1 110011101=>0 110011110=>3 110011111=>6 110100000=>3 110100001=>2 110100010=>2 110100011=>1 110100100=>2 110100101=>3 110100110=>1 110100111=>0 110101000=>2 110101001=>3 110101010=>3 110101011=>0 110101100=>1 110101101=>0 110101110=>0 110101111=>2 110110000=>0 110110001=>1 110110010=>1 110110011=>2 110110100=>1 110110101=>0 110110110=>2 110110111=>3 110111000=>1 110111001=>0 110111010=>0 110111011=>3 110111100=>3 110111101=>2 110111110=>6 110111111=>1 111000000=>3 111000001=>2 111000010=>2 111000011=>1 111000100=>2 111000101=>3 111000110=>1 111000111=>0 111001000=>2 111001001=>3 111001010=>3 111001011=>0 111001100=>1 111001101=>0 111001110=>0 111001111=>2 111010000=>2 111010001=>3 111010010=>3 111010011=>0 111010100=>3 111010101=>2 111010110=>0 111010111=>1 111011000=>1 111011001=>0 111011010=>0 111011011=>3 111011100=>0 111011101=>1 111011110=>2 111011111=>7 111100000=>1 111100001=>0 111100010=>0 111100011=>3 111100100=>0 111100101=>1 111100110=>3 111100111=>2 111101000=>0 111101001=>1 111101010=>1 111101011=>2 111101100=>3 111101101=>2 111101110=>2 111101111=>0 111110000=>4 111110001=>5 111110010=>5 111110011=>6 111110100=>5 111110101=>4 111110110=>6 111110111=>7 111111000=>3 111111001=>2 111111010=>2 111111011=>1 111111100=>2 111111101=>3 111111110=>1 111111111=>4 0000000001=>1 0010000001=>0 0010101101=>3 0010101011=>3 0010100111=>3 0010011101=>3 0010011011=>1 0010010111=>3 0010001111=>0 0001110101=>3 0001110011=>1 0001101101=>1 0001101011=>1 0001100111=>1 0001011101=>3 0001011011=>1 0001010111=>3 0001001111=>0 0000111101=>0 0000111011=>1 0000110111=>1 0000101111=>0 0000011111=>4 0000000000=>0 0000000110=>2 0000011000=>2 0000011110=>1 0000010010=>0 0001100000=>2 0001100110=>0 0001111000=>1 0001111110=>3 0001110010=>2 0001001000=>0 0001001110=>2 0001000010=>0 0001011010=>2 0010101010=>0 0010101000=>1 0010101100=>2 0010100100=>1 0010010100=>1 0001010100=>1 0010101110=>3 0010100000=>0 0010000100=>0 0000010100=>0 0000000010=>1 0000001110=>3 0000111110=>4 0000000100=>1 0000001010=>0 0000010110=>3 0000101110=>2 0001011110=>0 0001101110=>1 0001110110=>1 0001111010=>0 0001111100=>4 0000000101=>0 0000001101=>3 0000011101=>2 0001111101=>5 0000001001=>0 0000010101=>1 0000101101=>2 0010110001=>2 0001100001=>3 0001000001=>0 0000111111=>3 0001110100=>2 0001101100=>0 0010011100=>2 0001011100=>2 0000111100=>1 0001101010=>2 0010011010=>2 0000111010=>2 0010100110=>2 0010010110=>2 0001010110=>2 0000110110=>0 0010001110=>2 0001101001=>2 0000110001=>3 0000010001=>0 0000001100=>2 0000011001=>3 0000110011=>0 0000110100=>3 0000000011=>2 0000011011=>0 0001101111=>3 0000110101=>2 0010000010=>0 0001000011=>3 0010011110=>0 0001010000=>0 0001000100=>0 0001010101=>0 0001001011=>2 0001100011=>0 0001110111=>0 0010000000=>1 0000000111=>3 0000001111=>1 0001111111=>2 0000001011=>3 0000010111=>2 0001011111=>5 0001111001=>0 0000011010=>3 0000101010=>1 0000100101=>1 0010010001=>1 0010000101=>1 0010010101=>0 0010010011=>2 0000101011=>2 0001000111=>2 0000100111=>2 0010010010=>1 0001001010=>1 0001001001=>1 0001000101=>1 0000100011=>3 0000100100=>0 0000100010=>0 0010110000=>3 0000011100=>3 0000001000=>1 0000101100=>3 0010101111=>1 0000101000=>0 0010001000=>0 0001001100=>3 0001000110=>3 0001100100=>3 0001100010=>3 0000100110=>3 0000110010=>3 0010010000=>0 0001010010=>1 0001000000=>1 0001011001=>2 0001100101=>2 0001110001=>2 0010011001=>2 0010100101=>0 0010101001=>0 0010011111=>5 0000111001=>2 0010001101=>2 0001001101=>2 0001010001=>1 0001010011=>2 0001111011=>3 0010011000=>3 0001011000=>3 0000100000=>1 0010001011=>2 0010000111=>2 0000010011=>3 0010001100=>3 0010000110=>3 0000110000=>2 0000100001=>0 0010100011=>2 0010100001=>1 0000111000=>3 0001101000=>3 0000010000=>1 0000101001=>1 0010100010=>1 0010001001=>1 0010000011=>3 0010001010=>1 0001110000=>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