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Identifier
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 100111=>6 101000=>1 101001=>4 101010=>1 101011=>5 101100=>2 101101=>4 101110=>3 101111=>4 110000=>0 110001=>3 110010=>2 110011=>0 110100=>1 110101=>1 110110=>2 110111=>3 111000=>0 111001=>2 111010=>1 111011=>2 111100=>0 111101=>1 111110=>0 111111=>0 0000000=>0 0000001=>6 0000010=>5 0000011=>2 0000100=>4 0000101=>1 0000110=>0 0000111=>4 0001000=>3 0001001=>4 0001010=>5 0001011=>3 0001100=>6 0001101=>2 0001110=>1 0001111=>4 0010000=>2 0010001=>3 0010010=>6 0010011=>6 0010100=>5 0010101=>7 0010110=>0 0010111=>3 0011000=>0 0011001=>0 0011010=>7 0011011=>6 0011100=>6 0011101=>5 0011110=>0 0011111=>2 0100000=>1 0100001=>6 0100010=>3 0100011=>5 0100100=>4 0100101=>0 0100110=>7 0100111=>2 0101000=>1 0101001=>7 0101010=>0 0101011=>7 0101100=>5 0101101=>0 0101110=>5 0101111=>1 0110000=>2 0110001=>2 0110010=>1 0110011=>0 0110100=>4 0110101=>7 0110110=>6 0110111=>4 0111000=>3 0111001=>2 0111010=>3 0111011=>3 0111100=>4 0111101=>6 0111110=>5 0111111=>6 1000000=>0 1000001=>5 1000010=>4 1000011=>0 1000100=>3 1000101=>5 1000110=>6 1000111=>1 1001000=>2 1001001=>6 1001010=>5 1001011=>0 1001100=>0 1001101=>7 1001110=>6 1001111=>0 1010000=>1 1010001=>3 1010010=>4 1010011=>7 1010100=>1 1010101=>0 1010110=>5 1010111=>5 1011000=>2 1011001=>1 1011010=>4 1011011=>6 1011100=>3 1011101=>3 1011110=>4 1011111=>5 1100000=>0 1100001=>4 1100010=>3 1100011=>6 1100100=>2 1100101=>5 1100110=>0 1100111=>6 1101000=>1 1101001=>4 1101010=>1 1101011=>5 1101100=>2 1101101=>4 1101110=>3 1101111=>4 1110000=>0 1110001=>3 1110010=>2 1110011=>0 1110100=>1 1110101=>1 1110110=>2 1110111=>3 1111000=>0 1111001=>2 1111010=>1 1111011=>2 1111100=>0 1111101=>1 1111110=>0 1111111=>0 00000000=>0 00000001=>7 00000010=>6 00000011=>0 00000100=>5 00000101=>1 00000110=>2 00000111=>7 00001000=>4 00001001=>2 00001010=>1 00001011=>6 00001100=>0 00001101=>5 00001110=>4 00001111=>0 00010000=>3 00010001=>3 00010010=>4 00010011=>5 00010100=>5 00010101=>6 00010110=>3 00010111=>1 00011000=>6 00011001=>7 00011010=>2 00011011=>2 00011100=>1 00011101=>3 00011110=>4 00011111=>7 00100000=>2 00100001=>4 00100010=>3 00100011=>4 00100100=>6 00100101=>3 00100110=>6 00100111=>2 00101000=>5 00101001=>2 00101010=>7 00101011=>1 00101100=>0 00101101=>4 00101110=>3 00101111=>6 00110000=>0 00110001=>1 00110010=>0 00110011=>0 00110100=>7 00110101=>5 00110110=>6 00110111=>5 00111000=>6 00111001=>6 00111010=>5 00111011=>4 00111100=>0 00111101=>3 00111110=>2 00111111=>0 01000000=>1 01000001=>5 01000010=>6 01000011=>3 01000100=>3 01000101=>4 01000110=>5 01000111=>3 01001000=>4 01001001=>1 01001010=>0 01001011=>4 01001100=>7 01001101=>1 01001110=>2 01001111=>5 01010000=>1 01010001=>2 01010010=>7 01010011=>5 01010100=>0 01010101=>0 01010110=>7 01010111=>6 01011000=>5 01011001=>7 01011010=>0 01011011=>3 01011100=>5 01011101=>4 01011110=>1 01011111=>1 01100000=>2 01100001=>7 01100010=>2 01100011=>6 01100100=>1 01100101=>7 01100110=>0 01100111=>7 01101000=>4 01101001=>0 01101010=>7 01101011=>2 01101100=>6 01101101=>1 01101110=>4 01101111=>2 01110000=>3 01110001=>1 01110010=>2 01110011=>1 01110100=>3 01110101=>2 01110110=>3 01110111=>3 01111000=>4 01111001=>7 01111010=>6 01111011=>4 01111100=>5 01111101=>5 01111110=>6 01111111=>7 10000000=>0 10000001=>6 10000010=>5 10000011=>2 10000100=>4 10000101=>1 10000110=>0 10000111=>4 10001000=>3 10001001=>4 10001010=>5 10001011=>3 10001100=>6 10001101=>2 10001110=>1 10001111=>4 10010000=>2 10010001=>3 10010010=>6 10010011=>6 10010100=>5 10010101=>7 10010110=>0 10010111=>3 10011000=>0 10011001=>0 10011010=>7 10011011=>6 10011100=>6 10011101=>5 10011110=>0 10011111=>2 10100000=>1 10100001=>6 10100010=>3 10100011=>5 10100100=>4 10100101=>0 10100110=>7 10100111=>2 10101000=>1 10101001=>7 10101010=>0 10101011=>7 10101100=>5 10101101=>0 10101110=>5 10101111=>1 10110000=>2 10110001=>2 10110010=>1 10110011=>0 10110100=>4 10110101=>7 10110110=>6 10110111=>4 10111000=>3 10111001=>2 10111010=>3 10111011=>3 10111100=>4 10111101=>6 10111110=>5 10111111=>6 11000000=>0 11000001=>5 11000010=>4 11000011=>0 11000100=>3 11000101=>5 11000110=>6 11000111=>1 11001000=>2 11001001=>6 11001010=>5 11001011=>0 11001100=>0 11001101=>7 11001110=>6 11001111=>0 11010000=>1 11010001=>3 11010010=>4 11010011=>7 11010100=>1 11010101=>0 11010110=>5 11010111=>5 11011000=>2 11011001=>1 11011010=>4 11011011=>6 11011100=>3 11011101=>3 11011110=>4 11011111=>5 11100000=>0 11100001=>4 11100010=>3 11100011=>6 11100100=>2 11100101=>5 11100110=>0 11100111=>6 11101000=>1 11101001=>4 11101010=>1 11101011=>5 11101100=>2 11101101=>4 11101110=>3 11101111=>4 11110000=>0 11110001=>3 11110010=>2 11110011=>0 11110100=>1 11110101=>1 11110110=>2 11110111=>3 11111000=>0 11111001=>2 11111010=>1 11111011=>2 11111100=>0 11111101=>1 11111110=>0 11111111=>0 000000000=>0 000000001=>8 000000010=>7 000000011=>14 000000100=>6 000000101=>13 000000110=>0 000000111=>10 000001000=>5 000001001=>12 000001010=>1 000001011=>9 000001100=>2 000001101=>8 000001110=>7 000001111=>12 000010000=>4 000010001=>11 000010010=>2 000010011=>12 000010100=>1 000010101=>13 000010110=>6 000010111=>11 000011000=>0 000011001=>14 000011010=>5 000011011=>10 000011100=>4 000011101=>9 000011110=>0 000011111=>12 000100000=>3 000100001=>10 000100010=>3 000100011=>11 000100100=>4 000100101=>14 000100110=>5 000100111=>14 000101000=>5 000101001=>13 000101010=>6 000101011=>15 000101100=>3 000101101=>8 000101110=>1 000101111=>11 000110000=>6 000110001=>8 000110010=>7 000110011=>8 000110100=>2 000110101=>15 000110110=>2 000110111=>14 000111000=>1 000111001=>14 000111010=>3 000111011=>13 000111100=>4 000111101=>8 000111110=>7 000111111=>10 001000000=>2 001000001=>9 001000010=>4 001000011=>14 001000100=>3 001000101=>11 001000110=>4 001000111=>13 001001000=>6 001001001=>12 001001010=>3 001001011=>8 001001100=>6 001001101=>15 001001110=>2 001001111=>10 001010000=>5 001010001=>9 001010010=>2 001010011=>15 001010100=>7 001010101=>8 001010110=>1 001010111=>15 001011000=>0 001011001=>13 001011010=>4 001011011=>8 001011100=>3 001011101=>13 001011110=>6 001011111=>9 001100000=>0 001100001=>10 001100010=>1 001100011=>10 001100100=>0 001100101=>9 001100110=>0 001100111=>8 001101000=>7 001101001=>12 001101010=>5 001101011=>15 001101100=>6 001101101=>14 001101110=>5 001101111=>12 001110000=>6 001110001=>11 001110010=>6 001110011=>10 001110100=>5 001110101=>11 001110110=>4 001110111=>11 001111000=>0 001111001=>12 001111010=>3 001111011=>14 001111100=>2 001111101=>13 001111110=>0 001111111=>14 010000000=>1 010000001=>8 010000010=>5 010000011=>13 010000100=>6 010000101=>12 010000110=>3 010000111=>8 010001000=>3 010001001=>11 010001010=>4 010001011=>13 010001100=>5 010001101=>14 010001110=>3 010001111=>9 010010000=>4 010010001=>10 010010010=>1 010010011=>14 010010100=>0 010010101=>13 010010110=>4 010010111=>8 010011000=>7 010011001=>8 010011010=>1 010011011=>15 010011100=>2 010011101=>14 010011110=>5 010011111=>8 010100000=>1 010100001=>9 010100010=>2 010100011=>11 010100100=>7 010100101=>12 010100110=>5 010100111=>15 010101000=>0 010101001=>9 010101010=>0 010101011=>8 010101100=>7 010101101=>13 010101110=>6 010101111=>13 010110000=>5 010110001=>10 010110010=>7 010110011=>9 010110100=>0 010110101=>12 010110110=>3 010110111=>14 010111000=>5 010111001=>11 010111010=>4 010111011=>11 010111100=>1 010111101=>12 010111110=>1 010111111=>13 011000000=>2 011000001=>8 011000010=>7 011000011=>12 011000100=>2 011000101=>11 011000110=>6 011000111=>14 011001000=>1 011001001=>10 011001010=>7 011001011=>13 011001100=>0 011001101=>8 011001110=>7 011001111=>14 011010000=>4 011010001=>9 011010010=>0 011010011=>12 011010100=>7 011010101=>9 011010110=>2 011010111=>13 011011000=>6 011011001=>10 011011010=>1 011011011=>12 011011100=>4 011011101=>11 011011110=>2 011011111=>12 011100000=>3 011100001=>8 011100010=>1 011100011=>11 011100100=>2 011100101=>10 011100110=>1 011100111=>8 011101000=>3 011101001=>9 011101010=>2 011101011=>9 011101100=>3 011101101=>10 011101110=>3 011101111=>11 011110000=>4 011110001=>8 011110010=>7 011110011=>10 011110100=>6 011110101=>9 011110110=>4 011110111=>10 011111000=>5 011111001=>8 011111010=>5 011111011=>9 011111100=>6 011111101=>8 011111110=>7 011111111=>8 100000000=>0 100000001=>7 100000010=>6 100000011=>0 100000100=>5 100000101=>1 100000110=>2 100000111=>7 100001000=>4 100001001=>2 100001010=>1 100001011=>6 100001100=>0 100001101=>5 100001110=>4 100001111=>0 100010000=>3 100010001=>3 100010010=>4 100010011=>5 100010100=>5 100010101=>6 100010110=>3 100010111=>1 100011000=>6 100011001=>7 100011010=>2 100011011=>2 100011100=>1 100011101=>3 100011110=>4 100011111=>7 100100000=>2 100100001=>4 100100010=>3 100100011=>4 100100100=>6 100100101=>3 100100110=>6 100100111=>2 100101000=>5 100101001=>2 100101010=>7 100101011=>1 100101100=>0 100101101=>4 100101110=>3 100101111=>6 100110000=>0 100110001=>1 100110010=>0 100110011=>0 100110100=>7 100110101=>5 100110110=>6 100110111=>5 100111000=>6 100111001=>6 100111010=>5 100111011=>4 100111100=>0 100111101=>3 100111110=>2 100111111=>0 101000000=>1 101000001=>5 101000010=>6 101000011=>3 101000100=>3 101000101=>4 101000110=>5 101000111=>3 101001000=>4 101001001=>1 101001010=>0 101001011=>4 101001100=>7 101001101=>1 101001110=>2 101001111=>5 101010000=>1 101010001=>2 101010010=>7 101010011=>5 101010100=>0 101010101=>0 101010110=>7 101010111=>6 101011000=>5 101011001=>7 101011010=>0 101011011=>3 101011100=>5 101011101=>4 101011110=>1 101011111=>1 101100000=>2 101100001=>7 101100010=>2 101100011=>6 101100100=>1 101100101=>7 101100110=>0 101100111=>7 101101000=>4 101101001=>0 101101010=>7 101101011=>2 101101100=>6 101101101=>1 101101110=>4 101101111=>2 101110000=>3 101110001=>1 101110010=>2 101110011=>1 101110100=>3 101110101=>2 101110110=>3 101110111=>3 101111000=>4 101111001=>7 101111010=>6 101111011=>4 101111100=>5 101111101=>5 101111110=>6 101111111=>7 110000000=>0 110000001=>6 110000010=>5 110000011=>2 110000100=>4 110000101=>1 110000110=>0 110000111=>4 110001000=>3 110001001=>4 110001010=>5 110001011=>3 110001100=>6 110001101=>2 110001110=>1 110001111=>4 110010000=>2 110010001=>3 110010010=>6 110010011=>6 110010100=>5 110010101=>7 110010110=>0 110010111=>3 110011000=>0 110011001=>0 110011010=>7 110011011=>6 110011100=>6 110011101=>5 110011110=>0 110011111=>2 110100000=>1 110100001=>6 110100010=>3 110100011=>5 110100100=>4 110100101=>0 110100110=>7 110100111=>2 110101000=>1 110101001=>7 110101010=>0 110101011=>7 110101100=>5 110101101=>0 110101110=>5 110101111=>1 110110000=>2 110110001=>2 110110010=>1 110110011=>0 110110100=>4 110110101=>7 110110110=>6 110110111=>4 110111000=>3 110111001=>2 110111010=>3 110111011=>3 110111100=>4 110111101=>6 110111110=>5 110111111=>6 111000000=>0 111000001=>5 111000010=>4 111000011=>0 111000100=>3 111000101=>5 111000110=>6 111000111=>1 111001000=>2 111001001=>6 111001010=>5 111001011=>0 111001100=>0 111001101=>7 111001110=>6 111001111=>0 111010000=>1 111010001=>3 111010010=>4 111010011=>7 111010100=>1 111010101=>0 111010110=>5 111010111=>5 111011000=>2 111011001=>1 111011010=>4 111011011=>6 111011100=>3 111011101=>3 111011110=>4 111011111=>5 111100000=>0 111100001=>4 111100010=>3 111100011=>6 111100100=>2 111100101=>5 111100110=>0 111100111=>6 111101000=>1 111101001=>4 111101010=>1 111101011=>5 111101100=>2 111101101=>4 111101110=>3 111101111=>4 111110000=>0 111110001=>3 111110010=>2 111110011=>0 111110100=>1 111110101=>1 111110110=>2 111110111=>3 111111000=>0 111111001=>2 111111010=>1 111111011=>2 111111100=>0 111111101=>1 111111110=>0 111111111=>0 0000000001=>9 0010000001=>10 0010101101=>10 0010101011=>1 0010100111=>4 0010011101=>15 0010011011=>0 0010010111=>5 0010001111=>6 0001110101=>12 0001110011=>3 0001101101=>11 0001101011=>4 0001100111=>1 0001011101=>14 0001011011=>5 0001010111=>0 0001001111=>7 0000111101=>13 0000111011=>6 0000110111=>7 0000101111=>0 0000011111=>1 0000000000=>0 0000000110=>14 0000011000=>2 0000011110=>12 0000010010=>12 0001100000=>6 0001100110=>8 0001111000=>4 0001111110=>10 0001110010=>14 0001001000=>4 0001001110=>14 0001000010=>10 0001011010=>8 0010101010=>8 0010101000=>7 0010101100=>1 0010100100=>2 0010010100=>3 0001010100=>6 0010101110=>15 0010100000=>5 0010000100=>4 0000010100=>1 0000000010=>8 0000001110=>10 0000111110=>12 0000000100=>7 0000001010=>13 0000010110=>9 0000101110=>11 0001011110=>11 0001101110=>14 0001110110=>13 0001111010=>8 0001111100=>7 0000000101=>13 0000001101=>11 0000011101=>11 0001111101=>9 0000001001=>14 0000010101=>12 0000101101=>12 0010110001=>14 0001100001=>13 0001000001=>9 0000111111=>0 0001110100=>3 0001101100=>2 0010011100=>2 0001011100=>1 0000111100=>0 0001101010=>15 0010011010=>15 0000111010=>9 0010100110=>15 0010010110=>8 0001010110=>15 0000110110=>10 0010001110=>13 0001101001=>8 0000110001=>15 0000010001=>11 0000001100=>0 0000011001=>13 0000110011=>0 0000110100=>5 0000000011=>0 0000011011=>6 0001101111=>6 0000110101=>9 0010000010=>9 0001000011=>1 0010011110=>10 0001010000=>5 0001000100=>3 0001010101=>8 0001001011=>4 0001100011=>2 0001110111=>3 0010000000=>2 0000000111=>5 0000001111=>0 0001111111=>5 0000001011=>4 0000010111=>5 0001011111=>5 0001111001=>9 0000011010=>8 0000101010=>13 0000100101=>9 0010010001=>9 0010000101=>15 0010010101=>15 0010010011=>2 0000101011=>5 0001000111=>3 0000100111=>6 0010010010=>12 0001001010=>14 0001001001=>15 0001000101=>10 0000100011=>2 0000100100=>2 0000100010=>11 0010110000=>0 0000011100=>7 0000001000=>6 0000101100=>6 0010101111=>5 0000101000=>1 0010001000=>3 0001001100=>5 0001000110=>11 0001100100=>7 0001100010=>8 0000100110=>12 0000110010=>14 0010010000=>6 0001010010=>13 0001000000=>3 0001011001=>15 0001100101=>15 0001110001=>13 0010011001=>8 0010100101=>8 0010101001=>15 0010011111=>6 0000111001=>10 0010001101=>14 0001001101=>13 0001010001=>8 0001010011=>1 0001111011=>4 0010011000=>6 0001011000=>3 0000100000=>4 0010001011=>3 0010000111=>4 0000010011=>3 0010001100=>4 0010000110=>14 0000110000=>0 0000100001=>12 0010100011=>1 0010100001=>12 0000111000=>4 0001101000=>2 0000010000=>5 0000101001=>8 0010100010=>9 0010001001=>10 0010000011=>0 0010001010=>11 0001110000=>1
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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 $1\dots 10\dots 0$.
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