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A374910
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Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+4) = v(1)v(2)...v(k+4) with a (1^k,0,1,0,1)-weight assignment is a second-player winning game.
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3
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1, 25, 26, 31, 32, 37, 38, 63, 64, 69, 70, 76, 101, 102, 139, 145, 177, 189, 215, 235, 252, 253, 267, 284, 290, 305, 311, 328, 360, 668
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OFFSET
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1,2
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COMMENTS
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The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.
A Toggle move consists of selecting a vertex v and switching its weight as well as the weight of each of its neighbors. This move is only legal provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
In the special case G=P(k+4), a (1^k,0,1,0,1)-weight assignment is one in which vertices v(k+1) and v(k+3) are assigned weight 0 and all remaining vertices are assigned weight 1.
The path P(k+4m) where vertices v(k+1), v(k+3), ..., v(k+4m-1) are assigned weight 0 and all remaining vertices are assigned weight 1 will have the same Grundy numbers as G.
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REFERENCES
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E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.
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LINKS
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K. Barker, M. DeStefano, E. Fiorini, M. Gohn, J. Miller, J. Roeder, and T. W. H. Wong, Generalized Impartial Two-player Pebbling Games on K3 and C4, Journal of Integer Sequences, 27(5), 2024.
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PROG
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(Python) # See Cohen link.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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