OFFSET
1,2
COMMENTS
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 legal only provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
In the special case G = P(k+2), a (1^k, 0, 1)-weight assignment is one in which vertex v(k+1) is assigned weight 0 and all remaining vertices are assigned weight 1.
REFERENCES
E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.
LINKS
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.
Matthew Cohen, Python
E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6), 2021.
EXAMPLE
For n = 6, the {0,1}-Toggle game on P(8) with a (1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.
For n = 12, the {0,1}-Toggle game on P(14) with a (1,1,1,1,1,1,1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.
PROG
(Python) # See Cohen link.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Patrick G. Cesarz, Matthew Cohen, Eugene Fiorini, Joshua Lowrance, Emily Riley, Angel Pedro Torres and Andrew Woldar, Jul 20 2024
STATUS
approved