A374920 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+6) = v(1)v(2)...v(k+6) with a (1^k,0,1,0,1,0,1)-weight assignment is a second-player winning game.

1, 6, 7, 12, 13, 18, 23, 24, 39, 44, 45, 50, 51, 57, 62, 77, 115, 281, 319, 350, 389

Offset: 1

Author

Patrick G. Cesarz, Matthew Cohen, Eugene Fiorini, Joshua Lowrance, Emily Riley, Angel Pedro Torres and Andrew Woldar, Jul 20 2024

Keywords

• nonn
• more

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 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+6), a (1^k,0,1,0,1,0,1)-weight assignment is one in which vertices v(k+1), v(k+3), and v(k+5) are assigned weight 0 and all remaining vertices are assigned weight 1.

The path P(k+4m+2) 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.

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.

Crossrefs

Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374810, A374910

Programs

Python

# See Cohen link.