2010/04/29 by Larsson, Urban
#91A46 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1005.1555
In this paper we study a family of 2-pile Take Away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose sets of options and P-positions are \\0,t\| t∈ \N\ and \(t,t)| t∈ \M \ respectively. If we to 2-pile Nim adjoin the main-diagonal \(t,t)| t∈ \N\ as options, the new game is Wythoff Nim. It is well-known that the P-positions of this game lie on two 'beams' originating at the origin with slopes Φ= (1+√(5))/(2)>1 and \frac1Φ < 1. Hence one may think of this as if, in the process of going from Nim to Wythoff Nim, the set of P-positions has split and landed some distance off the main diagonal. This geometrical observation has motivated us to ask the following intuitive question. Does this splitting of the set of P-positions continue in some meaningful way if we, to the game of Wythoff Nim, adjoin some new generalized diagonal move, that is a move of the form \pt, qt\, where 0 < p < q are fixed positive integers and t > 0? Does the answer perhaps depend on the specific values of p and q? We state three conjectures of which the weakest form is: limt∈ \N(bt)/(at) exists, and equals Φ, if and only if (p, q) is a certain non-splitting pair, and where \\at, bt\\ represents the set of P-positions of the new game. Then we prove this conjecture for the special case (p,q) = (1,2) (a splitting pair). We prove the other direction whenever q / p < Φ. In the Appendix, a variety of experimental data is included, aiming to point out some directions for future work on GDWN games.