A Generalized Diagonal Wythoff Nim.

The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope and respectively. We study extensions to this game where a player may also remove simultaneously tokens from either of the piles and from the other, where are given positive integers and where t ranges over the positive integers. We prove that for certain pairs ( p, q) the P-positions are identical to those of Wythoff Nim, but for they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs ( p, q) for which Wythoff Nim's beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim. [ABSTRACT FROM AUTHOR]