Maharaja Nim: Wythoff’s Queen Meets the Knight

We introduce the impartial game of Maharaja Nim, an extension of the classical game of Wytho↵ Nim. In the latter game, two players take turns in moving a single Queen of Chess on a large board, attempting to be the first to put her in the lower left corner, position (0, 0). Here, in addition to the classical rules, a player may also move the Queen as the Knight of Chess moves, still taking into consideration that, by moving no coordinate increases. We prove that the second player’s winning positions are close to those of Wytho↵Nim, namely, they are within a bounded distance to the half-lines, starting at the origin, of slope p 5+1 2 and p 5 1 2 respectively. This result is made possible via a generalization of a recent result of Fraenkel and Peled on complementary sequences of positive integers, and via an encoding of the patterns of P-positions by means of a certain dictionary process; thus we here also present two methods for analyzing games related to Wytho↵ Nim. Via Post’s tag productions, we also prove that, in general, such dictionary processes are algorithmically undecidable.