Maharaja Nim: Wythoff's Queen meets the Knight

Larsson, Urban,1965Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper, matematik,Department of Mathematical Sciences, Mathematics (författare)

Maharaja Nim : Wythoff's Queen meets the Knight

We relax the hypothesis of a recent result of A. S. Fraenkel and U. Peled on certain complementary sequences of positive integers. The motivation is to understand to asymptotic behavior of the impartial game of \emph{Maharaja Nim}, an extension of the classical game of Wythoff Nim. In the latter game, two players take turn 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 Wythoff Nim, namely they are within a bounded distance to the half-lines, starting at the origin, of slope $\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$ respectively. We encode the patterns of the P-positions by means of a certain \emph{dictionary process}, thus introducing a new method for analyzing games related to Wythoff Nim. Via Post's Tag productions, we also prove that, in general, such dictionary processes are algorithmically undecidable.

Wästlund, Johan,1971Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper, matematik,Department of Mathematical Sciences, Mathematics (författare)
Göteborgs universitetInstitutionen för matematiska vetenskaper, matematik (creator_code:org_t)

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