Circular Nim games \(\mathrm{CN}(7,4)\). (English) Zbl 1490.91033
Summary: Circular Nim is a two-player impartial combinatorial game consisting of \(n\) stacks of tokens placed in a circle. A move consists of choosing \(k\) consecutive stacks and taking at least one token from one or more of the stacks. The last player able to make a move wins. The question of interest is: Who can win from a given position if both players play optimally? This question is answered by determining the set of \(\mathcal{P}\)-positions, those from which the next player is bound to lose, no matter what moves s/he makes. We will completely characterize the set of \(\mathcal{P}\)-positions for \(n= 7\) and \(k= 4\), adding to the known results for other games in this family. The interesting feature of the set of \(\mathcal{P}\)-positions of this game is that it splits into different subsets, unlike the structures for the previously solved games in this family.
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