Maximum Nim and Chocolate Bar Games: Discrete and Computational Geometry, Graphs, and Games - Miyadera, Ryohei; Kannan, Sohta; Manabe, Hikaru (Thai Journal of Mathematics, 2023)
The authors studied maximum Nim wherein each player can remove at most f(m) stones from a pile of m stones, where f(m)= md ceil or f(m)= m+1dfloor with a positive number d that is larger than 1, and present function h_d such that G(h_d(x)) = G(x), where G(x) is the Grundy number of the pile of x stones. The authors apply function h_d to the study of chocolate bar games, wherein two players take turns and cut a chocolate bar in a straight line and eat the pieces. These games can be considered the sum of two maximum Nim with a restriction on the size of the chocolate bar that can be eaten by the players. Therefore, using h_d, the authors devise formulas to calculate the winning position of the previous player for a rectangular chocolate bar game with a restriction on the size of the chocolate bar piece to be eaten. In addition, the authors present conjectures about rectangular chocolate bar games with some missing parts.