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Cubic chess

Source: 2022 Mexican Mathematics Olympiad P2

November 8, 2022
3dchesscombinatorics

Problem Statement

Let nn be a positive integer. David has six n×nn\times n chessboards which he arranges in an n×n×nn\times n\times n cube. Two cells are "aligned" if they can be connected by a path of cells a=c1, c2, …, cm=ba=c_1,\ c_2,\ \dots,\ c_m=b such that all consecutive cells in the path share a side, and the sides that the cell cic_i shares with its neighbors are on opposite sides of the square for i=2, 3, … m−1i=2,\ 3,\ \dots\ m-1.
Two towers attack each other if the cells they occupy are aligned. What is the maximum amount of towers he can place on the board such that no two towers attack each other?
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