Abstract In the chapter “Magic with a Matrix” in Hexaflexagons and Other Mathematical Diversions (1988); Martin Gardner describes a delightful “party trick” to fill the squares of a d-by-d chessboard with nonnegative integers such that the sum of the numbers covered by any placement of d nonthreatening rooks is a given number N. We consider such chessboards from a geometric perspective which gives rise to a family of lattice polytopes. The polyhedral structure of these Gardner polytopes explains the underlying trick and enables us to count such chessboards for given N in three different ways. We also observe a curious duality that relates Gardner polytopes to Birkhoff polytopes

中文翻译：

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马丁加德纳多面体

摘要 在 Hexaflexagons and Other Mathematical Diversions (1988) 的“Magic with a Matrix”一章中；Martin Gardner 描述了一个令人愉快的“派对技巧”，用非负整数填充 d-by-d 棋盘的方格，使得 d 个非威胁车的任何放置所覆盖的数字总和是给定的数字 N。我们考虑这样的棋盘产生格子多胞体家族的几何透视图。这些加德纳多面体的多面体结构解释了潜在的技巧，并使我们能够以三种不同的方式计算给定 N 的此类棋盘。我们还观察到一种奇怪的二元性，它将加德纳多胞体与伯克霍夫多胞体联系起来