The Martin Gardner Polytopes

TL;DR

This paper explores a geometric perspective on Martin Gardner's chessboard trick, introducing Gardner polytopes, analyzing their structure, counting configurations, and revealing a duality with Birkhoff polytopes.

Contribution

It defines Gardner polytopes from a geometric viewpoint, provides methods to count configurations, and uncovers a duality with Birkhoff polytopes, advancing understanding of these combinatorial structures.

Findings

Polyhedral structure explains the chessboard trick.

Three methods to count configurations for given N.

Discovery of a duality with Birkhoff polytopes.

Abstract

In the chapter "Magic with a Matrix" in \emph{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.