2018/08/27 by Fritsch, Kristin, Heuer, Janin, Sanyal, Raman +1
#05Axx #52B12 #52B20 #52B35 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1808.08797
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.