Extensions of Golomb’s Tromino Theorem

A polyomino is a finite edge–connected collection of equal-sized squares in the plane [1]. Often the stipulation is added that the union of the collection of squares have no holes. A tromino is a polyomino consisting of just three squares. One of the simplest and most beautiful theorems about polyominos is Solomon Golomb’s Tromino Theorem, which states that if you start with a chess board of size 2N x 2N and remove one of the squares, then the remaining board can always be covered by trominos of the type shown in Figure 1, which we call the “basic tromino.” In this paper we show how Golomb’s Theorem can be generalized to three and higher dimensions and then give versions of Golomb’s Theorem that hold on boards of size 3N x 3N and 4N x 4N .

By: Arthur Befumo, Jonathan Lenchner

Published in: RC25498 in 2014


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