4-tangrams are 4-avoidable

Abstract

A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $k\ge 1$, let $t(k)$ denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most~$k$. The existence of infinite ternary square-free words shows that $t(1)=t(2)=3$. We show that $t(3)=t(4)=4$, answering a question from Dębski, Grytczuk, Pawlik, Przybyło, and Śleszyńska-Nowak.