For linear orders \(A\) and \(B\), their ordered sum \(A + B\) is obtained by placing a copy of \(B\) to the right of \(A\). This operation is non-commutative in general, so one might ask: which pairs of linear orders do additively commute? That is, for which linear orders \(A\) and \(B\) do we have \(A + B \cong B + A\)?
Tarski studied this question seriously at various times in his career from the 1920s through the 1950s. In this talk, I'll discuss recent work with Eric Paul in which we use a novel Euclidean algorithm for pairs of linear orders that "almost" additively commute to solve two problems posed by Tarski in 1956. Along the way, I'll draw connections between the classical combinatorial perspective for studying sums of linear orders and a more modern perspective that translates such questions into the language of generalized interval exchange transformations.