When can you tile a box with translates of two given rectangular bricks? (Q1883692)

Summary: When can a \(d\)-dimensional rectangular box \(R\) be tiled by translates of two given \(d\)-dimensional rectangular bricks \(B_1\) and \(B_2\)? We prove that \(R\) can be tiled by translates of \(B_1\) and \(B_2\) if and only if \(R\) can be partitioned by a hyperplane into two subboxes \(R_1\) and \(R_2\) such that \(R_i\) can be tiled by translates of the brick \(B_i\) alone \((i= 1,2)\). Thus an obvious sufficient condition for a tiling is also a necessary condition. (However, there may be tilings that do not give rise to a bipartition of \(R\).) There is an equivalent formulation in terms of the (not necessarily integer) edge lengths of \(R\), \(B_1\), and \(B_2\). Let \(R\) be of size \(z_1\times z_2\times\cdots\times z_d\), and let \(B_1\) and \(B_2\) be of respective sizes \(v_1\times v_2\times\cdots\times v_d\) and \(w_1\times w_2\times\cdots\times w_d\). Then there is a tiling of the box \(R\) with translates of the bricks \(B_1\) and \(B_2\) if and only if (a) \(z_i/v_i\) is an integer for \(i= 1,2,\dots, d\); or (b) \(z_i/w_i\) is an integer for \(i= 1,2,\dots, d\); or (c) there is an index \(k\) such that \(z_i/v_i\) and \(z_i/w_i\) are integers for all \(i\neq k\), and \(z_k= \alpha v_k+ \beta w_k\) for some nonnegative integers \(\alpha\) and \(\beta\). Our theorem extends some well-known results (due to de Bruijn and Klarner) on tilings of rectangles by rectangles with integer edge lengths.