Unfolding and dissection of multiple cubes

A polyomino is a “simply connected” set of unit squares introduced by Solomon W. Golomb in 1954. Since then, a set of polyominoes has been playing an important role in puzzle society (see, e.g., [3, 1]). In Figure 82 in [1], it is shown that a set of 12 pentominoes exactly covers a cube that is the square root of 10 units on the side. In 1962, Golomn also proposed an interesting notion called “rep-tiles”: a polygon is a reptile of order k if it can be divided into k replicas congruent to one another and similar to the original (see [2, Chap 19]). These notions lead us to the following natural question: is there any polyomino that can be folded to a cube and divided into k polyominoes such that each of them can be folded to a (smaller) cube for some k? That is, a polyomino is a rep-cube of order k if it is a net of a cube, and it can be divided into k polyominoes such that each of them can be folded to a cube. If each of these k polyominoes has the same size, we call the original polyomino regular rep-cube of order k. In this paper, we give an affirmative answer. We first give some regular rep-cubes of order k for some specific k. Based on this idea, we give a constructive proof for a series of regular rep-cubes of order 36gk'^2 for any positive integer k' and an integer g in {2; 4; 5; 8; 10; 50}. That is, there are infinitely many k that allow regular rep-cubes of order k. We also give some non-regular rep-cubes and its variants.