We show that it is possible to use a greedy algorithm to test whether a set of cubes forms a solution. For every n≥2 we find at least n perfect solutions, and we use our greedy algorithm to count the perfect solutions for n≤6. Perfect solutions turn out to be precisely those which can be reached using only balanced moves. Every perfect solution corresponds naturally to a Latin square. However, we show that almost all Latin squares do not correspond to solutions.

We construct an infinite family of imperfect solutions and show that the total size of its three orthogonal projections is asymptotic to the minimum possible value.

Our results solve several conjectures posed in a proceedings paper by Barát, Korondi and Varga.

If our dismantling process is reversed we get a build-up process very similar to well-studied models of bootstrap percolation.