Covering a square with consecutive squares

In this article we address the following problem. Given are a \(1\times 1\) square, a \(2\times 2\) square, and so on, finally a \(n\times n\) square. What is the biggest square that can be covered completely by this given set of “small” squares? It is assumed that the small squares must stand parallel to the sides of the big square, and overlap is allowed. In contrast to the packing version of the problem (asking for the smallest square that can accommodate all small squares without overlap) which has been studied in several papers since the 1960’s, the covering version of the problem seems new. We construct optimal coverings for small values of n. For moderately bigger n values we solve the problem optimally by a commercial mathematical programming solver, and for even bigger n values we give a heuristic algorithm that can find near optimal solutions. We also provide an expansion-algorithm, that from a given good cover using consecutive squares up to size n, can generate a cover for a larger square using small squares up to size \(n+1\). Finally we prove that a simple covering policy can generate an asymptotically optimal covering.