Square packing in a square


Square packing in a square is a packing problem where the objective is to determine how many squares of side one can be packed into a square of side. If is an integer, the answer is, but the precise, or even asymptotic, amount of wasted space for non-integer is an open question.

Small numbers of squares

The smallest value of that allows the packing of unit squares is known when is a perfect square, as well as for 2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 24, 34, 35, 46, 47, and 48. For most of these numbers, the packing is the natural one with axis-aligned squares, and is.
The figure shows the optimal packings for 5 and 10 squares, the two smallest numbers of squares for which the optimal packing involves tilted squares.
The smallest unresolved case involves packing 11 unit squares into a larger square.
11 unit squares cannot be packed in a square of side less than. By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084,
slightly improving a similar packing found earlier by Walter Trump.

Asymptotic results

For larger values of the side length, the exact number of unit squares that can pack an square remains unknown.
It is always possible to pack a grid of axis-aligned unit squares,
but this may leave a large area, approximately, uncovered and wasted.
Instead, Paul Erdős and Ronald Graham showed that for a different packing by tilted unit squares, the wasted space could be significantly reduced to .
In an unpublished manuscript, Graham and Fan Chung further reduced the wasted space, to.
However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least. In particular, when is a half-integer, the wasted space is at least proportional to its square root. The precise asymptotic growth rate of the wasted space, even for half-integer side lengths, remains an open problem.
Some numbers of unit squares are never the optimal number in a packing. In particular,
if a square of size allows the packing of unit squares, then it must be the case that
and that a packing of unit squares is also possible.