Square packing

Square packing is a packing problem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square or circle.

Square packing in a square
Square packing in a square is the problem of determining the maximum number of unit squares (squares of side length one) that can be packed inside a larger square of side length $$a$$. If $$a$$ is an integer, the answer is $$a^2,$$ but the precise – or even asymptotic – amount of unfilled space for an arbitrary non-integer $$a$$ is an open question.

The smallest value of $$a$$ that allows the packing of $$n$$ unit squares is known when $$n$$ is a perfect square (in which case it is $$\sqrt{n}$$), as well as for $$n={}$$2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 24, 34, 35, 46, 47, and 48. For most of these numbers (with the exceptions only of 5 and 10), the packing is the natural one with axis-aligned squares, and $$a$$ is $$\lceil\sqrt{n}\,\rceil$$, where $$\lceil\,\ \rceil$$ is the ceiling (round up) function. 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 length less than $$\textstyle 2+2\sqrt{4/5} \approx 3.789$$. By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084 found by Walter Trump.

Asymptotic results
For larger values of the side length $$a$$, the exact number of unit squares that can pack an $$a\times a$$ square remains unknown. It is always possible to pack a $$\lfloor a\rfloor \!\times\! \lfloor a\rfloor$$ grid of axis-aligned unit squares, but this may leave a large area, approximately $$2a(a-\lfloor a\rfloor)$$, 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 $$o(a^{7/11})$$ (here written in little o notation). Later, Graham and Fan Chung further reduced the wasted space to $$O(a^{3/5})$$. However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least $$\Omega\bigl(a^{1/2}(a-\lfloor a\rfloor)\bigr)$$. In particular, when $$a$$ 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 $$a\times a$$ allows the packing of $$n^2-2$$ unit squares, then it must be the case that $$a\ge n$$ and that a packing of $$n^2$$ unit squares is also possible.

Square packing in a circle
Square packing in a circle is a related problem of packing n unit squares into a circle with radius as small as possible. For this problem, good solutions are known for n up to 35. Here are minimum solutions for n up to 12: