Square packing in a square

Short description: Two-dimensional packing problem

Unsolved problem in mathematics: What is the asymptotic growth rate of wasted space for square packing in a half-integer square? (more unsolved problems in mathematics)

Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side $a$. If $a$ is an integer, the answer is $a^2$, but the precise, or even asymptotic, amount of wasted space for non-integer $a$ is an open question.

Small numbers of squares

5 unit squares in a square of side length $2+1/\sqrt{2}\approx 2.707$

10 unit squares in a square of side length $3+1/\sqrt{2}\approx 3.707$

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 less than $2+2\sqrt{4/5}\approx 3.789$. By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084, slightly improving a similar packing found earlier 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 $\mathrm{\Omega }(a^{1/2}(a-\lfloor a\rfloor ))$. 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

A related problem is that 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:

Number of squares	Circle radius
1	0.707...
2	1.118...
3	1.288...
4	1.414...
5	1.581...
6	1.688...
7	1.802...
8	1.978...
9	2.077...
10	2.121...
11	2.214...
12	2.236...

See also

• Circle packing in a square
• Squaring the square
• Rectangle packing
• Moving sofa problem

External links

• Friedman, Erich, Squares in Squares, Erich's Packing Center, https://erich-friedman.github.io/packing/squinsqu/