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.^[1]

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

11 unit squares in a square of side length

a\approx 3.877084

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.^[2]^[3] 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.^[4]^[5]

The smallest unresolved case is $n=11$ . It is known that 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.^[4]^[6]

The smallest case where the best known packing involves squares at three different angles is $n=17$ . It was discovered in 1998 by John Bidwell, an undergraduate student at the University of Hawaiʻi, and has side length $a\approx 4.6756$ .^[4]

Below are the minimum solutions for values up to n=12:^[7] (The case for n=11 remains unresolved)

Number of unit squares $n$	Minimal side length $a$ of big square
1	1
2	2
3	2
4	2
5	2.707...	$2+{\frac {\sqrt {2}}{2}}$
6	3
7	3
8	3
9	3
10	3.707...	$3+{\frac {\sqrt {2}}{2}}$
11	3.877... ?
12	4

Asymptotic results

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)

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.^[4] 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).^[8] Later, Graham and Fan Chung further reduced the wasted space to $O(a^{3/5})$ .^[9] 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.^[10] The precise asymptotic growth rate of the wasted space, even for half-integer side lengths, remains an open problem.^[1]

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 $a\geq n$ and that a packing of $n^{2}$ unit squares is also possible.^[2]

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 the minimum known solutions for up to n=12:^[11] (Only the cases n=1 and n=2 are known to be optimal)

Number of squares	Circle radius
1	${\frac {\sqrt {2}}{2}}$ ≈ 0.707...
2	${\frac {\sqrt {5}}{2}}$ ≈ 1.118...
3	${\frac {5{\sqrt {17}}}{16}}$ ≈ 1.288...
4	${\sqrt {2}}$ ≈ 1.414...
5	${\frac {\sqrt {10}}{2}}$ ≈ 1.581...
6	1.688...
7	${\frac {\sqrt {13}}{2}}$ ≈ 1.802...
8	1.978...
9	${\frac {\sqrt {1105}}{16}}$ ≈ 2.077...
10	${\frac {3{\sqrt {2}}}{2}}$ ≈ 2.121...
11	2.214...
12	${\sqrt {5}}$ ≈ 2.236...