Square packing

Template:Short description Template:CS1 config 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.

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.

Template:Multiple image 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 is $n=11$. It is known that 11 unit squares cannot be packed in a square of side length less than $2+\frac{4}{\sqrt{5}}\approx 3.789$. By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084 found by Walter Trump.

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 $a\approx 4.6756$.

A mutilated chessboard, an optimal packing for Template:Math squares

Below are the minimum solutions for values up to $n=16$; the case $n=11$ remains unresolved.

Number of unit squares $n$	Minimal side length $a$ of big square
1	1
2, 3, 4	2
5	$2+\frac{\sqrt{2}}{2}\approx 2.707\dots$
6, 7, 8, 9	3
10	$3+\frac{\sqrt{2}}{2}\approx 3.707\dots$
11	3.877... ?
12, ..., 16	4

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.

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 $O(a^{7/11})\subseteq o(a^{0.637})$ (the latter written in little o notation). Later, Graham and Fan Chung further reduced the wasted space to $O(a^{0.631})$, and subsequent work reduced the wasted space to $O(a^{0.625})$, and then $O(a^{0.6})$. However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least $\mathrm{\Omega }((a\cdot |a-\mathrm{round}a|{)}^{1/2})$. 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.

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 $n=12$ (although only the cases $n=1,2$ and $n=4$ are known to be optimal):

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

In other shapes

Packing squares into other shapes can have high computational complexity: testing whether a given number of axis-parallel unit squares can fit into a given polygon is NP-complete. It remains NP-complete even for a simple polygon (with no holes) that is orthogonally convex, with axis-parallel sides, and with half-integer vertex coordinates.

See also

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

External links

• Template:Citation