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 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 ${\displaystyle a}$. If ${\displaystyle a}$ is an integer, the answer is ${\displaystyle a^{2},}$ but the precise – or even asymptotic – amount of unfilled space for an arbitrary non-integer ${\displaystyle a}$ is an open question.^[1]

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

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

11 unit squares in a square of side length ${\displaystyle a\approx 3.877084}$

The smallest value of ${\displaystyle a}$ that allows the packing of ${\displaystyle n}$ unit squares is known when ${\displaystyle n}$ is a perfect square (in which case it is ${\displaystyle {\sqrt {n}}}$), as well as for ${\displaystyle 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 ${\displaystyle a}$ is ${\displaystyle \lceil {\sqrt {n}}\,\rceil }$, where ${\displaystyle \lceil \,\ \rceil }$ 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 ${\displaystyle n=11}$. It is known that 11 unit squares cannot be packed in a square of side length less than ${\displaystyle \textstyle 2+2{\sqrt {4/5}}\approx 3.789}$. 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 ${\displaystyle n=17}$. It was discovered in 1998 by John Bidwell, an undergraduate student at the University of Hawaiʻi, and has side length ${\displaystyle a\approx 4.6756}$.^[4]

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

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

For larger values of the side length ${\displaystyle a}$, the exact number of unit squares that can pack an ${\displaystyle a\times a}$ square remains unknown. It is always possible to pack a ${\displaystyle \lfloor a\rfloor \!\times \!\lfloor a\rfloor }$ grid of axis-aligned unit squares, but this may leave a large area, approximately ${\displaystyle 2a(a-\lfloor a\rfloor )}$, 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 ${\displaystyle o(a^{7/11})}$ (here written in little o notation).^[8] Later, Graham and Fan Chung further reduced the wasted space to ${\displaystyle O(a^{3/5})}$.^[9] However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least ${\displaystyle \Omega {\bigl (}a^{1/2}(a-\lfloor a\rfloor ){\bigr )}}$. In particular, when ${\displaystyle a}$ 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 ${\displaystyle a\times a}$ allows the packing of ${\displaystyle n^{2}-2}$ unit squares, then it must be the case that ${\displaystyle a\geq n}$ and that a packing of ${\displaystyle n^{2}}$ unit squares is also possible.^[2]

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

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

• Friedman, Erich, "Squares in Squares", Github, Erich's Packing Center