Square Packing In A Square

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.

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 $\sqrt$), 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 $\lceil\sqrt\,\rceil$, 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 $\textstyle 2+2\sqrt \approx 3.789$. 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 $a\approx 4.6756$.

Below are the minimum solutions for values up to $n = 12$; the case $n=11$ remains unresolved:

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 $o(a^)$ (here written in little o notation).
Later, Graham and Fan Chung further reduced the wasted space to $O(a^)$.
However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least $\Omega\bigl(a^(a-\lfloor a\rfloor)\bigr)$. 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.

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 $n = 1$ and $n = 2$ are known to be optimal):

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

References

External links

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{{Packing_problems

Packing problems
Unsolved problems in geometry