Square packing in a square

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 . If is an integer, the answer is , but the precise, or even asymptotic, amount of wasted space for non-integer is an open question.

Small numbers of squares

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 involves packing 11 unit squares into a larger square.
11 unit squares cannot be packed in a square of side 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,
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 $o(a^)$ (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..

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:

See also

*Circle packing in a square
*Squaring the square
*Rectangle packing

References

{{reflist

External links

Squares in Squares
Erich's Packing Center, Erich Friedman, Stetson University

Packing problems
Unsolved problems in geometry