# Square packing in a square

 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)

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

## Small numbers of squares

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 ${\displaystyle 3+1/{\sqrt {2}}\approx 3.707}$

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 ${\displaystyle \lceil {\sqrt {n}}\rceil }$ .[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 involves packing 11 unit squares into a larger square. 11 unit squares cannot be packed in a square of side 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, slightly improving a similar packing found earlier by Walter Trump.[6]

## Asymptotic results

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).[7] In an unpublished manuscript, Graham and Fan Chung further reduced the wasted space, to ${\displaystyle O(a^{3/5})}$ .[8] 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.[9] 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]