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
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 . If  is an integer, the answer is  but the precise – or even asymptotic – amount of unfilled space for an arbitrary non-integer  is an open question. The smallest value of  that allows the packing of  unit squares is known when  is a perfect square (in which case it is ), as well as for 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  is , where  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 length less than . By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084 found by Walter Trump. Asymptotic results

For larger values of the side length , the exact number of unit squares that can pack an  square remains unknown. It is always possible to pack a  grid of axis-aligned unit squares,
but this may leave a large area, approximately , 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  (here written in little o notation). Later, Graham and Fan Chung further reduced the wasted space to . However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least . In particular, when  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  allows the packing of  unit squares, then it must be the case that 
and that a packing of  unit squares is also possible. Square packing 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 minimum solutions for n up to 12:

See also
Circle packing in a square
Squaring the square
Rectangle packing
Moving sofa problem

References

External links

 
Packing problems
Unsolved problems in geometry