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{n}), 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{n}\,\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+\frac{4}{\sqrt{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. 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.
Asymptotic results , an optimal packing for squares 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. Later, Graham and
Fan Chung further reduced the wasted space to O(a^{0.631}), and subsequent work reduced the wasted space to O(a^{0.625}), and then O(a^{0.6}). However, as
Klaus Roth and
Bob Vaughan proved, all solutions must waste space at least \Omega\bigl((a \cdot |a-\operatorname{round} a|)^{1/2}\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. == In a circle ==