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

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

11 unit squares in a square of side length ${\displaystyle a\approx 3.877084}$

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 ${\displaystyle \lceil {\sqrt {n}}\,\rceil }$, where ${\displaystyle \lceil \,\ \rceil }$ is the ceiling (round up) function.^[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 length 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 found by Walter Trump.^[6][4]

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:^[10]

See also

• Circle packing in a square
• Squaring the square
• Rectangle packing
• Moving sofa problem

References

External links

• Friedman, Erich, "Squares in Squares", Github, Erich's Packing Center