# 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 1 (unit squares) can be packed into a square of side a. Obviously, if a is an integer, the answer is a2, but the precise, or even asymptotic, amount of wasted space for non-integer a is an open question.

Proven minimum solutions:[1]

Number of squares Square size
1 1
2 2
3 2
4 2
5 2.707 (2 + 2 −1/2)
6 3
7 3
8 3
9 3
10 3.707 (3 + 2 −1/2)

Other results:

• If it is possible to pack n2 − 2 unit squares in a square of side a, then an.[2]
• The naive approach in which all squares are parallel to the coordinate axes, and are placed touching edge-to-edge, leaves wasted space of less than 2a + 1.[1]
• The wasted space of an optimal solution is asymptotically o(a7/11).[3]
• All solutions must waste space at least Ω(a1/2) for some values of a.[4]
• 11 unit squares cannot be packed in a square of side less than ${\displaystyle 2+2{\sqrt {4/5}}}$.[5]