Square packing in a square - Alchetron, the free social encyclopedia

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:

Other results:

If it is possible to pack n² − 2 unit squares in a square of side a, then a ≥ n.

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.

The wasted space of an optimal solution is asymptotically o(a^7/11).

All solutions must waste space at least Ω(a^1/2) for some values of a.

11 unit squares cannot be packed in a square of side less than 2 + 2 4 / 5 .

References

Square packing in a square Wikipedia

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