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. 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.

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 √n), as well as for n = 2, 3, 5, 6, 7, 8, 10, 14, 15, 24, and 35. The table below shows the optimal value of a for n ≤ 10, with some example packings.[1][2]

Number of squares Square size
1 1
2 2
3 2
4 2
5 2 + 1/√2 ≈ 2.707

6 3
7 3

8 3
9 3
10 3 + 1/√2 ≈ 3.707

Other results that do not establish exact optimal packings are known. For example:

  • If it is possible to pack n2 − 2 unit squares in a square of side a, then a ≥ n.[3]
  • 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 in a square of side a.[1]
  • The wasted space of an optimal solution is asymptotically o(a7/11) (here written in little o notation).[4]
  • All solutions must waste space at least Ω(a1/2) for some values of a.[5]
  • 11 unit squares cannot be packed in a square of side less than . By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.8772.[2]

See also

References

  1. 1 2 Friedman, Erich (2009), "Packing unit squares in squares: a survey and new results", Electronic Journal of Combinatorics, Dynamic Survey 7, MR 1668055 .
  2. 1 2 Stromquist, Walter (2003), "Packing 10 or 11 unit squares in a square", Electronic Journal of Combinatorics, 10, Research paper 8, 11pp., MR 2386538 .
  3. ↑ Kearney, Michael J.; Shiu, Peter (2002), "Efficient packing of unit squares in a square", Electronic Journal of Combinatorics, 9 (1), Research Paper 14, 14 pp., MR 1912796 .
  4. ↑ Erdős, P.; Graham, R. L. (1975), "On packing squares with equal squares" (PDF), Journal of Combinatorial Theory, Series A, 19: 119–123, doi:10.1016/0097-3165(75)90099-0, MR 0370368 .
  5. ↑ Roth, K. F.; Vaughan, R. C. (1978), "Inefficiency in packing squares with unit squares", Journal of Combinatorial Theory, Series A, 24 (2): 170–186, doi:10.1016/0097-3165(78)90005-5, MR 0487806 .


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