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Question, Web Fundamentals.svg Unsolved problem in mathematics:
What is the asymptotic growth rate of wasted space for square packing in a half-integer square?
(more unsolved problems in mathematics)

Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side . If is an integer, the answer is , but the precise, or even asymptotic, amount of wasted space for non-integer is an open question.[1]


Small numbers of squaresEdit

5 unit squares in a square of side length  
10 unit squares in a square of side length  

The smallest value of   that allows the packing of   unit squares is known when   is a perfect square (in which case it is  ), as well as for  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   is  .[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 less than  . By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084, slightly improving a similar packing found earlier by Walter Trump.[6]

Asymptotic resultsEdit

For larger values of the side length  , the exact number of unit squares that can pack an   square remains unknown. It is always possible to pack a   grid of axis-aligned unit squares, but this may leave a large area, approximately  , uncovered and wasted.[4] Instead, Paul Erdős and Ronald Graham showed that for a different packing by tilted unit squares, the wasted space could be significantly reduced to   (here written in little o notation).[7] In an unpublished manuscript, Graham and Fan Chung further reduced the wasted space, to  .[8] However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least  . In particular, when   is a half-integer, the wasted space is at least proportional to its square root.[9] The precise asymptotic growth rate of the wasted space, even for half-integer side lengths, remains an open problem.[1]

Some numbers of unit squares are never the optimal number in a packing. In particular, if a square of size   allows the packing of   unit squares, then it must be the case that   and that a packing of   unit squares is also possible.[2]

See alsoEdit


  1. ^ a b Brass, Peter; Moser, William; Pach, János (2005), Research Problems in Discrete Geometry, New York: Springer, p. 45, ISBN 978-0387-23815-9, MR 2163782
  2. ^ a b 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.
  3. ^ Bentz, Wolfram (2010), "Optimal packings of 13 and 46 unit squares in a square", Electronic Journal of Combinatorics, 17 (1), Research Paper 126, MR 2729375
  4. ^ a b Friedman, Erich (2009), "Packing unit squares in squares: a survey and new results", Electronic Journal of Combinatorics, Dynamic Survey 7, MR 1668055.
  5. ^ Stromquist, Walter (2003), "Packing 10 or 11 unit squares in a square", Electronic Journal of Combinatorics, 10, Research Paper 8, MR 2386538.
  6. ^ Gensane, Thierry; Ryckelynck, Philippe (2005), "Improved dense packings of congruent squares in a square", Discrete & Computational Geometry, 34 (1): 97–109, doi:10.1007/s00454-004-1129-z, MR 2140885
  7. ^ 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.
  8. ^ *Chung, Fan; Graham, Ron, Efficient packings of unit squares in a large square (PDF), retrieved 2019-04-28
  9. ^ 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.

External linksEdit