https://erich-friedman.github.io/packing/squinsqu/ Squares in Squares The following pictures show n unit squares packed inside the smallest known square (of side length s). For the n not pictured, the trivial packing (with no tilted squares) is the best known packing. --------------------------------------------------------------------- 1. 2. 3. [s1] [s2] [s3] s = 2 s = 2 s = 1 Proved by Frits Proved by Frits Trivial. Gobel Gobel in 1979. in 1979. --------------------------------------------------------------------- 4. 5. 6. [s4] [s5] [s6] s = 2 + 1 / [?]2 = s = 3 s = 2 2.707+ Proved by Michael Trivial. Proved by Frits Kearney Gobel and Peter Shiu in in 1979. April 2002. --------------------------------------------------------------------- 7. 8. 9. [s7] [s8] [s9] s = 3 s = 3 Proved by Erich Proved by Erich s = 3 Friedman Friedman Trivial. in 1999. in 1999. --------------------------------------------------------------------- 10. 11. 14. [s10] [s11] [s14] s = 3 + 1 / [?]2 = s = 3.877+ s = 4 3.707+ Found by Proved by Erich Proved by Walter Walter Trump Friedman Stromquist in 1979. in 1999. in 2003. --------------------------------------------------------------------- 15. 17. 18. [s15] [s17] [s18] s = 4 s = 4.675+ s = (7 + [?]7) / 2 Proved by Erich Found by John = 4.822+ Friedman Bidwell Found by Pertti in 1999. in 1997. Hamalainen in 1979. --------------------------------------------------------------------- 19. 24. 26. [s19] [s24] [s26] s = 3 + 4 [?]2 / s = 5 s = 7 / 2 + 3 / 3 = 4.885+ Proved by [?]2 = 5.621+ Found by Robert Erich Friedman Found by Erich Wainwright in 1999. Friedman in 1979. in 1997. --------------------------------------------------------------------- 27. 28. 29. [s27] [s28] [s29] s = 5 + 1 / [?]2 s = 3 + 2 [?]2 = s = 5.934+ = 5.707+ 5.828+ Found by Thierry Found by Frits Found by Frits Gensane Gobel Gobel and Philippe in 1979. in 1979. Ryckelynck in April 2004. --------------------------------------------------------------------- 37. 38. 39. [s37] [s38] [s39] s = 6.598+ s = 6 + 1 / [?]2 s = 6.818+ Found by David = 6.707+ Found by David W. Cantrell Found by Frits W. Cantrell in September Gobel in August 2002. 2002. in 1979. --------------------------------------------------------------------- 40. 41. 50. [s40] [s41] [s50] s = 4 + 2 [?]2 s = 6.937+ s = 7.598+ = 6.828+ Found by Joe Found by David W. Found by DeVincentis Cantrell Frits Gobel in April 2014. in September in 1979. 2002. --------------------------------------------------------------------- 51. 52. 53. [s51] [s52] [s53] s = 7.704+ s = 7 + 1 / [?]2 s = 7.823+ Found by = 7.707+ Found by David W. Karoly Hajba Found by Frits Cantrell in July 2009. Gobel in September in 1979. 2002. --------------------------------------------------------------------- 54. 55. 65. [s54] [s55] [s65] s = 7.846+ s = 7.954+ s = 5 + 5 / [?]2 Found by Joe Found by Joe = 8.535+ DeVincentis DeVincentis Found by Frits in April 2014. in April 2014. Gobel in 1979. --------------------------------------------------------------------- 66. 67. 68. [s66] [s67] [s68] s = 3 + 4 [?]2 = s = 8 + 1 / [?]2 s = 15/2 + [?]7/2 8.657+ = 8.707+ = 8.822+ Found by Evert Found by Frits Found by David Stenlund Gobel W. Cantrell in 1980. in 1979. in September 2002. --------------------------------------------------------------------- 69. 70. 71. [s69] [s70] [s71] s = 8.828+ s = 8.881+ s = 8.960+ Found by Found by Joe Found by Joe Maurizio DeVincentis DeVincentis Morandi in April 2014. in April 2014. in June 2010. --------------------------------------------------------------------- 82. 83. 84. [s82] [s83] [s84] s = 6 + 5 / [?]2 s = 4 + 4 [?]2 = s = 9 + 1 / [?]2 = 9.535+ 9.657+ = 9.707+ Found by Frits Found by Evert Found by Frits Gobel Stenlund Gobel in 1979. in 1980. in 1979. --------------------------------------------------------------------- 85. 86. 87. [s85] [s86] [s87] s = 11 / 2 + 3 s = 17 / 2 + [?]7 s = 9.851+ [?]2 = 9.742+ / 2 = 9.822+ Found by David Found by Erich Found by Erich W. Cantrell Friedman Friedman in August 2002. in 1997. in 1997. --------------------------------------------------------------------- 88. 89. [s88] [s89] s = 9.901+ s = 5 + 7 / [?]2 = Found by David W. 9.950+ Cantrell Found by Evert in August 2002. Stenlund in 1980. --------------------------------------------------------------------- For more details, see my paper on the subject: Packing Unit Squares in Squares: A Survey and New Results.