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. |
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| s = 1 Trivial. |
s = 2 Proved by Frits Göbel in 1979. |
s = 2 Proved by Frits Göbel in 1979. |
4. |
5. |
6. |
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| s = 2 Trivial. |
s = 2 + 1 / √2 = 2.707+ Proved by Frits Göbel in 1979. |
s = 3 Proved by Michael Kearney and Peter Shiu in April 2002. |
7. |
8. |
9. |
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| s = 3 Proved by Erich Friedman in 1999. |
s = 3 Proved by Erich Friedman in 1999. |
s = 3 Trivial. |
10. |
11. |
14. |
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| s = 3 + 1 / √2 = 3.707+ Proved by Walter Stromquist in 2003. |
s = 3.877+ Found by Walter Trump in 1979. |
s = 4 Proved by Erich Friedman in 1999. |
15. |
17. |
18. |
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| s = 4 Proved by Erich Friedman in 1999. |
s = 4.675+ Found by John Bidwell in 1997. |
s = (7 + √7) / 2 = 4.822+ Found by Pertti Hamalainen in 1979. |
19. |
24. |
26. |
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| s = 3 + 4 √2 / 3 = 4.885+ Found by Robert Wainwright in 1979. |
s = 5 Proved by Erich Friedman in 1999. |
s = 7 / 2 + 3 / √2 = 5.621+ Found by Erich Friedman in 1997. |
27. |
28. |
29. |
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| s = 5 + 1 / √2 = 5.707+ Found by Frits Göbel in 1979. |
s = 3 + 2 √2 = 5.828+ Found by Frits Göbel in 1979. |
s = 5.934+ Found by Thierry Gensane and Philippe Ryckelynck in April 2004. |
37. |
38. |
39. |
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| s = 6.598+ Found by David W. Cantrell in September 2002. |
s = 6 + 1 / √2 = 6.707+ Found by Frits Göbel in 1979. |
s = 6.818+ Found by David W. Cantrell in August 2002. |
40. |
41. |
50. |
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| s = 4 + 2 √2 = 6.828+ Found by Frits Göbel in 1979. |
s = 6.937+ Found by Joe DeVincentis in April 2014. |
s = 7.598+ Found by David W. Cantrell in September 2002. |
51. |
52. |
53. |
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| s = 7.704+ Found by Károly Hajba in July 2009. |
s = 7 + 1 / √2 = 7.707+ Found by Frits Göbel in 1979. |
s = 7.823+ Found by David W. Cantrell in September 2002. |
54. |
55. |
65. |
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| s = 7.846+ Found by Joe DeVincentis in April 2014. |
s = 7.954+ Found by Joe DeVincentis in April 2014. |
s = 5 + 5 / √2 = 8.535+ Found by Frits Göbel in 1979. |
66. |
67. |
68. |
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| s = 3 + 4 √2 = 8.657+ Found by Evert Stenlund in 1980. |
s = 8 + 1 / √2 = 8.707+ Found by Frits Göbel in 1979. |
s = 15/2 + √7/2 = 8.822+ Found by David W. Cantrell in September 2002. |
69. |
70. |
71. |
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| s = 8.828+ Found by Maurizio Morandi in June 2010. |
s = 8.881+ Found by Joe DeVincentis in April 2014. |
s = 8.960+ Found by Joe DeVincentis in April 2014. |
82. |
83. |
84. |
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| s = 6 + 5 / √2 = 9.535+ Found by Frits Göbel in 1979. |
s = 4 + 4 √2 = 9.657+ Found by Evert Stenlund in 1980. |
s = 9 + 1 / √2 = 9.707+ Found by Frits Göbel in 1979. |
85. |
86. |
87. |
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| s = 11 / 2 + 3 √2 = 9.742+ Found by Erich Friedman in 1997. |
s = 17 / 2 + √7 / 2 = 9.822+ Found by Erich Friedman in 1997. |
s = 9.851+ Found by David W. Cantrell in August 2002. |
88. |
89. |
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| s = 9.901+ Found by David W. Cantrell in August 2002. |
s = 5 + 7 / √2 = 9.950+ Found by Evert Stenlund in 1980. |
For more details, see my paper on the subject:
Packing Unit Squares in Squares: A Survey and New Results.
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