David W. Cantrell improved my best packings for n=9, 10, 12, 13, 14, 15, 16, 17, 18, 20, and 21, and then did n=22, 23, and 24!

Then in 2011, Maurizio Morandi improved several packings.

Here are the best known packings:

s(1) = √1 = 1.000 | s(2) = √2+√1 = 2.414+ | s(3) = √3+√2 = 3.146+ | ||

s(4)=√4+√3=3.732+ | s(5) = √5+√4 = 4.236+ | s(6) = √4+√3+√2 = 5.146+ | ||

s(7) ≤ √6+√5+√(3/2) = 5.910+ | s(8) ≤ √7+√5+√3 = 6.613+ | s(9) ≤ 7.246+ (DC) | ||

s(10) ≤ 7.829+ (DC) | s(11) ≤ √9+√8+√7 = 8.474+ | s(12) ≤ √10+√9+√8 = 8.990+ (DC) | ||

s(13) ≤ √12+√10+√3+√2 = 9.772+ (MM) | s(14) ≤ 10.510+ (MM) | s(15) ≤ √13+√9+√6+√5 = 11.291+ (MM) | ||

s(16) ≤ √14+√10+√7+√6 = 11.999+ (MM) | s(17) ≤ √14+√10+√9+√8 = 12.732+ (MM) | s(18) ≤ √15+√11+√10+√9 = 13.351+ (MM) | ||

s(19) ≤ √19+√18+√12+√4 = 14.065+ (MM) | s(20) ≤ √17+√15+√8+√6+√2 = 14.688+ (MM) | s(21) ≤ √21+√20+√11+√9 = 15.371+ (MM) | ||

s(22) ≤ √19+√18+√16+√12 = 16.065+ (MM) | s(23) ≤ √23+√22+√17+√10 = 16.771+ (MM) | s(24) ≤ √24+√23+√20+√11 = 17.483+ (MM) |

Sasha Ravsky notes that s(n) must be at least √n+√(n-1), and this is enough to prove the values of s(n) for n≤5. He also notes that a collection of squares with total area S and the side of the largest square x can always be packed inside a square of side x+√(S-x^{2}) without tilting any squares. Thus √(n(n+1)/2) ≤ s(n) ≤ √n + √(n(n-1)/2).

Philippe Fondanaiche managed to pack squares of areas 1 through 100 inside a square of side 71.647+, with less than 2% wasted area. But then David Cantrell packed them inside a square of side 71.344+, with less than .8% wasted area!

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/8/11.