# Problem of the Month(January 2013)

What is the smallest square that k squares each of areas 1-n can be packed into?

Here are the best known results:

Squares of Area 1 and 2
 1s = 1 + √2 = 2.414+ 2s = 2√2 = 2.828+ 3s = 2 + √2 = 3.414+ 4s = 1 + 2√2 = 3.828+ 5s = 3√2 = 4.242+
 6s = 4.676+(Maurizio Morandi) 7s = 2 + 2√2 = 4.828+ 8s = 5.215+(Joe DeVincentis) 9s = 1 + 3√2 = 5.242+ 10s = 3 + 2√2 = 5.828+
 11s = 6.179+(Maurizio Morandi) 12s = 2 + 3√2 = 6.242+ 13s = 6.630+(Joe DeVincentis) 14s = 1 + 4√2 = 6.656+ 15s = 5√2 = 7.071+
 16s = 5√2 = 7.071+ 17s = 3 + 3√2 = 7.242+(Maurizio Morandi) 18s = 2 + 4√2 = 7.656+ 19s = 5 + 2√2 = 7.828+ 20s = 1 + 5√2 = 8.071+
 21s = 8.213+(Maurizio Morandi) 22s = 4 + 3√2 = 8.242+(Maurizio Morandi) 23s = 3 + 4√2 = 8.656+(Joe DeVincentis) 24s = 3 + 4√2 = 8.656+(Joe DeVincentis) 25s = 2 + 5√2 = 9.071+(Joe DeVincentis)
 26s = 2 + 5√2 = 9.071+(Joe DeVincentis) 27s = 2 + 5√2 = 9.071+(Maurizio Morandi) 28s = 1 + 6√2 = 9.485+(Joe DeVincentis) 29s = 4 + 4√2 = 9.656+(Joe DeVincentis) 30s = 4 + 4√2 = 9.656+(Maurizio Morandi)

Squares of Area 1, 2, and 3
 1s = √2 + √3 = 3.146+ 2s = 1 + 2√2 = 3.828+ 3s = 2√2 + √3 = 4.560+ 4s = 2+√2+√3 = 5.146+ 5s=1+√2+2√3=5.878+
 6s = 2√2+2√3 = 6.292+ 7s = √2 + 3√3 = 6.610+(Maurizio Morandi) 8s = 2 + 3√3 = 7.196+(Maurizio Morandi) 9s=1+√2+3√3=7.610+ 10s=2+3√2+√3=7.974+(Maurizio Morandi)
 11s=2+2√2+2√3=8.292+(Maurizio Morandi) 12s=2+√2+3√3=8.610+(Maurizio Morandi) 13s=1+2√2+3√3=9.024+ 14s=1+4√3+√2=9.342+(Maurizio Morandi) 15s = 1 + 5√3 = 9.660+(Maurizio Morandi)
 16s=4+√3+3√2=9.974+(Maurizio Morandi) 17s=4+2√3+2√2=10.292+(Maurizio Morandi) 18s=1+3√3+3√2=10.438+(Joe DeVincentis) 19s=1+4√3+2√2=10.756+(Maurizio Morandi) 20s=2+2√3+4√2=11.120+(Maurizio Morandi)

Squares of Area 1, 2, 3, and 4
 1s = 2 + √3 = 3.732+ 2s = √2 + 2√3 = 4.878+ 3s = 4 + √3 = 5.732+(Maurizio Morandi) 4s=2+2√2+√3=6.560+(Maurizio Morandi) 5s = 3 + 3√2 = 7.242+(Maurizio Morandi)
 6s = 1 + 4√3 = 7.928+ 7s=4+2√2+√3=8.560+(Maurizio Morandi) 8s = 4√2+2√3 = 9.120+(Maurizio Morandi) 9s = 4 + 4√2 = 9.656+(Maurizio Morandi) 10s=7+√3+√2=10.146+(Maurizio Morandi)
 11s=4+√2+3√3=10.610+(Maurizio Morandi) 12s=4+5√2=11.071+(Maurizio Morandi) 13s=7+√3+2√2=11.560+(Maurizio Morandi) 14s=6+√3+3√2=11.974+(Joe DeVincentis) 15s=5+√3+4√2=12.388+(Joe DeVincentis)

Squares of Area 1, 2, 3, 4, and 5
 1s = 2 + √5 = 4.236+ 2s = 4 + √3 = 5.732+ 3s=2+2√2+√5=7.064+(Maurizio Morandi) 4s = 2√5+2√3 = 7.936+(Maurizio Morandi) 5s=√2+3√3+√5=8.846+
 6s = 6+√5+√2 = 9.650+ 7s=9+√2=10.414+(Maurizio Morandi) 8s=2+2√3+4√2=11.120+(Maurizio Morandi) 9s=1+2√5+2√3+2√2=11.764+(Maurizio Morandi) 10s=2+4√5+√2=12.358+(Joe DeVincentis)
 11s = 4+4√5 = 12.944+(Maurizio Morandi) 12s=2+2√5+5√2=13.543+(Joe DeVincentis)

Squares of Area 1, 2, 3, 4, 5, and 6
 1s = 2+√3+√2 = 5.146+ 2s = 2+√6+√5 = 6.685+ 3s = 4+√6+√3 = 8.181+(Maurizio Morandi) 4s=4+√6+2√2=9.277+(Maurizio Morandi) 5s=4+√6+√5+√3=10.417+(Joe DeVincentis)
 6s=2+2√6+2√5=11.371+(Joe DeVincentis) 7s=2+√5+3√3+2√2=12.260+(Maurizio Morandi) 8s=2+2√6+2√5+√3=13.103+(Joe DeVincentis) 9s=6+√5+4√2=13.892+(Maurizio Morandi) 10s=1+3√2+2√5+2√6=14.613+(Maurizio Morandi)

Squares of Area 1, 2, 3, 4, 5, 6, and 7
 1s=√6+√5+√(3/2)=5.910+ 2s=√7+√5+2√2=7.710+(Maurizio Morandi) 3s=2+√7+√6+√5=9.331+(Joe DeVincentis) 4s=2+√2+√5+√6+√7=10.745+(Maurizio Morandi) 5s=2√7+3√5=11.999+(Joe DeVincentis)
 6s=2+√7+3√5+√3=13.086+(Maurizio Morandi) 7s=4+√7+√6+√5+2√2=14.159+(Joe DeVincentis)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/30/17.