Problem of the Month
(January 2013)

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

ANSWERS

Here are the best known results:

Squares of Area 1 and 2
1

s = 1 + √2 = 2.414+ 		2

s = 2√2 = 2.828+ 		3

s = 2 + √2 = 3.414+ 		4

s = 1 + 2√2 = 3.828+ 		5

s = 3√2 = 4.242+
6

s = 4.676+
(Maurizio Morandi) 		7

s = 2 + 2√2 = 4.828+
  		8

s = 5.215+
(Joe DeVincentis) 		9

s = 1 + 3√2 = 5.242+
  		10

s = 3 + 2√2 = 5.828+
 
11

s = 6.179+
(Maurizio Morandi) 		12

s = 2 + 3√2 = 6.242+
  		13

s = 6.630+
(Joe DeVincentis) 		14

s = 1 + 4√2 = 6.656+
  		15

s = 5√2 = 7.071+
 
16

s = 5√2 = 7.071+
  		17

s = 3 + 3√2 = 7.242+
(Maurizio Morandi) 		18

s = 2 + 4√2 = 7.656+
  		19

s = 5 + 2√2 = 7.828+
  		20

s = 1 + 5√2 = 8.071+
 
21

s = 8.213+
(Maurizio Morandi) 		22

s = 4 + 3√2 = 8.242+
(Maurizio Morandi) 		23

s = 3 + 4√2 = 8.656+
(Joe DeVincentis) 		24

s = 3 + 4√2 = 8.656+
(Joe DeVincentis) 		25

s = 2 + 5√2 = 9.071+
(Joe DeVincentis)
26

s = 2 + 5√2 = 9.071+
(Joe DeVincentis) 		27

s = 2 + 5√2 = 9.071+
(Maurizio Morandi) 		28

s = 1 + 6√2 = 9.485+
(Joe DeVincentis) 		29

s = 4 + 4√2 = 9.656+
(Joe DeVincentis) 		30

s = 4 + 4√2 = 9.656+
(Maurizio Morandi)
Squares of Area 1, 2, and 3
1

s = √2 + √3 = 3.146+ 		2

s = 1 + 2√2 = 3.828+ 		3

s = 2√2 + √3 = 4.560+ 		4

s = 2+√2+√3 = 5.146+ 		5

s=1+√2+2√3=5.878+
6

s = 2√2+2√3 = 6.292+
  		7

s = √2 + 3√3 = 6.610+
(Maurizio Morandi) 		8

s = 2 + 3√3 = 7.196+
(Maurizio Morandi) 		9

s=1+√2+3√3=7.610+
  		10

s=2+3√2+√3=7.974+
(Maurizio Morandi)
11

s=2+2√2+2√3=8.292+
(Maurizio Morandi) 		12

s=2+√2+3√3=8.610+
(Maurizio Morandi) 		13

s=1+2√2+3√3=9.024+
  		14

s=1+4√3+√2=9.342+
(Maurizio Morandi) 		15

s = 1 + 5√3 = 9.660+
(Maurizio Morandi)
16

s=4+√3+3√2=9.974+
(Maurizio Morandi) 		17

s=4+2√3+2√2=10.292+
(Maurizio Morandi) 		18

s=1+3√3+3√2=10.438+
(Joe DeVincentis) 		19

s=1+4√3+2√2=10.756+
(Maurizio Morandi) 		20

s=2+2√3+4√2=11.120+
(Maurizio Morandi)
Squares of Area 1, 2, 3, and 4
1

s = 2 + √3 = 3.732+
  		2

s = √2 + 2√3 = 4.878+
  		3

s = 4 + √3 = 5.732+
(Maurizio Morandi) 		4

s=2+2√2+√3=6.560+
(Maurizio Morandi) 		5

s = 3 + 3√2 = 7.242+
(Maurizio Morandi)
6

s = 1 + 4√3 = 7.928+
  		7

s=4+2√2+√3=8.560+
(Maurizio Morandi) 		8

s = 4√2+2√3 = 9.120+
(Maurizio Morandi) 		9

s = 4 + 4√2 = 9.656+
(Maurizio Morandi) 		10

s=7+√3+√2=10.146+
(Maurizio Morandi)
11

s=4+√2+3√3=10.610+
(Maurizio Morandi) 		12

s=4+5√2=11.071+
(Maurizio Morandi) 		13

s=7+√3+2√2=11.560+
(Maurizio Morandi) 		14

s=6+√3+3√2=11.974+
(Joe DeVincentis) 		15

s=5+√3+4√2=12.388+
(Joe DeVincentis)
Squares of Area 1, 2, 3, 4, and 5
1

s = 2 + √5 = 4.236+
  		2

s = 4 + √3 = 5.732+
  		3

s=2+2√2+√5=7.064+
(Maurizio Morandi) 		4

s = 2√5+2√3 = 7.936+
(Maurizio Morandi) 		5

s=√2+3√3+√5=8.846+
 
6

s = 6+√5+√2 = 9.650+
  		7

s=9+√2=10.414+
(Maurizio Morandi) 		8

s=2+2√3+4√2=11.120+
(Maurizio Morandi) 		9

s=1+2√5+2√3+2√2=11.764+
(Maurizio Morandi) 		10

s=2+4√5+√2=12.358+
(Joe DeVincentis)
11

s = 4+4√5 = 12.944+
(Maurizio Morandi) 		12

s=2+2√5+5√2=13.543+
(Joe DeVincentis)
Squares of Area 1, 2, 3, 4, 5, and 6
1

s = 2+√3+√2 = 5.146+
  		2

s = 2+√6+√5 = 6.685+
  		3

s = 4+√6+√3 = 8.181+
(Maurizio Morandi) 		4

s=4+√6+2√2=9.277+
(Maurizio Morandi) 		5

s=4+√6+√5+√3=10.417+
(Joe DeVincentis)
6

s=2+2√6+2√5=11.371+
(Joe DeVincentis) 		7

s=2+√5+3√3+2√2=12.260+
(Maurizio Morandi) 		8

s=2+2√6+2√5+√3=13.103+
(Joe DeVincentis) 		9

s=6+√5+4√2=13.892+
(Maurizio Morandi) 		10

s=1+3√2+2√5+2√6=14.613+
(Maurizio Morandi)
Squares of Area 1, 2, 3, 4, 5, 6, and 7
1

s=√6+√5+√(3/2)=5.910+
  		2

s=√7+√5+2√2=7.710+
(Maurizio Morandi) 		3

s=2+√7+√6+√5=9.331+
(Joe DeVincentis) 		4

s=2+√2+√5+√6+√7=10.745+
(Maurizio Morandi) 		5

s=2√7+3√5=11.999+
(Joe DeVincentis)
6

s=2+√7+3√5+√3=13.086+
(Maurizio Morandi) 		7

s=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.