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 = 6.17279+
(Yanis Gomes) 		12

s = 2 + 3√2 = 6.242+
  		13

s = 6.60880+
(Yanis Gomes) 		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)
31

s = 10.07106+
(Yanis Gomes) 		32

s = 3 + 5√2 = 10.07106+
(Yanis Gomes) 		33

s = 10.41611+
(Yanis Gomes) 		34

s = 10.48528+
(Yanis Gomes) 		35

s = 10.65685+
(Yanis Gomes)
36

s = 10.65688+
(Yanis Gomes) 		37

s = 10.93528+
(Yanis Gomes) 		38

s = 11.07107+
(Yanis Gomes) 		39

s = 11.23737+
(Yanis Gomes) 		40

s = 11.48528+
(Yanis Gomes)
41

s = 11.48528+
(Yanis Gomes) 		42

s = 11.65685+
(Yanis Gomes) 		43

s = 11.86467+
(Yanis Gomes) 		44

s = 11.89950+
(Yanis Gomes) 		45

s = 12.05766+
(Yanis Gomes)
46

s = 12.12141+
(Yanis Gomes) 		47

s = 12.31372+
(Yanis Gomes) 		48

s = 12.48529+
(Yanis Gomes) 		49

s = 12.48530+
(Yanis Gomes) 		50

s = 12.80926+
(Yanis Gomes)
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 = 5.13032+
(Yanis Gomes) 		5

s=1+√2+2√3=5.878+
 
6

s = 6.26782+
(Yanis Gomes) 		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)
21

s = 11.43879+
(Yanis Gomes) 		22

s = 11.78437+
(Yanis Gomes) 		23

s = 12.07448+
(Yanis Gomes) 		24

s = 12.38890+
(Yanis Gomes) 		25

s = 12.53516+
(Yanis Gomes)
26

s = 12.85301+
(Yanis Gomes) 		27

s = 13.12440+
(Yanis Gomes) 		28

s = 13.26733+
(Yanis Gomes) 		29

s = 13.57211+
(Yanis Gomes) 		30

s = 13.85312+
(Yanis Gomes)
31

s = 14.07107+
(Yanis Gomes) 		32

s = 14.22082+
(Yanis Gomes) 		33

s = 14.55882+
(Yanis Gomes) 		34

s = 14.72697+
(Yanis Gomes) 		35

s = 14.90293+
(Yanis Gomes)
36

s = 15.19655+
(Yanis Gomes) 		37

s = 15.33788+
(Yanis Gomes) 		38

s = 15.55027+
(Yanis Gomes) 		39

s = 15.84438+
(Yanis Gomes) 		40

s = 15.95282+
(Yanis Gomes)
41

s = 16.15749+
(Yanis Gomes) 		42

s = 16.32052+
(Yanis Gomes) 		43

s = 16.57536+
(Yanis Gomes) 		44

s = 16.68168+
(Yanis Gomes) 		45

s = 16.95197+
(Yanis Gomes)
46

s = 17.14563+
(Yanis Gomes) 		47

s = 17.31679+
(Yanis Gomes) 		48

s = 17.50847+
(Yanis Gomes) 		49

s = 17.82762+
(Yanis Gomes) 		50

s = 18.04927+
(Yanis Gomes)
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 = 7.23956+
(Yanis Gomes)
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)
16

s = 12.92820+
(Yanis Gomes) 		17

s = 13.29253+
(Yanis Gomes) 		18

s = 13.70675+
(Yanis Gomes) 		19

s = 14.07447+
(Yanis Gomes) 		20

s = 14.54709+
(Yanis Gomes)
21

s = 14.87833+
(Yanis Gomes) 		22

s = 15.14631+
(Yanis Gomes) 		23

s = 15.56024+
(Yanis Gomes) 		24

s = 15.87836+
(Yanis Gomes) 		25

s = 16.17088+
(Yanis Gomes)
26

s = 16.63511+
(Yanis Gomes) 		27

s = 16.87832+
(Yanis Gomes) 		28

s = 17.10993+
(Yanis Gomes) 		29

s = 17.58249+
(Yanis Gomes) 		30

s = 17.80315+
(Yanis Gomes)
31

s = 18.07448+
(Yanis Gomes) 		32

s = 18.39564+
(Yanis Gomes) 		33

s = 18.70627+
(Yanis Gomes) 		34

s = 19.05986+
(Yanis Gomes) 		35

s = 19.21953+
(Yanis Gomes)
36

s = 19.52358+
(Yanis Gomes) 		37

s = 19.82939+
(Yanis Gomes) 		38

s = 20.12117+
(Yanis Gomes) 		39

s = 20.38160+
(Yanis Gomes) 		40

s = 20.75684+
(Yanis Gomes)
41

s = 20.94215+
(Yanis Gomes) 		42

s = 21.15740+
(Yanis Gomes) 		43

s = 21.41422+
(Yanis Gomes) 		44

s = 21.63786+
(Yanis Gomes) 		45

s = 22.16421+
(Yanis Gomes)
46

s = 22.12040+
(Yanis Gomes) 		47

s = 22.38095+
(Yanis Gomes) 		48

s = 22.85694+
(Yanis Gomes) 		49

s = 22.91860+
(Yanis Gomes) 		50

s = 23.03948+
(Yanis Gomes)
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+2√5+√2=7.88634+
(Yanis Gomes) 		5

s = 8.84559+
(Yanis Gomes)
6

s = 9.61840+
(Yanis Gomes) 		7

s = 10.38233+
(Yanis Gomes) 		8

s = 11.11438+
(Yanis Gomes) 		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) 		13

s = 14.20418+
(Yanis Gomes) 		14

s = 14.78688+
(Yanis Gomes) 		15

s = 15.47562+
(Yanis Gomes)
16

s = 15.85479+
(Yanis Gomes) 		17

s = 16.38233+
(Yanis Gomes) 		18

s = 16.84644+
(Yanis Gomes) 		19

s = 17.24812+
(Yanis Gomes) 		20

s = 17.88762+
(Yanis Gomes)
21

s = 18.20421+
(Yanis Gomes) 		22

s = 18.63642+
(Yanis Gomes) 		23

s = 19.05890+
(Yanis Gomes) 		24

s = 19.50121+
(Yanis Gomes) 		25

s = 19.99030+
(Yanis Gomes)
26

s = 20.25672+
(Yanis Gomes) 		27

s = 20.94113+
(Yanis Gomes) 		28

s = 21.20781+
(Yanis Gomes) 		29

s = 21.50476+
(Yanis Gomes) 		30

s = 21.83056+
(Yanis Gomes)
31

s = 22.23257+
(Yanis Gomes) 		32

s = 22.61908+
(Yanis Gomes) 		33

s = 22.90436+
(Yanis Gomes) 		34

s = 23.29612+
(Yanis Gomes) 		35

s = 23.65904+
(Yanis Gomes)
36

s = 24.05859+
(Yanis Gomes) 		37

s = 24.36334+
(Yanis Gomes) 		38

s = 24.66704+
(Yanis Gomes) 		39

s = 25.08451+
(Yanis Gomes) 		40

s = 25.67833+
(Yanis Gomes)
41

s = 25.95870+
(Yanis Gomes) 		42

s = 26.26427+
(Yanis Gomes) 		43

s = 26.39978+
(Yanis Gomes) 		44

s = 26.69178+
(Yanis Gomes) 		45

s = 26.95809+
(Yanis Gomes)
46

s = 27.57467+
(Yanis Gomes) 		47

s = 27.83748+
(Yanis Gomes) 		48

s = 28.14053+
(Yanis Gomes) 		49

s = 28.51973+
(Yanis Gomes) 		50

s = 28.57719+
(Yanis Gomes)
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 = 8.14965+
(Yanis Gomes) 		4

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

s = 10.38572+
(Yanis Gomes)
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=4√2+4√5 = 14.60112+
(Yanis Gomes)
11

s = 15.50787+
(Yanis Gomes) 		12

s = 16.16857+
(Yanis Gomes) 		13

s = 16.87746+
(Yanis Gomes) 		14

s = 17.55659+
(Yanis Gomes) 		15

s = 18.18029+
(Yanis Gomes)
16

s = 18.76002+
(Yanis Gomes) 		17

s = 19.33925+
(Yanis Gomes) 		18

s = 20.00940+
(Yanis Gomes) 		19

s = 20.69495+
(Yanis Gomes) 		20

s = 21.30217+
(Yanis Gomes)
21

s = 21.75684+
(Yanis Gomes) 		22

s = 22.35661+
(Yanis Gomes) 		23

s = 22.86927+
(Yanis Gomes) 		24

s = 23.39268+
(Yanis Gomes) 		25

s = 23.84593+
(Yanis Gomes)
26

s = 24.25283+
(Yanis Gomes) 		27

s = 24.63167+
(Yanis Gomes) 		28

s = 25.32382+
(Yanis Gomes) 		29

s = 25.75428+
(Yanis Gomes) 		30

s = 26.14505+
(Yanis Gomes)
31

s = 26.42365+
(Yanis Gomes) 		32

s = 27.12422+
(Yanis Gomes) 		33

s = 27.32067+
(Yanis Gomes) 		34

s = 27.89135+
(Yanis Gomes) 		35

s = 28.45804+
(Yanis Gomes)
36

s = 28.93212+
(Yanis Gomes) 		37

s = 29.22561+
(Yanis Gomes) 		38

s = 29.72547+
(Yanis Gomes) 		39

s = 30.06666+
(Yanis Gomes) 		40

s = 30.37010+
(Yanis Gomes)
41

s = 30.78454+
(Yanis Gomes) 		42

s = 31.19596+
(Yanis Gomes) 		43

s = 31.37970+
(Yanis Gomes) 		44

s = 31.94138+
(Yanis Gomes) 		45

s = 32.34431+
(Yanis Gomes)
46

s = 32.47625+
(Yanis Gomes) 		47

s = 33.09208+
(Yanis Gomes) 		48

s = 33.47054+
(Yanis Gomes) 		49

s = 33.74155+
(Yanis Gomes) 		50

s = 34.01151+
(Yanis Gomes)
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+√2+2√3+√6 = 9.32780+
(Yanis Gomes) 		4

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

s= 11.99422+
(Yanis Gomes)
6

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

s=4+√7+√6+√5+2√2=14.159+
(Joe DeVincentis) 		8

s = 15.23482+
(Yanis Gomes) 		9

s=4+√2+2√3+3√6 = 16.22678+
(Yanis Gomes) 		10

s = 17.12332+
(Yanis Gomes)
11

s = 17.93655+
(Yanis Gomes) 		12

s = 18.86091+
(Yanis Gomes) 		13

s = 19.55378+
(Yanis Gomes) 		14

s = 20.29778+
(Yanis Gomes) 		15

s = 21.19015+
(Yanis Gomes)
16

s = 21.87504+
(Yanis Gomes) 		17

s = 22.39002+
(Yanis Gomes) 		18

s = 23.06401+
(Yanis Gomes) 		19

s = 23.93594+
(Yanis Gomes) 		20

s = 24.31968+
(Yanis Gomes)
21

s = 25.19122+
(Yanis Gomes) 		22

s = 25.72248+
(Yanis Gomes) 		23

s = 26.50724+
(Yanis Gomes) 		24

s = 27.08466+
(Yanis Gomes) 		25

s = 27.52157+
(Yanis Gomes)
26

s = 27.93874+
(Yanis Gomes) 		27

s = 28.78988+
(Yanis Gomes) 		28

s = 29.30965+
(Yanis Gomes) 		29

s = 29.61415+
(Yanis Gomes) 		30

s = 30.09420+
(Yanis Gomes)
31

s = 30.73658+
(Yanis Gomes) 		32

s = 31.35778+
(Yanis Gomes) 		33

s = 31.54139+
(Yanis Gomes) 		34

s = 32.01978+
(Yanis Gomes) 		35

s = 32.54827+
(Yanis Gomes)
36

s = 33.31955+
(Yanis Gomes) 		37

s = 33.69411+
(Yanis Gomes) 		38

s = 34.02786+
(Yanis Gomes) 		39

s = 34.68970+
(Yanis Gomes) 		40

s = 35.17626+
(Yanis Gomes)
41

s = 35.44249+
(Yanis Gomes) 		42

s = 35.82917+
(Yanis Gomes) 		43

s = 36.50052+
(Yanis Gomes) 		44

s = 36.73389+
(Yanis Gomes) 		45

s = 37.38236+
(Yanis Gomes)
46

s = 37.88375+
(Yanis Gomes) 		47

s = 38.13484+
(Yanis Gomes) 		48

s = 38.51107+
(Yanis Gomes) 		49

s = 38.76154+
(Yanis Gomes) 		50

s = 39.30802+
(Yanis Gomes)

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