Problem of the Month (July 2012)

Define an n-sliced square as a unit square with n consecutive corners (right isosceles triangles with area 1/8) cut off. What are the smallest squares that some number of n-sliced squares can be packed into? What about n-sliced squares packed into n-sliced squares? What about squares packed into n-sliced squares?

ANSWERS

Here are the best-known packings:

1-Sliced Squares in Squares
1

s = 1 		2

s = 3/2 = 1.5 		3

s = 3(1+√2)/4 = 1.810+ 		4

s = 2
5

s = (3+√2)/2 = 2.207+
  		6

s = 5/2 = 2.5
  		7

s = 2.700+
(DC after MM) 		8

s = 2√2 = 2.828+
(Haym Hirsh)
9

s = 3
  		10

s = 3.174+
(David W. Cantrell) 		11

s =(50+21√2)/24= 3.320+
(Maurizio Morandi) 		12

s =(29+√31)/10= 3.456+
(David W. Cantrell)
13

s=(52+19√2)/22=3.585+
(Maurizio Morandi) 		14

s = 3 + 1/√2 = 3.707+
(Maurizio Morandi) 		15

s=(18+9√2)/8 = 3.840+
(Yanis Gomes) 		16

s = (6+7√2)/4 = 3.974+
(Maurizio Morandi)
17

s = 4.067+
(Yanis Gomes) 		18

s = (18+11√2)/8 = 4.194+
(Yanis Gomes) 		19

s = 4.289+
(Yanis Gomes) 		20

s = 4.389+
(Yanis Gomes)
21

s = (38+5√2)/10 = 4.507+
(Yanis Gomes) 		22

s = 4.606+
(Yanis Gomes) 		23

s = 4.733+
(Yanis Gomes) 		24

s = 4.810+
(Yanis Gomes)
25

s = (47+12√2)/13 = 4.920+
(Yanis Gomes) 		26

s = (23+5√2)/6 = 5.011+
(Yanis Gomes) 		27

s = 5.128+
(Yanis Gomes) 		28

s = 5.206+
(Yanis Gomes)
29

s = 5.285+
(Yanis Gomes) 		30

s = 5.373+
(Yanis Gomes) 		31

s = (50+11√2)/12 = 5.463+
(Yanis Gomes) 		32

s = 5.648+
(Yanis Gomes)
33

s = 5.765+
(Yanis Gomes) 		34

s = 5.373+
(Yanis Gomes) 		35

s = 5.841+
(Yanis Gomes) 		36

s = 5.912+
(Yanis Gomes)
37

s = 6.003+
(Yanis Gomes) 		38

s = 6.075+
(Yanis Gomes) 		39

s = 6.133+
(Yanis Gomes) 		40

s = 6.221+
(Yanis Gomes)
41

s = 6.286+
(Yanis Gomes) 		42

s = 6.403+
(Yanis Gomes) 		43

s = 6.477+
(Yanis Gomes) 		44

s = 6.536+
(Yanis Gomes)
45

s = 6.583+
(Yanis Gomes) 		46

s = 6.710+
(Yanis Gomes) 		47

s = 6.750+
(Yanis Gomes) 		48

s = 6.856+
(Yanis Gomes)
49

s = 6.938+
(Yanis Gomes) 		50

s = 7
(Yanis Gomes)

1-Sliced Squares in 1-Sliced Squares
1

s = 1 		2

s = 5/3 = 1.666+ 		3

s = 1+2√2/3 = 1.942+ 		4

s = (3+√2)/2 = 2.207+
5

s = 1 + √2 = 2.414+
(Maurizio Morandi) 		6

s = (5+4√2)/4 = 2.664+
(Yanis Gomes) 		7

s = (3+20√2)/11 = 2.844+
(Yanis Gomes) 		8

s = 3
 
9

s = (5+√2)/2 = 3.207+
(Bryce Herdt) 		10

s = (56+13√2)/22 = 3.381+
(Yanis Gomes) 		11

s = (20+41√2)/22 = 3.544+
(Yanis Gomes) 		12

s = 3.701+
(Yanis Gomes)
13

s = (30+14√2)/13 = 3.830+
(Yanis Gomes) 		14

s = 3.982+
(Yanis Gomes) 		15

s = 4.092+
(Yanis Gomes) 		16

s = (21+15√2)/10 = 4.221+
(Yanis Gomes)
17

s = 4.346+
(Yanis Gomes) 		18

s = 4.438+
(Yanis Gomes) 		19

s = (82+267√2)/100 = 4.595+
(Yanis Gomes) 		20

s = 4.708+
(Yanis Gomes)
21

s = 4.830+
(Yanis Gomes) 		22

s = 4.962+
(Yanis Gomes) 		23

s = 5.054+
(Yanis Gomes) 		24

s = 5.173+
(Yanis Gomes)
25

s = (60+21√2)/17 = 5.276+
(Yanis Gomes) 		26

s = 5.346+
(Yanis Gomes) 		27

s = 5.457+
(Yanis Gomes) 		28

s = 5.555+
(Yanis Gomes)
29

s = 5.678+
(Yanis Gomes) 		30

s = 5.750+
(Yanis Gomes) 		31

s = 5.865+
(Yanis Gomes) 		32

s = 5.996+
(Yanis Gomes)
33

s = 6.041+
(Yanis Gomes) 		34

s = 6.125+
(Yanis Gomes) 		35

s = 6.264+
(Yanis Gomes) 		36

s = 6.325+
(Yanis Gomes)
37

s = 6.380+
(Yanis Gomes) 		38

s = 6.494+
(Yanis Gomes) 		39

s = 6.590+
(Yanis Gomes) 		40

s = 6.696+
(Yanis Gomes)
41

s = (375+97√2)/76 = 6.739+
(Yanis Gomes) 		42

s = 6.828+
(Yanis Gomes) 		43

s = 6.931+
(Yanis Gomes) 		44

s = 7.003+
(Yanis Gomes)
45

s = 7.047+
(Yanis Gomes) 		46

s = 7.185+
(Yanis Gomes) 		47

s = 7.221+
(Yanis Gomes) 		48

s = 7.300+
(Yanis Gomes)
49

s = 7.409+
(Yanis Gomes) 		50

s = 7.433+
(Yanis Gomes)

Squares in 1-Sliced Squares
1

s = 4/3 = 1.333+ 		2

s = 2 		3

s = 2 		4

s = 8/3 = 2.666+
5

s = 2 + 2√2/3 = 2.942+
(Maurizio Morandi) 		6

s = 3
  		7

s = 4(1+√2)/3 = 3.218+
  		8

s = 10/3 = 3.333+
 
9

s = 3.750+
(Yanis Gomes) 		10

s = 2+4√2/3 = 3.885+
(Maurizio Morandi) 		11

s = 4
  		12

s = 4
 
13

s = 4
  		14

s = 3 + √2 = 4.414+
  		15

s = 14/3 = 4.666+
(David W. Cantrell) 		16

s = 2 + 2√2 = 4.828+
(David W. Cantrell)
17

s = 4.935+
(Yanis Gomes) 		18

s = 5
(Yanis Gomes) 		19

s = 5
(Yanis Gomes) 		20

s=(8+5√2)/3=5.023+
(Yanis Gomes)
21

s = 5.329+
(Yanis Gomes) 		22

s = 16/3 = 5.333+
(Yanis Gomes) 		23

s = 5.581+
(Yanis Gomes) 		24

s=(16+9√2)/5=5.745+
(Yanis Gomes)
25

s = 5.828+
(Yanis Gomes) 		26

s = 5.979+
(Yanis Gomes) 		27

s = 6
(Yanis Gomes) 		28

s = 6
(Yanis Gomes)
29

s = 6
(Yanis Gomes) 		30

s = 6
(Yanis Gomes) 		31

s = 6.276+
(Yanis Gomes) 		32

s = 6.535+
(Yanis Gomes)
33

s = 20/3 = 6.666+
(Yanis Gomes) 		34

s = 6+1/√2 = 6.707+
(Yanis Gomes) 		35

s = 4+2√2 = 6.828+
(Yanis Gomes) 		36

s = 6.942+
(Yanis Gomes)
37

s = 7
(Yanis Gomes) 		38

s = 7
(Yanis Gomes) 		39

s = 7
(Yanis Gomes) 		40

s = 7
(Yanis Gomes)
41

s = 7.218+
(Yanis Gomes) 		42

s = 7.299+
(Yanis Gomes) 		43

s = 22/3 = 7.333+
(Yanis Gomes) 		44

s = 7.535+
(Yanis Gomes)
45

s = 7.685+
(Yanis Gomes) 		46

s = 7.771+
(Yanis Gomes) 		47

s = 7.868+
(Yanis Gomes) 		48

s = 7.886+
(Yanis Gomes)
49

s = 8
(Yanis Gomes) 		50

s = 8
(Yanis Gomes)

2-Sliced Squares in Squares
1

s = 1
  		2

s = √2 = 1.414+
  		3

s = 1 + 1/√2 = 1.707+
  		4

s = 1.989+
(Yanis Gomes)
5

s = 3√2/2 = 2.121+
  		6

s = 2 + √2/4 = 2.353+
(David W. Cantrell) 		7

s = 1+11√2/10 = 2.555+
(Maurizio Morandi) 		8

s = 2 + 1/√2 = 2.707+
(Maurizio Morandi)
9

s = 2√2 = 2.828+
(Maurizio Morandi) 		10

s =(2+7√2)/4= 2.974+
(Maurizio Morandi) 		11

s = 1 + 3/√2 = 3.121+
(Maurizio Morandi) 		12

s =(6+5√2)/4= 3.267+
(Maurizio Morandi)
13

s = 3.395+
(Yanis Gomes) 		14

s = (9+6√2)/5 = 3.497+
(Yanis Gomes) 		15

s=(50+47√2)/32=3.639+
(Yanis Gomes) 		16

s = 3.742+
(Yanis Gomes)
17

s=(18+9√2)/8=3.840+
(Yanis Gomes) 		18

s=(111+67√2)/52=3.956+
(Yanis Gomes) 		19

s = 4.057+
(Yanis Gomes) 		20

s=(134+81√2)/60=4.142+
(Yanis Gomes)
21

s=(290+119√2)/108=4.243+
(Yanis Gomes) 		22

s=(104+37√2)/36=4.342+
(Yanis Gomes) 		23

s=(503+196√2)/176=4.432+
(Yanis Gomes) 		24

s = 9/2 = 4.5
(Yanis Gomes)
25

s=(145+41√2)/44=4.613+
(Yanis Gomes) 		26

s=(160+33√2)/44=4.697+
(Yanis Gomes) 		27

s=(138+119√2)/64=4.785+
(Yanis Gomes) 		28

s = 4.857+
(Yanis Gomes)
29

s=(27+37√2)/16=4.957+
(Yanis Gomes) 		30

s = 5.038+
(Yanis Gomes) 		31

s = 5.146+
(Yanis Gomes) 		32

s = 5.202+
(Yanis Gomes)
33

s = 5.274+
(Yanis Gomes) 		34

s = 5.357+
(Yanis Gomes) 		35

s=(197+168√2)/80=5.432+
(Yanis Gomes) 		36

s = 5.502+
(Yanis Gomes)
37

s = 5.588+
(Yanis Gomes) 		38

s = 5.668+
(Yanis Gomes) 		39

s = 5.726+
(Yanis Gomes) 		40

s=(32921+875√2)/5884=5.805+
(Yanis Gomes)
41

s = 5.849+
(Yanis Gomes) 		42

s = 5.895+
(Yanis Gomes) 		43

s=(2894+245√2)/544=5.956+
(Yanis Gomes) 		44

s = 6
(Yanis Gomes)
45

s = 6.099+
(Yanis Gomes) 		46

s = 6.175+
(Yanis Gomes) 		47

s = 6.250+
(Yanis Gomes) 		48

s = 6.322+
(Yanis Gomes)
49

s = 6.403+
(Yanis Gomes) 		50

s = 6.462+
(Yanis Gomes)

2-Sliced Squares in 2-Sliced Squares
1

s = 1
  		2

s = 1.873+
(Maurizio Morandi) 		3

s = 5/4 + 1/√2 = 1.957+
(Maurizio Morandi) 		4

s = 2(2+√2)/3 = 2.276+
 
5

s = 5/2 = 2.5
(Maurizio Morandi) 		6

s = 2 + 1/√2 = 2.707+
(Maurizio Morandi) 		7

s=(32+19√2)/20=2.943+
(Maurizio Morandi) 		8

s=(25+13√2)/14=3.098+
(Maurizio Morandi)
9

s = 4(1+√2)/3 = 3.218+
(Maurizio Morandi) 		10

s=(20+15√2)/12=3.434+
(Yanis Gomes) 		11

s=(22+15√2)/12=3.601+
(Yanis Gomes) 		12

s=(92+83√2)/56=3.738+
(Yanis Gomes)
13

s = 3.901+
(Yanis Gomes) 		14

s=(14+47√2)/20=4.023+
(Yanis Gomes) 		15

s=(53+45√2)/28=4.165+
(Yanis Gomes) 		16

s=(45+41√2)/24=4.290+
(Yanis Gomes)
17

s=(98+90√2)/51=4.417+
(Yanis Gomes) 		18

s=(136+127√2)/70=4.508+
(Yanis Gomes) 		19

s=(45+47√2)/24=4.644+
(Yanis Gomes) 		20

s = 4.767+
(Yanis Gomes)
21

s = 4.863+
(Yanis Gomes) 		22

s = 4.989+
(Yanis Gomes) 		23

s=(775+277√2)/228=5.117+
(Yanis Gomes) 		24

s=(73+22√2)/20=5.205+
(Yanis Gomes)
25

s = 5.306+
(Yanis Gomes) 		26

s = 5.409+
(Yanis Gomes) 		27

s = 5.499+
(Yanis Gomes) 		28

s = 5.603+
(Yanis Gomes)
29

s = 5.723+
(Yanis Gomes) 		30

s=(113+35√2)/28=5.803+
(Yanis Gomes) 		31

s = 5.872+
(Yanis Gomes) 		32

s=(41+26√2)/13=5.982+
(Yanis Gomes)
33

s = 6.068+
(Yanis Gomes) 		34

s = 6.168+
(Yanis Gomes) 		35

s = 6.257+
(Yanis Gomes) 		36

s = 6.337+
(Yanis Gomes)
37

s = 6.425+
(Yanis Gomes) 		38

s = 6.491+
(Yanis Gomes) 		39

s = 6.574+
(Yanis Gomes) 		40

s = 6.683+
(Yanis Gomes)
41

s = 6.745+
(Yanis Gomes) 		42

s = 6.838+
(Yanis Gomes) 		43

s = 6.901+
(Yanis Gomes) 		44

s=(2082+323√2)/364=6.974+
(Yanis Gomes)
45

s = 7.056+
(Yanis Gomes) 		46

s = 7.146+
(Yanis Gomes) 		47

s = 7.216+
(Yanis Gomes) 		48

s=(1842+367√2)/324=7.287+
(Yanis Gomes)
49

s = 7.351+
(Yanis Gomes) 		50

s = 7.418+
(Yanis Gomes)

Squares in 2-Sliced Squares
1

s = √2 = 1.414+
(Andrew Bayly) 		2

s = 2
  		3

s = 2(2+√2)/3 = 2.276+
  		4

s = 2√2 = 2.828+
 
5

s = 3
  		6

s = 1 + 3/√2 = 3.121+
(Maurizio Morandi) 		7

s = 7/2 = 3.5
  		8

s = 1 + 2√2 = 3.828+
(David W. Cantrell)
9

s = 5/2 + √2 = 3.914+ 		10

s = 4 		11

s = 2 + 3/√2 = 4.121+ 		12

s = 9/2 = 4.5
13

s = 10√2/3 = 4.714+ 		14

s = 2 + 2√2 = 4.828+ 		15

s = 2 + 2√2 = 4.828+ 		16

s = 5
17

s=(6+3√2)/2=5.121+
(Yanis Gomes) 		18

s=(2+10√2)/3=5.380+
(Yanis Gomes) 		19

s = 11/2 = 5.5
(Yanis Gomes) 		20

s=(4+9√2)/3=5.575+
(Yanis Gomes)
21

s=(6+8√2)/3=5.771+
(Yanis Gomes) 		22

s=(6+8√2)/3=5.771+
(Yanis Gomes) 		23

s = 6
(Yanis Gomes) 		24

s = 6
(Yanis Gomes)
25

s = 6.121+
(Yanis Gomes) 		26

s = 6.407+
(Yanis Gomes) 		27

s=(8+8√2)/3=6.437+
(Yanis Gomes) 		28

s=(8+8√2)/3=6.437+
(Yanis Gomes)
29

s=(6+10√2)/3=6.714+
(Yanis Gomes) 		30

s=(6+10√2)/3=6.714+
(Yanis Gomes) 		31

s = 6.925+
(Yanis Gomes) 		32

s = 6.990+
(Yanis Gomes)
33

s = 7
(Yanis Gomes) 		34

s = 7.121+
(Yanis Gomes) 		35

s = 7.242+
(Yanis Gomes) 		36

s = 7.242+
(Yanis Gomes)
37

s=(8+10√2)/3=7.380+
(Yanis Gomes) 		38

s = 7.535+
(Yanis Gomes) 		39

s = 2+4√2 = 7.656+
(Yanis Gomes) 		40

s = 7.810+
(Yanis Gomes)
41

s = 7.914+
(Yanis Gomes) 		42

s = 8
(Yanis Gomes) 		43

s = 8
(Yanis Gomes) 		44

s = 8
(Yanis Gomes)
45

s = 8.047+
(Yanis Gomes) 		46

s = 8.242+
(Yanis Gomes) 		47

s = 8.2426+
(Yanis Gomes) 		48

s=(4+9√2)/2=8.363+
(Yanis Gomes)
49

s=(10+5√2)/2=8.535+
(Yanis Gomes) 		50

s = 8.599+
(Yanis Gomes)

3-Sliced Squares in Squares
1

s = 1 		2

s = √2 = 1.414+ 		3

s = 2(1+√2)/3 = 1.609+ 		4

s = 5√2/4 = 1.767+
5

s = √2+√10/5 = 2.046+
(Maurizio Morandi) 		6

s = 3/√2 = 2.121+
  		7

s =(4+7√2)/6= 2.316+
(Maurizio Morandi) 		8

s = 1 + √2 = 2.414+
(David W. Cantrell)
9

s = 11√2/6 = 2.592+
(Maurizio Morandi) 		10

s =1+5√2/4= 2.767+
(Maurizio Morandi) 		11

s = 2√2 = 2.828+
  		12

s = 2√2 = 2.828+
(Maurizio Morandi)
13

s=(6+35√2)/18=3.083+
(David W. Cantrell) 		14

s = (6+7√2)/5 = 3.179+
(Yanis Gomes) 		15

s=(4+53√2)/24=3.289+
(David W. Cantrell) 		16

s = 3.366+
(Yanis Gomes)
17

s = 3.490+
(Yanis Gomes) 		18

s = 5/√2 = 3.535+
(Yanis Gomes) 		19

s = 3.661+
(Yanis Gomes) 		20

s = 3.745+
(Yanis Gomes)
21

s = 3.840+
(Yanis Gomes) 		22

s = 3.911+
(Yanis Gomes) 		23

s=(13+22√2)/11=4.010+
(Yanis Gomes) 		24

s = 4.101+
(Yanis Gomes)
25

s = 4.194+
(Yanis Gomes) 		26

s = 3√2 = 4.242+
(Yanis Gomes) 		27

s = 4.334+
(Yanis Gomes) 		28

s = 4.421+
(Yanis Gomes)
29

s = 4.517+
(Yanis Gomes) 		30

s = 4.558+
(Yanis Gomes) 		31

s = 13√2/4 = 4.596+
(Yanis Gomes) 		32

s = 4.704+
(Yanis Gomes)
33

s = 4.775+
(Yanis Gomes) 		34

s = 4.842+
(Yanis Gomes) 		35

s = 4.927+
(Yanis Gomes) 		36

s = 5.012+
(Yanis Gomes)
37

s = 5.052+
(Yanis Gomes) 		38

s = 5.133+
(Yanis Gomes) 		39

s = 5.206+
(Yanis Gomes) 		40

s = 5.246+
(Yanis Gomes)
41

s = 277√2/74 = 5.293+
(Yanis Gomes) 		42

s = 15√2/4 = 5.303+
(Yanis Gomes) 		43

s = 5.417+
(Yanis Gomes) 		44

s = 5.523+
(Yanis Gomes)
45

s = 5.626+
(Yanis Gomes) 		46

s = 4√2 = 5.656+
(Yanis Gomes) 		47

s = 5.760+
(Yanis Gomes) 		48

s = 5.806+
(Yanis Gomes)
49

s = 5.821+
(Yanis Gomes) 		50

s = 5.944+
(Yanis Gomes)

3-Sliced Squares in 3-Sliced Squares
1

s = 1
  		2

s = (2+2√2)/3 = 1.609+
  		3

s = 2
  		4

s = (3+√2)/2 = 2.207+
(Maurizio Morandi)
5

s = 2.498+
(Yanis Gomes) 		6

s = 2.742+
(Maurizio Morandi) 		7

s = (9+4√2)/5 = 2.931+
(Maurizio Morandi) 		8

s = 3
(Maurizio Morandi)
9

s = 11/6 + √2 = 3.247+
(Maurizio Morandi) 		10

s = 2 + √2 = 3.414+
(Maurizio Morandi) 		11

s = 4(3+√2)/5 = 3.531+
(Maurizio Morandi) 		12

s=(13+4√2)/5=3.731+
(Yanis Gomes)
13

s = 3.886+
(Yanis Gomes) 		14

s=(33+2√2)/9=3.980+
(Maurizio Morandi) 		15

s = 3+4√2/5 = 4.131+
(Maurizio Morandi) 		16

s=(27+11√2)/10=4.255+
(Yanis Gomes)
17

s = 4.358+
(Yanis Gomes) 		18

s=(55+15√2)/17=4.483+
(Yanis Gomes) 		19

s = 4.597+
(Yanis Gomes) 		20

s = 4.692+
(Yanis Gomes)
21

s = 4.850+
(Yanis Gomes) 		22

s = 4.932+
(Yanis Gomes) 		23

s=(51+10√2)/13=5.010+
(Yanis Gomes) 		24

s = 5.131+
(Yanis Gomes)
25

s = 5.259+
(Yanis Gomes) 		26

s = 5.381+
(Yanis Gomes) 		27

s = 5.470+
(Yanis Gomes) 		28

s = 5.571+
(Yanis Gomes)
29

s = 5.670+
(Yanis Gomes) 		30

s=(82+36√2)/23=5.778+
(Yanis Gomes) 		31

s=(39+14√2)/10=5.879+
(Yanis Gomes) 		32

s = 5.968+
(Yanis Gomes)
33

s = 6.017+
(Yanis Gomes) 		34

s = 6.131+
(Yanis Gomes) 		35

s = 6.288+
(Yanis Gomes) 		36

s = 6.311+
(Yanis Gomes)
37

s = 6.385+
(Yanis Gomes) 		38

s = 6.480+
(Yanis Gomes) 		39

s = 6.564+
(Yanis Gomes) 		40

s = 6.667+
(Yanis Gomes)
41

s = 6.754+
(Yanis Gomes) 		42

s = 6.849+
(Yanis Gomes) 		43

s = 6.938+
(Yanis Gomes) 		44

s = 6.994+
(Yanis Gomes)
45

s = 7.076+
(Yanis Gomes) 		46

s = 7.159+
(Yanis Gomes) 		47

s = 7.205+
(Yanis Gomes) 		48

s = 7.279+
(Yanis Gomes)
49

s = 7.308+
(Yanis Gomes) 		50

s = 7.385+
(Yanis Gomes)

Squares in 3-Sliced Squares
1

s = √2 = 1.414+ 		2

s = 2(2+√2)/3 = 2.276+ 		3

s = 2√2 = 2.828+ 		4

s = 2√2 = 2.828+
5

s = 1 + 3/√2 = 3.121+
  		6

s = 3.665+
(David W. Cantrell) 		7

s = 3.912+
(David W. Cantrell) 		8

s = 2 + 3/√2 = 4.121+
 
9

s = 3√2 = 4.242+ 		10

s = 3√2 = 4.242+ 		11

s = 10√2/3 = 4.714+ 		12

s = 2 + 2√2 = 4.828+
13

s = 11√2/3 = 5.185+
(Maurizio Morandi) 		14

s = 5.385+
(Yanis Gomes) 		15

s = 5.544+
(Yanis Gomes) 		16

s = 4√2 = 5.656+
 
17

s = 4√2 = 5.656+
(Yanis Gomes) 		18

s = 4√2 = 5.656+
(Yanis Gomes) 		19

s = 5.771+
(Yanis Gomes) 		20

s = 6.121+
(Yanis Gomes)
21

s = 6.407+
(Yanis Gomes) 		22

s=(8+8√2)/3=6.437+
(Yanis Gomes) 		23

s=(6+5√2)/2=6.535+
(Yanis Gomes) 		24

s = 4+2√2 = 6.828+
(Yanis Gomes)
25

s = 6.942+
(Yanis Gomes) 		26

s = 5√2 = 7.071+
(Yanis Gomes) 		27

s = 5√2 = 7.071+
(Yanis Gomes) 		28

s = 5√2 = 7.071+
(Yanis Gomes)
29

s = 5√2 = 7.071+
(Yanis Gomes) 		30

s = 7.242+
(Yanis Gomes) 		31

s = 7.380+
(Yanis Gomes) 		32

s = 2+4√2 = 7.656+
(Yanis Gomes)
33

s = 7.823+
(Yanis Gomes) 		34

s=(18+4√2)/3=7.885+
(Yanis Gomes) 		35

s = 8.047+
(Yanis Gomes) 		36

s=(12+3√2)/2=8.121+
(Yanis Gomes)
37

s = 8.242+
(Yanis Gomes) 		38

s = 8.357+
(Yanis Gomes) 		39

s = 6√2 = 8.485+
(Yanis Gomes) 		40

s = 6√2 = 8.485+
(Yanis Gomes)
41

s = 6√2 = 8.485+
(Yanis Gomes) 		42

s = 6√2 = 8.485+
(Yanis Gomes) 		43

s = 8.599+
(Yanis Gomes) 		44

s = 8.872+
(Yanis Gomes)
45

s = 8.949+
(Yanis Gomes) 		46

s=(8+7√2)/2=8.949+
(Yanis Gomes) 		47

s = 9.242+
(Yanis Gomes) 		48

s = 9.242+
(Yanis Gomes)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/15/12.