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 = 2.834+
(David W. Cantrell)
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=(66+19√2)/24=3.869+
(Maurizio Morandi)
16

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

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 = 8/3 = 2.666+
 
7

s = 1+4√2/3 = 2.885+
(David W. Cantrell)
8

s = 3
 
9

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

s=3(9+4√2)/13=3.382+
(Maurizio Morandi)
11

s = 3.550+
(David W. Cantrell)
12

s =(19+5√2)/7= 3.724+
(Maurizio Morandi)
13

s=(59+37√2)/29=3.838+
(Maurizio Morandi)
14

s=(36+31√2)/20=3.992+
(Maurizio Morandi)
15

s=(61+21√2)/22=4.122+
(Maurizio Morandi)
16

s = 4.247+
(David W. Cantrell)

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/√3 = 2.942+
(Maurizio Morandi)
6

s = 3
 
7

s = 4(1+√2)/3 = 3.218+
 
8

s = 10/3 = 3.333+
 
9

s = 2+5√2/4 = 3.767+
 
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)


2-Sliced Squares in Squares
1

s = 1
2

s = √2 = 1.414+
3

s = (1+√2)/2 = 1.707+
4

s = 2
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 = 2 + √2 = 3.414+
(Maurizio Morandi)
14

s = 5/√2 = 3.535+
(Maurizio Morandi)
15

s =1+15√2/8= 3.651+
(Maurizio Morandi)
16

s=(15+16√2)/10=3.762+
(David W. Cantrell)

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 =(11+2√2)/4= 3.457+
(Maurizio Morandi)
11

s= 2(4+√2)/3= 3.609+
(Maurizio Morandi)
12

s = 7/3 + √2 = 3.747+
(Maurizio Morandi)
13

s = 47/12 = 3.916+
(Maurizio Morandi)
14

s=(30+19√2)/14=4.062+
(Maurizio Morandi)
15

s =(7+4√2)/3= 4.218+
(Maurizio Morandi)
16

s =(16+7√2)/6= 4.316+
(Maurizio Morandi)

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


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 = 9√2/4 = 3.181+
 
15

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

s = 3.384+
(Maurizio Morandi)

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 = 2.499+
(David W. Cantrell)
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=(29+6√2)/10=3.748+
(Maurizio Morandi)
13

s=(23+3√2)/7=3.891+
(Maurizio Morandi)
14

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

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

s =2(5+√2)/3= 4.276+
(Maurizio Morandi)

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 = 5.388+
(Maurizio Morandi)
15

s = 4/3+3√2 = 5.575+
(David W. Cantrell)
16

s = 4√2 = 5.656+
 


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