# 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?

Here are the best-known packings:

1-Sliced Squares in Squares
 1s = 1 2s = 3/2 = 1.5 3s = 3(1+√2)/4 = 1.810+ 4s = 2 5s = (3+√2)/2 = 2.207+ 6s = 5/2 = 2.5 7s = 2.700+(DC after MM) 8s = 2.834+(David W. Cantrell) 9s = 3 10s = 3.174+(David W. Cantrell) 11s =(50+21√2)/24= 3.320+(Maurizio Morandi) 12s =(29+√31)/10= 3.456+(David W. Cantrell) 13s=(52+19√2)/22=3.585+(Maurizio Morandi) 14s = 3 + 1/√2 = 3.707+(Maurizio Morandi) 15s=(66+19√2)/24=3.869+(Maurizio Morandi) 16s = (6+7√2)/4 = 3.974+(Maurizio Morandi)

1-Sliced Squares in 1-Sliced Squares
 1s = 1 2s = 5/3 = 1.666+ 3s = 1+2√2/3 = 1.942+ 4s = (3+√2)/2 = 2.207+ 5s = 1 + √2 = 2.414+(Maurizio Morandi) 6s = 8/3 = 2.666+ 7s = 1+4√2/3 = 2.885+(David W. Cantrell) 8s = 3 9s = (5+√2)/2 = 3.207+(Bryce Herdt) 10s=3(9+4√2)/13=3.382+(Maurizio Morandi) 11s = 3.550+(David W. Cantrell) 12s =(19+5√2)/7= 3.724+(Maurizio Morandi) 13s=(59+37√2)/29=3.838+(Maurizio Morandi) 14s=(36+31√2)/20=3.992+(Maurizio Morandi) 15s=(61+21√2)/22=4.122+(Maurizio Morandi) 16s = 4.247+(David W. Cantrell)

Squares in 1-Sliced Squares
 1s = 4/3 = 1.333+ 2s = 2 3s = 2 4s = 8/3 = 2.666+ 5s = 2 + 2/√3 = 2.942+(Maurizio Morandi) 6s = 3 7s = 4(1+√2)/3 = 3.218+ 8s = 10/3 = 3.333+ 9s = 2+5√2/4 = 3.767+ 10s = 2+4√2/3 = 3.885+(Maurizio Morandi) 11s = 4 12s = 4 13s = 4 14s = 3 + √2 = 4.414+ 15s = 14/3 = 4.666+(David W. Cantrell) 16s = 2 + 2√2 = 4.828+(David W. Cantrell)

2-Sliced Squares in Squares
 1s = 1 2s = √2 = 1.414+ 3s = (1+√2)/2 = 1.707+ 4s = 2 5s = 3√2/2 = 2.121+ 6s = 2 + √2/4 = 2.353+(David W. Cantrell) 7s = 1+11√2/10 = 2.555+(Maurizio Morandi) 8s = 2 + 1/√2 = 2.707+(Maurizio Morandi) 9s = 2√2 = 2.828+(Maurizio Morandi) 10s =(2+7√2)/4= 2.974+(Maurizio Morandi) 11s = 1 + 3/√2 = 3.121+(Maurizio Morandi) 12s =(6+5√2)/4= 3.267+(Maurizio Morandi) 13s = 2 + √2 = 3.414+(Maurizio Morandi) 14s = 5/√2 = 3.535+(Maurizio Morandi) 15s =1+15√2/8= 3.651+(Maurizio Morandi) 16s=(15+16√2)/10=3.762+(David W. Cantrell)

2-Sliced Squares in 2-Sliced Squares
 1s = 1 2s = 1.873+(Maurizio Morandi) 3s = 5/4 + 1/√2 = 1.957+(Maurizio Morandi) 4s = 2(2+√2)/3 = 2.276+ 5s = 5/2 = 2.5(Maurizio Morandi) 6s = 2 + 1/√2 = 2.707+(Maurizio Morandi) 7s=(32+19√2)/20=2.943+(Maurizio Morandi) 8s=(25+13√2)/14=3.098+(Maurizio Morandi) 9s = 4(1+√2)/3 = 3.218+(Maurizio Morandi) 10s =(11+2√2)/4= 3.457+(Maurizio Morandi) 11s= 2(4+√2)/3= 3.609+(Maurizio Morandi) 12s = 7/3 + √2 = 3.747+(Maurizio Morandi) 13s = 47/12 = 3.916+(Maurizio Morandi) 14s=(30+19√2)/14=4.062+(Maurizio Morandi) 15s =(7+4√2)/3= 4.218+(Maurizio Morandi) 16s =(16+7√2)/6= 4.316+(Maurizio Morandi)

Squares in 2-Sliced Squares
 1s = √2 = 1.414+(Andrew Bayly) 2s = 2 3s = 2(2+√2)/3 = 2.276+ 4s = 2√2 = 2.828+ 5s = 3 6s = 1 + 3/√2 = 3.121+(Maurizio Morandi) 7s = 7/2 = 3.5 8s = 1 + 2√2 = 3.828+(David W. Cantrell) 9s = 5/2 + √2 = 3.914+ 10s = 4 11s = 2 + 3/√2 = 4.121+ 12s = 9/2 = 4.5 13s = 10√2/3 = 4.714+ 14s = 2 + 2√2 = 4.828+ 15s = 2 + 2√2 = 4.828+ 16s = 5

3-Sliced Squares in Squares
 1s = 1 2s = √2 = 1.414+ 3s = 2(1+√2)/3 = 1.609+ 4s = 5√2/4 = 1.767+ 5s = √2+√10/5 = 2.046+(Maurizio Morandi) 6s = 3/√2 = 2.121+ 7s =(4+7√2)/6= 2.316+(Maurizio Morandi) 8s = 1 + √2 = 2.414+(David W. Cantrell) 9s = 11√2/6 = 2.592+(Maurizio Morandi) 10s =1+5√2/4= 2.767+(Maurizio Morandi) 11s = 2√2 = 2.828+ 12s = 2√2 = 2.828+(Maurizio Morandi) 13s=(6+35√2)/18=3.083+(David W. Cantrell) 14s = 9√2/4 = 3.181+ 15s=(4+53√2)/24=3.289+(David W. Cantrell) 16s = 3.384+(Maurizio Morandi)

3-Sliced Squares in 3-Sliced Squares
 1s = 1 2s = (2+2√2)/3 = 1.609+ 3s = 2 4s = (3+√2)/2 = 2.207+(Maurizio Morandi) 5s = 2.499+(David W. Cantrell) 6s = 2.742+(Maurizio Morandi) 7s = (9+4√2)/5 = 2.931+(Maurizio Morandi) 8s = 3(Maurizio Morandi) 9s = 11/6 + √2 = 3.247+(Maurizio Morandi) 10s = 2 + √2 = 3.414+(Maurizio Morandi) 11s =4(3+√2)/5= 3.531+(Maurizio Morandi) 12s=(29+6√2)/10=3.748+(Maurizio Morandi) 13s=(23+3√2)/7=3.891+(Maurizio Morandi) 14s=(33+2√2)/9=3.980+(Maurizio Morandi) 15s = 3+4√2/5 = 4.131+(Maurizio Morandi) 16s =2(5+√2)/3= 4.276+(Maurizio Morandi)

Squares in 3-Sliced Squares
 1s = √2 = 1.414+ 2s = 2(2+√2)/3 = 2.276+ 3s = 2√2 = 2.828+ 4s = 2√2 = 2.828+ 5s = 1 + 3/√2 = 3.121+ 6s = 3.665+(David W. Cantrell) 7s = 3.912+(David W. Cantrell) 8s = 2 + 3/√2 = 4.121+ 9s = 3√2 = 4.242+ 10s = 3√2 = 4.242+ 11s = 10√2/3 = 4.714+ 12s = 2 + 2√2 = 4.828+ 13s = 11√2/3 = 5.185+(Maurizio Morandi) 14s = 5.388+(Maurizio Morandi) 15s = 4/3+3√2 = 5.575+(David W. Cantrell) 16s = 4√2 = 5.656+

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.