Problem of the Month (November 2017)

Start with a unit disk. Make n straight cuts so that the disk is cut into n+1 pieces. Rearrange the non-overlapping pieces so that 2 equal non-overlapping disks can be packed inside. How should the original disk be cut and rearranged so as to maximize the radius of the disks to be fit inside? What if n equal disks are to be packed in the original cut up disk? What about n equal squares or equilateral triangles?


ANSWERS

Here are the best-known results:

Two Circles
n=0

r = 1/2 = .500
n=1

r = 9/16 = .562+
Maurizio Morandi
n=2

r = .611+
Maurizio Morandi
n=3

r = .656+
Joe DeVincentis

Three Circles
n=0

r = 2√3–3 = .464+
n=1

r = .478+
Maurizio Morandi
n=2

r = 1/2 = .500
Maurizio Morandi
n=3

r = .510+
Jeremy Galvagni

Four Circles
n=0

r = √2–1 = .414+
n=1

r = .427+
Maurizio Morandi
n=2

r = .433+
Jeremy Galvagni

Five Circles
n=0

r = .370+
n=1

r = .382+
Maurizio Morandi

One Square
n=0, 1

s = √2 = 1.414+
n=2

s = 1.523+
Maurizio Morandi
n=3

s = 1.551+
Maurizio Morandi
n=4

s = 1.614+
Jeremy Galvagni

Two Squares
n=0

s = 2/√5 = .894+
n=1

s = .985+
Maurizio Morandi
n=2

s = 1.035+
Maurizio Morandi
n=3

s = 1.090+
Maurizio Morandi

Three Squares
n=0

s = 16/5√17 = .776+
n=1

s = .808+
Jeremy Galvagni
n=2

s = 2/√5 = .894+
Maurizio Morandi

Four Squares
n=0

s = 1/√2 = .707+
n=1

s = .736+
Maurizio Morandi

One Triangle
n=0

s = √3 = 1.732+
n=1

s = 1.882+
Maurizio Morandi
n=2

s = 2.122+
Jeremy Galvagni
n=3

s = 2.237+
Maurizio Morandi
n=4

s = 2.265+
Jeremy Galvagni

Two Triangles
n=0

s = 2/√3 = 1.154+
n=1

s = 1.462+
Maurizio Morandi
n=2

s = 4/√7 = 1.511+
Maurizio Morandi
n=3

s = 1.543+
Maurizio Morandi

Three Triangles
n=0

s = (√3+√2)/3 = 1.048+
n=1

s = 1.151+
Jeremy Galvagni

One Pentagon
n=0, 1

s = 1.175+
n=2

s = 1.203+
Jeremy Galvagni

One Hexagon
n=0, 1

s = 1
n=2

s = 1.007+
Maurizio Morandi
n=3

s = 1.018+
Jeremy Galvagni


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/19/17.