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.