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?

Here are the best-known results:

Two Circles
 n=0r = 1/2 = .500 n=1r = 9/16 = .562+Maurizio Morandi n=2r = .611+Maurizio Morandi n=3r = .656+Joe DeVincentis

Three Circles
 n=0r = 2√3–3 = .464+ n=1r = .478+Maurizio Morandi n=2r = 1/2 = .500 Maurizio Morandi n=3r = .510+ Jeremy Galvagni

Four Circles
 n=0r = √2–1 = .414+ n=1r = .427+Maurizio Morandi n=2r = .433+Jeremy Galvagni

Five Circles
 n=0r = .370+ n=1r = .382+Maurizio Morandi

One Square
 n=0, 1s = √2 = 1.414+ n=2s = 1.523+Maurizio Morandi n=3s = 1.551+Maurizio Morandi n=4s = 1.614+Jeremy Galvagni

Two Squares
 n=0s = 2/√5 = .894+ n=1s = .985+Maurizio Morandi n=2s = 1.035+Maurizio Morandi n=3s = 1.090+Maurizio Morandi

Three Squares
 n=0s = 16/5√17 = .776+ n=1s = .808+Jeremy Galvagni n=2s = 2/√5 = .894+Maurizio Morandi

Four Squares
 n=0s = 1/√2 = .707+ n=1s = .736+Maurizio Morandi

One Triangle
 n=0s = √3 = 1.732+ n=1s = 1.882+Maurizio Morandi n=2s = 2.122+Jeremy Galvagni n=3s = 2.237+Maurizio Morandi n=4s = 2.265+Jeremy Galvagni

Two Triangles
 n=0s = 2/√3 = 1.154+ n=1s = 1.462+Maurizio Morandi n=2s = 4/√7 = 1.511+Maurizio Morandi n=3s = 1.543+Maurizio Morandi

Three Triangles
 n=0s = (√3+√2)/3 = 1.048+ n=1s = 1.151+Jeremy Galvagni

One Pentagon
 n=0, 1s = 1.175+ n=2s = 1.203+Jeremy Galvagni

One Hexagon
 n=0, 1s = 1 n=2s = 1.007+Maurizio Morandi n=3s = 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.