Problem of the Month (May 2018)

Given an integer n≥2, consider a configuration of circles so that every circle of radius r>1 touches n other circles of radius r–1. How large can the largest circle be? What about other shapes such as squares, triangles, and hexagons?

ANSWERS

Gordon Atkinson pointed out that n=2 is trivial, as r can be arbitrarily large if we make a "triangle" of circles as shown below. The same trick can be used for other shapes as well.

The best known solutions are shown below.

Circles

n=3 r=6.193+ (Joe DeVincentis) r=6.000+ (Maurizio Morandi) r=5.957+ (Joe DeVincentis) r=5.15+ (Erich Friedman)	n=4 r=4.326+ (Joe DeVincentis) r=3.75+ (Erich Friedman)
n=5 r=3.515+ (Joe DeVincentis) r=3.12+ (Erich Friedman)	n=6 r=2.882+ (Maurizio Morandi) r=2.83+ (Erich Friedman)
n=7 r=2.494+ (Erich Friedman)	n=8 r=2.330+ (Maurizio Morandi) r=2.329+ (Erich Friedman)
n=9 r=2.082+ (Erich Friedman)	n=10 r=(5+√5)/4=1.809+ (Erich Friedman)

Squares

n=3 s=8 (Maurizio Morandi) s=7.5 (Joe DeVincentis) s=7 (Maurizio Morandi)	n=4 s=5 (Maurizio Morandi)
n=5 s=4 (Maurizio Morandi) s=3.5 (Jeremy Galvagni)	n=6 s=2+√2=3.414+ (Maurizio Morandi)
n=7 s=3 (Bryce Herdt)	n=8 s=2.658+ (Maurizio Morandi) s=2.658+ (Bryce Herdt) s=2.544+ (Joe DeVincentis) s=2.5 (Erich Friedman)
n=9 s=1+√2=2.414+ (Jeremy Galvagni)	n=10 s=2.276+ (Maurizio Morandi) s=2.25 (Joe DeVincentis) s=2.25 (Jeremy Galvagni)
n=11 s=2.042+ (Maurizio Morandi) s=2.013+ (Jeremy Galvagni)	n=12 s=2 (George Sicherman)

Triangles

n=3 s=12 (Maurizio Morandi) s=11 (Jean Hoffman)	n=4 s=7.5 (Maurizio Morandi) s=7 (Jean Hoffman)
n=5 s=5 (Jean Hoffman)	n=6 s=9/2=4.5 (Maurizio Morandi) s=4 (Jean Hoffman)
n=7 s=4 (Maurizio Morandi)	n=8 s=7/2=3.5 (Maurizio Morandi)
n=9 s=10/3=3.333+ (Bryce Herdt) s=3 (Erich Friedman)	n=10 s=11/4=2.75 (Maurizio Morandi)
n=12 s=5/2=2.5 (Maurizio Morandi)	n=18 s=2 (Erich Friedman)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/1/18.