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=2.25 (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=9

s=10/3=3.333+ (Bryce Herdt)
s=3 (Erich Friedman)
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.