Problem of the Month (March 2011)

The problem of packing unit circles into various shapes has been well studied. This month we consider the problem of packing circles of integer radii and total area nπ into the smallest squares, circles, and triangles. What are the answers for small n?

ANSWERS

Answers were received this month from Joe DeVincentis, Maurizio Morandi, Jeremy Galvagni, Bryce Herdt, and Jon Palin.

Here are the best known answers for squares:

Best-Known Packings of Integer Circles in Squares
  n  best known packings
1

s = 2
2

s = 2+√2
= 3.414+
3

s = (4+√2+√6)/2
= 3.931+
4

s = 4
s = 4
5

s = 2+2√2
= 4.828+
s = 3+3√2/2
= 5.121+
6

s = 5.328+
 
s = (14+4√11)/5
= 5.453+
7

s = 4+√3
= 5.732+
s = 5.774+
 
8

s = 2+√2+√6
= 5.863+
s = 3+2√2
= 5.828+
s = 4+2√2
= 6.828+
9

s = 6
 
s = 5.987+
 
s = 4+2√2
= 6.828+
s = 6
 
10

s = 6.747+
 
s = 6.644+
 
s = 4+2√2
= 6.828+
s = 4+2√2
= 6.828+
11

s = 7.022+
 
s = 7.026+ (MM)
 
s = 7.099+
 
s = (18+4√19)/5
= 7.087+
12

s = 2+15√34/17
= 7.144+
s = 7.291+ (MM)
 
s = 3+3√2
= 7.242+ (JD)
s = 4+√2+√6
= 7.863+
s = 7.454+
 
13

s = 7.463+
 
s = 7.403+
 
s = 7.619+ (MM)
 
s = 4+√2+√6
= 7.863+
s = 4+2√3
= 7.464+
s = 5+5√2/2
= 8.535+
14

s = 6+√3
= 7.732+
s = 7.626+ (JP)
 
s = 7.667+ (MM)
 
s = 7.938+
 
s = 5+√7
= 7.645+
s = 5+5√2/2
= 8.535+
15

s = 4+√2+√6
= 7.863+
s = 7.855+ (MM)
 
s = 7.752+ (MM)
 
s = 7.958+ (MM)
 
s = 7.950+ (MM)
 
s = 5+5√2/2
= 8.535+
16

s = 8
 
s = 7.986+
 
s = 5+2√2
= 7.828+
s = 7.994+
 

s = 8
 
s = 7.997+
 
s = 5+5√2/2
= 8.535+
s = 8
 
17

s = 8.532+
 
s = 8.449+
 
s = 8.378+ (MM)
 
s = 8.380+
 
s = 4+3√2
= 8.242+

s = 8.491+
 
s = 5+5√2/2
= 8.535+
s = (24+4√31)/5
= 9.254+
s = 5+5√2/2
= 8.535+
18

s = 8.656+
 
s = 8.729+ (MM)
 
s = 8.656+ (JD)
 
s = 8.719+ (MM)
 
s = 8.791+
 

s = 8.732+ (MM)
 
s = 8.807+ (MM)
 
s = (24+4√31)/5
= 9.254+
s = 6+3√2
= 10.242+ (JD)
s = (22+4√29)/5
= 8.708+
19

s = 8.907+
 
s = 8.917+ (JD)
 
s = 8.920+ (MM)
 
s = 4+7/√2
= 8.949+ (MM)
s = 9.033+ (MM)
 

s = 8.877+ (MM)
 
s = 8.981+
 
s = (24+4√31)/5
= 9.254+
s = 6+3√2
= 10.242+ (JD)
s = 9
 
20

s = 8.978+
 
s = 9.097+ (JD)
 
s = 9.070+ (MM)
 
s = 9.149+ (MM)
 
s = 9.318+ (MM)
 
s = 4+4√2
= 9.656+

s = 9.173+ (MM)
 
s = 9.208+ (MM)
 
s = 9.284+
 
s = 6+3√2
= 10.242+ (JD)
s = 2+5√2
= 9.071+
s = 6+3√2
= 10.242+
21

s = 9.358+
 
s = 9.278+ (MM)
 
s = 9.256+ (MM)
 
s = 9.305+ (MM)
 
s = 9.429+ (JD)
 

s = 4+4√2
= 9.656+ (JD)
s = 4+2√7
= 9.291+ (MM)
s = 9.337+ (MM)
 
s = (202+128√2)/41
= 9.341+ (MM)

s = 9.883+ (MM)
 
s = 6+3√2
= 10.242+
s = 6+√46/2
= 9.391+ (MM)
s = 6+3√2
= 10.242+
22

s = 9.463+
 
s = 9.445+ (JD)
 
s = 9.455+ (MM)
 
s = 9.503+ (MM)
 
s = 9.548+ (MM)
 

s = 4+4√2
= 9.656+ (JD)
s = 9.374+ (JD)
 
s = 9.454+ (MM)
 
s = 9.474+ (JP)
 
s = 9.883+ (MM)
 

s = 6+3√2
= 10.242+
s = 6+3√2
= 10.242+
s = 9.666+ (JD)
 
s = 6+3√2
= 10.242+
23

s = 2+2√2+2√6
= 9.727+
s = 9.706+ (JP)
 
s = 9.596+ (JD)
 
s = 9.583+ (MM)
 
s = 9.613+ (MM)
 

s = 4+4√2
= 9.656+ (JD)
s = 9.622+ (JD)
 
s = 9.569+ (MM)
 
s = 9.679+ (MM)
 
s = 9.883+ (MM)
 

s = 6+3√2
= 10.242+
s = 6+3√2
= 10.242+
s = 9.834+ (MM)
 
s = 6+3√2
= 10.242+
24

s = 6+√2+√6
= 9.863+
s = 9.851+ (JD)
 
s = 9.777+ (MM)
 
s = 9.622+ (JP)
 

s = 9.644+ (MM)
 
s = 4+4√2
= 9.656+ (JD)
s = 4+24/√13
= 10.656+
s = 9.873+ (MM)
 

s = 9.798+ (MM)
 
s = 9.779+ (MM)
 
s = 9.883+ (MM)
 
s = 6+3√2
= 10.242+

s = 6+3√2
= 10.242+
s = 9.855+ (JD)
 
s = 6+3√2
= 10.242+
s = (28+8√11)/5
= 10.906+
25

s = 10
 
s = 9.986+ (MM)
 
s = 9.937+ (JD)
 
s = 9.821+ (JD)
 
s = 4+4√2
= 9.656+ (JD)

s = 10.267+ (MM)
 
s = 4+24/√13
= 10.656+
s = 9.992+ (JD)
 
s = 9.960+ (JD)
 
s = 9.886+ (JD)
 

s = 9.896+ (JD)
 
s = 10.739+ (JD)
 
s = 6+3√2
= 10.242+
s = 6+3√2
= 10.242+
s = 9.892+ (MM)
 

s = 6+3√2
= 10.242+
s = (28+8√11)/5
= 10.906+
s = 7+7/√2
= 11.949+
s = 10
 

Here are the best known answers for circles:

Best-Known Packings of Integer Circles in Circles
  n  best known packings
1

r = 1
2

r = 2
3

r = 1+2/√3
= 2.154+
4

r = 1+√2
= 2.414+
r = 2
 
5

r = 2.701+
r = 3
6

r = 3
 
r = (10+8√2)/7
= 3.044+
7

r = 3
r = 3.170+
8

r = 3.304+
r = 3.355+
r = 4
9

r = 3.613+
r = 3.581+
r = 4
r = 3
10

r = 3.813+
r = 3.802+ (MM)
r = 4
r = 4
11

r = 3.923+
 
r = 3.915+
 
r = 4
 
r = (21+6√15)/11
= 4.021+
12

r = 4.029+
 
r = 4
 
r = 4
 
r = 2+4/√3
= 4.309+
r = 4.086+
 
13

r = 2+√5
= 4.236+
r = 4
 
r = 4.210+
 
r = 4.323+
 
r = 4.192+
 
r = 5
 
14

r = 4.328+
 
r = 2+√5
= 4.236+
r = 4.248+ (MM)
 
r = 4.343+
 
r = 4.336+
 
r = 5
 
15

r = 4.521+
 
r = 4.432+ (MM)
 
r = 4.482+ (MM)
 
r = (3+√33)/2
= 4.372+
r = 4.514+ (MM)
 
r = 5
 
16

r = 4.615+
r = 4.659+ (MM)
r = 4.561+ (MM)
r = 4.620+ (MM)

r = 2+2√2
= 4.828+
r = 4.684+
 
r = 5
 
r = 4
 
17

r = 4.792+
 
r = 4.762+ (MM)
 
r = 4.776+ (JD)
 
r = 4.746+ (JD)
 
r = 4.907+ (JD)
 

r = 4.807+ (JD)
 
r = 5
 
r = 5.149+ (JD)
 
r = 5
 
18

r = 1+√2+√6
= 4.863+
r = 4.874+ (MM)
 
r = 4.886+ (JD)
 
r = 4.880+ (JD)
 
r = 5 (JD)
 

r = 4.932+ (MM)
 
r = 5.000+ (JD)
 
r = 5.149+ (JD)
 
r = 6 (JD)
 
r = 5.012+ (JD)
 
19

r = 1+√2+√6
= 4.863+ (JD)
r = 4.968+ (JD)
 
r = 4.932+ (JD)
 
r = 5 (JD)
 
r = 5 (JD)
 

r = 4.987+ (JD)
 
r = 5.023+ (MM)
 
r = 5.149+ (JD)
 
r = 6 (JD)
 
r = 5.051+ (JD)
 

Here are the best known answers for triangles:

Best-Known Packings of Integer Circles in Triangles
  n  best known packings
1

s = 2√3
= 3.464+
2

s = 2+2√3
= 5.464+
3

s = 2+2√3
= 5.464+
4

s = 4√3
= 6.928+
s = 4√3
= 6.928+
5

s = 4+2√3
= 7.464+
s = 3√3+2√2
= 8.024+
6

s = 4+2√3
= 7.464+
s = 3√3+2√2
= 8.024+
7

s = 2+4√3
= 8.928+
s = 5√3
= 8.660+
8

s = 9.293+
 
s = 9.411+ (MM)
 
s = 4+4√3
= 10.928+
9

s = 6+2√3
= 9.464+
s = 2+8/√3+8/√6
= 9.884+ (MM)
s = 4+4√3
= 10.928+
s = 6√3
= 10.392+
10

s = 6+2√3
= 9.464+
s = 2+2√2+3√3
= 10.024+ (MM)
s = 4+4√3
= 10.928+
s = 6√3
= 10.392+
11

s = 10.730+
 
s = 10.640+ (MM)
 
s = 4+4√3
= 10.928+
s = 6√3
= 10.392+
12

s = 4+4√3
= 10.928+
s = 11.159+ (MM)
 
s = 11.105+ (MM)
 
s = 4+4√3
= 10.928+
s = 6√3
= 10.392+
13

s = 11.406+
 
s = 3+2√3+2√6
= 11.363+ (MM)
s = 2√2+5√3
= 11.488+ (MM)

s = (9√3+3√7)/2
= 11.762+ (MM)
s = 11.919+ (MM)
 
s = 5√3+2√6
= 13.559+
14

s = 8+2√3
= 11.464+
s = 4+8/√3+8/√6
= 11.884+ (MM)
s = 11.959+ (MM)
 

s = 4+2√2+3√3
= 12.024+ (MM)
s = 11.919+ (MM)
 
s = 5√3+2√6
= 13.559+
15

s = 8+2√3
= 11.464+
s = 12.010+ (MM)
 
s = 4+2√2+3√3
= 12.024+ (MM)

s = 4+10/√3+10/√15
= 12.355+ (MM)
s = 12.361+ (MM)
 
s = 5√3+2√6
= 13.559+
16

s = 12.713+
 
s = 12.530+ (MM)
 
s = 12.379+ (MM)
 
s = 4√2+4√3
= 12.585+ (MM)

s = 8√3
= 13.856+
s = 2+6√3
= 12.392+ (MM)
s = 5√3+2√6
= 13.559+
s = 8√3
= 13.856+
17

s = 12.928+
 
s = 12.920+ (MM)
 
s = 12.916+ (MM)
 

s = 13.059+ (MM)
 
s = 8√3
= 13.856+
s = (11√3+√39)/2
= 12.648+ (MM)

s = 5√3+2√6
= 13.559+
s = 5√3+2√6
= 13.559+ (MM)
s = 8√3
= 13.856+
18

s = 13.293+
 
s = 13.168+ (JD)
 
s = 13.410+ (JD)
 
s = 13.377+ (MM)
 

s = 8√3
= 13.856+
s = 13.172+ (JD)
 
s = 5√3+2√6
= 13.559+

s = 5√3+2√6
= 13.559+
s = 6+6√3
= 16.392+
s = 8√3
= 13.856+
19

s = 13.448+
 
s = 13.458+ (JD)
 
s = 13.665+ (MM)
 
s = 13.681+ (JD)
 

s = 8√3
= 13.856+
s = 13.755+ (JD)
 
s = 5√3+2√6
= 13.559+

s = 5√3+2√6
= 13.559+
s = 6+6√3
= 16.392+
s = 8√3
= 13.856+

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/26/11.