Circle packing in a circle
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle.
Table of solutions, 1 ≤ ''n'' ≤ 20
If more than one optimal solution exists, all are shown.| Enclosing circle radius | Density | Optimality | Layout of the circles | |
| 1 | 1 | 1.0 | Trivially optimal. | |
| 2 | 2 | 0.5 | Trivially optimal. | |
| 3 | 2.155... | 0.6466... | Trivially optimal. | |
| 4 | 2.414... | 0.6864... | Trivially optimal. | |
| 5 | 2.701... | 0.6854... | Proved optimal by Graham | |
| 6 | 3 | 0.6666... | Proved optimal by Graham | |
| 7 | 3 | 0.7777... | Trivially optimal. | |
| 8 | 3.304... | 0.7328... | Proved optimal by Pirl | |
| 9 | 3.613... | 0.6895... | Proved optimal by Pirl | |
| 10 | 3.813... | 0.6878... | Proved optimal by Pirl | |
| 11 | 3.923... | 0.7148... | Proved optimal by Melissen | |
| 12 | 4.029... | 0.7392... | Proved optimal by Fodor | |
| 13 | 4.236... | 0.7245... | Proved optimal by Fodor | |
| 14 | 4.328... | 0.7474... | Proved optimal by Ekanayake and LaFountain . | |
| 15 | 4.521... | 0.7339... | Conjectured optimal by Pirl . | |
| 16 | 4.615... | 0.7512... | Conjectured optimal by Goldberg . | |
| 17 | 4.792... | 0.7403... | Conjectured optimal by Reis . | |
| 18 | 4.863... | 0.7609... | Conjectured optimal by Pirl, with additional arrangements by Graham, Lubachevsky, Nurmela, and Östergård. | |
| 19 | 4.863... | 0.8032... | Proved optimal by Fodor | |
| 20 | 5.122... | 0.7623... | Conjectured optimal by Goldberg. |
Special cases
Only 26 optimal packings are thought to be rigid. Numbers in bold are prime:- Proven for n = 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 19
- Conjectured for n = 15, 16, 17, 18, 22, 23, 27, 30, 31, 33, 37, 61, 91