Cover the largest circle, square or triangle with n unit disks (the page lists the target size). Validated by whole-domain critical-point analysis and by exact rational provers.
| Problem | n | Listed on the page | Our certified value | Margin | Margin % | Status | Listed by |
|---|---|---|---|---|---|---|---|
| Disks covering a circle | 1 | r = 1 | 0.999999980 | −2.00e-08 | -0.000 | reproduced | |
| Disks covering a circle | 3 | r = 2 / √3 = 1.154+ | 1.154700514 | −2.47e-08 | -0.000 | reproduced | |
| Disks covering a circle | 4 | r = √2 = 1.414+ | 1.414213534 | −2.83e-08 | -0.000 | reproduced | |
| Disks covering a circle | 7 | r = 2 | 1.999999960 | −4.00e-08 | -0.000 | reproduced | |
| Disks covering a circle | 11 | r = 2.631+ | 2.631690720 | +6.91e-04 | 0.026 | reproduced | Found by Hans Melissen in 1997. |
| Disks covering a circle | 12 | r = 2.769+ | 2.769293194 | +2.93e-04 | 0.011 | reproduced | Found by Hans Melissen in 1997. |
| Disks covering a circle | 13 | r = 2.884+ | 2.884791824 | +7.92e-04 | 0.027 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 14 | r = 3.014+ | 3.014481198 | +4.81e-04 | 0.016 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 15 | r = 3.143+ | 3.143241238 | +2.41e-04 | 0.008 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 16 | r = 3.244+ | 3.244437813 | +4.38e-04 | 0.013 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 17 | r = 3.349+ | 3.349729194 | +7.29e-04 | 0.022 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 18 | r = 3.446+ | 3.446282681 | +2.83e-04 | 0.008 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 19 | r = √13 = 3.605+ | 3.605551203 | −7.21e-08 | -0.000 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 20 | r = 3.692+ | 3.692105244 | +1.05e-04 | 0.003 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 21 | r = 3.804+ | 3.804522454 | +5.22e-04 | 0.014 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 22 | r = 3.948+ | 3.948644484 | +6.44e-04 | 0.016 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 23 | r = 4.000+ | 4.000739557 | +7.40e-04 | 0.018 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 24 | r = 4.076+ | 4.076613342 | +6.13e-04 | 0.015 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a circle | 25 | r = √6 + √3 = 4.181+ | 4.181540411 | −1.39e-07 | -0.000 | reproduced | Found by Jeremy Tan in 2018. |
| Disks covering a square | 1 | s = √2 = 1.414+ | 1.414213534 | −2.83e-08 | -0.000 | reproduced | |
| Disks covering a square | 2 | s = 4 / √5 = 1.788+ | 1.788854346 | −3.58e-08 | -0.000 | reproduced | |
| Disks covering a square | 6 | s = 3.347+ | 3.347537289 | +5.37e-04 | 0.016 | reproduced | Found by Hans Melissen in 1997. |
| Disks covering a square | 8 | s = 3.841+ | 3.841719451 | +7.19e-04 | 0.019 | reproduced | Found by Hans Melissen and P. Schuur in 1997. |
| Disks covering a square | 9 | s = 4.335+ | 4.335819027 | +8.19e-04 | 0.019 | reproduced | Found by Kari Nurmela and Patric Östergård in 2000. |
| Disks covering a square | 10 | s = (18 + 24√3) / 13 = 4.582+ | 4.582247553 | −9.17e-08 | -0.000 | reproduced | Found by T. Tarnai and Z Gáspár in 1995. |
| Disks covering a square | 11 | s = 4.705+ | 4.705527595 | +5.28e-04 | 0.011 | reproduced | Found by Hans Melissen and P. Schuur in 1997. |
| Disks covering a square | 12 | s = 4.943+ | 4.943742845 | +7.43e-04 | 0.015 | reproduced | Found by Kari Nurmela and Patric Östergård in 2000. |
| Disks covering a triangle | 1 | s = √3 = 1.732+ | 1.732050773 | −3.46e-08 | -0.000 | reproduced | |
| Disks covering a triangle | 2 | s = 2 | 1.999999960 | −4.00e-08 | -0.000 | reproduced | |
| Disks covering a triangle | 3 | s = 2√3 = 3.464+ | 3.464101546 | −6.93e-08 | -0.000 | reproduced | |
| Disks covering a triangle | 7 | s = 5.398+ | 5.398078920 | +7.89e-05 | 0.001 | reproduced | Found by P. Schuur in 1997. |
| Disks covering a triangle | 8 | s = 5.649+ | 5.649951494 | +9.51e-04 | 0.017 | reproduced | Found by P. Schuur in 1997. |
| No cases match this filter. | |||||||
Data as of 2026-09-12 05:21 UTC. Every value is certified by an independent validator; none is a claim of optimality. Nothing has been submitted to the page.