Heilbronn's problem: place n points in the unit square, the unit right triangle, or any convex region (scaled to unit area) so that the smallest triangle formed by three of the points is as large as possible. Values are exact rationals of the exported coordinates; listed values are the published records for n at least 9.
| Problem | n | Listed on the page | Our certified value | Margin | Margin % | Status | Listed by |
|---|---|---|---|---|---|---|---|
| Heilbronn problem, convex region | 9 | A = 0.0640647523079378 | 0.064064752 | +3.86e-15 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, convex region | 10 | A = 0.0519930715229431 | 0.051993072 | +9.71e-17 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, convex region | 11 | A = 0.0425531914893617 | 0.042553191 | −1.39e-17 | -0.000 | reproduced | David Cantrell |
| Heilbronn problem, convex region | 12 | A = 0.0392156862745098 | 0.039215686 | −6.94e-18 | -0.000 | reproduced | David Cantrell |
| Heilbronn problem, convex region | 13 | A = 0.0309372079216206 | 0.030937208 | +1.63e-16 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, convex region | 14 | A = 0.0278355805175536 | 0.027835581 | +1.04e-16 | 0.000 | reproduced | AlphaEvolve |
| Heilbronn problem, convex region | 15 | A = 0.0245640556132301 | 0.024564056 | +7.60e-16 | 0.000 | reproduced | Tej Stead |
| Heilbronn problem, convex region | 16 | A = 0.0222728771289812 | 0.022272877 | +6.59e-17 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, convex region | 17 | A = 0.0190573327521721 | 0.019057333 | +3.47e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 18 | A = 0.018238895463491 | 0.018238895 | +4.16e-17 | 0.000 | reproduced | Tej Stead |
| Heilbronn problem, convex region | 19 | A = 0.0159310923501429 | 0.015931092 | +2.78e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 20 | A = 0.0144642050306691 | 0.014464205 | +2.95e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 21 | A = 0.0130641168423859 | 0.013064117 | +5.90e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 22 | A = 0.0122549627897763 | 0.012254963 | +3.82e-16 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 23 | A = 0.0110824852805632 | 0.011082485 | +2.05e-16 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 24 | A = 0.0107126406526936 | 0.010712641 | +2.78e-16 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 25 | A = 0.0093161059776902 | 0.009316106 | +6.47e-16 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 26 | A = 0.0087147876362939 | 0.008714788 | +4.02e-16 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 27 | A = 0.0079139226016312 | 0.007913923 | +8.67e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 28 | A = 0.0076277190031123 | 0.007627719 | +8.59e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 29 | A = 0.0069744865386578 | 0.006974487 | +8.85e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 30 | A = 0.0065617395367395 | 0.006561740 | −5.20e-18 | -0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 31 | A = 0.0060333454841533 | 0.006033345 | +3.99e-16 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 32 | A = 0.0058151883792375 | 0.005815188 | +3.12e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, convex region | 33 | A = 0.0051760597305179 | 0.005176060 | +1.69e-16 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 34 | A = 0.0048974842109679 | 0.004897484 | +2.67e-16 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, convex region | 35 | A = 0.0045849619786946 | 0.004584962 | +1.63e-16 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 10 | A = 0.0465374195825417 | 0.046537420 | +6.25e-17 | 0.000 | reproduced | F |
| Heilbronn problem, square | 11 | A = 0.037037037037037 | 0.037037037 | +2.08e-17 | 0.000 | reproduced | Michael Goldberg |
| Heilbronn problem, square | 12 | A = 0.0325988586918196 | 0.032598859 | +7.63e-17 | 0.000 | reproduced | F |
| Heilbronn problem, square | 13 | A = 0.0270199166691111 | 0.027019917 | +0.00e+00 | 0.000 | reproduced | Peter Karpov |
| Heilbronn problem, square | 14 | A = 0.0243039796209924 | 0.024303980 | +5.55e-17 | 0.000 | reproduced | Mark Beyleveld |
| Heilbronn problem, square | 15 | A = 0.0212105410054906 | 0.021210541 | −6.94e-18 | -0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, square | 16 | A = 0.0205278592375366 | 0.020527859 | +3.12e-17 | 0.000 | reproduced | Mark Beyleveld |
| Heilbronn problem, square | 17 | A = 0.0172616770725236 | 0.017261677 | +0.00e+00 | 0.000 | reproduced | Chouaieb Nemri (AlphaEvolve |
| Heilbronn problem, square | 18 | A = 0.0151252752243591 | 0.015125275 | +5.90e-17 | 0.000 | reproduced | Marc-Emmanuel Coupvent des Graviers |
| Heilbronn problem, square | 19 | A = 0.0139478620814832 | 0.013947862 | +3.99e-17 | 0.000 | reproduced | Tej Stead |
| Heilbronn problem, square | 20 | A = 0.0129115556839418 | 0.012911556 | +6.07e-17 | 0.000 | reproduced | William Shanley |
| Heilbronn problem, square | 21 | A = 0.0111734121303578 | 0.011173412 | +2.08e-17 | 0.000 | reproduced | Tej Stead |
| Heilbronn problem, square | 22 | A = 0.0104725344235016 | 0.010472534 | +5.20e-18 | 0.000 | reproduced | Chouaieb Nemri (AlphaEvolve |
| Heilbronn problem, square | 23 | A = 0.0091866155511547 | 0.009186616 | +8.50e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 24 | A = 0.0090890060783181 | 0.009089006 | +3.49e-14 | 0.000 | reproduced | Ilkyoo Choi |
| Heilbronn problem, square | 25 | A = 0.0078595631513419 | 0.007859563 | +1.56e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 26 | A = 0.0073387656898376 | 0.007338766 | +6.25e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 27 | A = 0.0069811806760772 | 0.006981181 | +6.42e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 28 | A = 0.0067758102393368 | 0.006775810 | +4.34e-18 | 0.000 | reproduced | Marc-Emmanuel Coupvent des Graviers |
| Heilbronn problem, square | 29 | A = 0.0062759392131717 | 0.006275939 | +5.03e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 30 | A = 0.0059141207209502 | 0.005914121 | +9.28e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 31 | A = 0.0055707077600467 | 0.005570708 | +7.20e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 32 | A = 0.0058109022076319 | 0.005810902 | +7.29e-17 | 0.000 | reproduced | Marc-Emmanuel Coupvent des Graviers |
| Heilbronn problem, square | 33 | A = 0.0047482138627896 | 0.004748214 | +7.98e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 34 | A = 0.0043214527304 | 0.004321453 | −1.13e-17 | -0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, square | 35 | A = 0.0041013052818516 | 0.004101305 | +1.11e-16 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 9 | A = 0.0548469387755102 | 0.054846939 | −2.08e-17 | -0.000 | reproduced | David Cantrell |
| Heilbronn problem, triangle | 10 | A = 0.0433767406780494 | 0.043376741 | +5.55e-17 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, triangle | 11 | A = 0.0365298898800298 | 0.036529890 | −6.94e-18 | -0.000 | reproduced | AlphaEvolve |
| Heilbronn problem, triangle | 12 | A = 0.0310047817435253 | 0.031004782 | +3.47e-18 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, triangle | 13 | A = 0.0265565289197382 | 0.026556529 | +6.25e-17 | 0.000 | reproduced | Tej Stead |
| Heilbronn problem, triangle | 14 | A = 0.0237757731075727 | 0.023775773 | +6.59e-17 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, triangle | 15 | A = 0.0210907694602666 | 0.021090769 | +4.86e-17 | 0.000 | reproduced | David Cantrell |
| Heilbronn problem, triangle | 16 | A = 0.0179762759872349 | 0.017976276 | −1.73e-17 | -0.000 | reproduced | David Cantrell |
| Heilbronn problem, triangle | 17 | A = 0.0155336750278416 | 0.015533675 | +2.43e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, triangle | 18 | A = 0.0148943429879589 | 0.014894343 | +7.63e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, triangle | 19 | A = 0.0134535434259641 | 0.013453543 | +6.59e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, triangle | 20 | A = 0.0120384477602191 | 0.012038448 | +1.91e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 21 | A = 0.011059924510497 | 0.011059925 | +2.43e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 22 | A = 0.0102661276072628 | 0.010266128 | +6.77e-17 | 0.000 | reproduced | Nathan Sudermann-Merx |
| Heilbronn problem, triangle | 23 | A = 0.0092697818729312 | 0.009269782 | +2.95e-17 | 0.000 | reproduced | Tej Stead |
| Heilbronn problem, triangle | 24 | A = 0.0089851540727786 | 0.008985154 | −2.08e-17 | -0.000 | reproduced | Tej Stead |
| Heilbronn problem, triangle | 25 | A = 0.0076273954183023 | 0.007627395 | −1.73e-18 | -0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 26 | A = 0.006946659873861 | 0.006946660 | +1.73e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 27 | A = 0.0065944287391307 | 0.006594429 | +6.77e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 28 | A = 0.0061394131478185 | 0.006139413 | +1.73e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 29 | A = 0.0057284527336897 | 0.005728453 | +1.21e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 30 | A = 0.0053432028761361 | 0.005343203 | +8.93e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 31 | A = 0.0049651624443372 | 0.004965162 | +1.99e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 32 | A = 0.0046570194545139 | 0.004657019 | −1.73e-18 | -0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 33 | A = 0.0042539806977079 | 0.004253981 | +8.59e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 34 | A = 0.0041022992898881 | 0.004102299 | +1.82e-17 | 0.000 | reproduced | Rhys Chappell |
| Heilbronn problem, triangle | 35 | A = 0.0038692507299291 | 0.003869251 | +4.94e-17 | 0.000 | reproduced | Rhys Chappell |
| 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.