Pangram verdict · v3.3
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Place n points in a unit-area region such that the smallest triangle determined by any three points achieves the largest possible area, A(n). This site collects the best known configurations across three classic containers: the square, the triangle, and optimal convex regions. Each entry includes exact coordinates, symmetry and congruence analysis, references to published proofs, and an in-browser rational arithmetic verifier.
Recent records dateentrynew valuegainfound by 2026-10-01 Triangle, n = 20 0.01260939 +2.37% Rob Gardiner 2026-10-01 Square, n = 23 0.00975728 +1.18% Rob Gardiner 2026-09-28 Square, n = 21 0.01147376 +1.85% Rob Gardiner 2026-09-28 Square, n = 23 0.00964331 +1.97% Rob Gardiner 2026-09-28 Square, n = 25 0.00823006 +3.74% Rob Gardiner 2026-09-28 Square, n = 35 0.00443228 +0.53% Rob Gardiner 2026-09-25 Convex, n = 31 0.00630967 +3.46% Alexandar Lackovic with help of Opus 5.5 2026-09-25 Convex, n = 33 0.00547013 +3.21% Alexandar Lackovic with help of Opus 5.5 2026-09-23 Triangle, n = 19 0.01348189 +0.21% Marc-Emmanuel Coupvent des Graviers 2026-09-23 Triangle, n = 29 0.00612369 +2.55% Marc-Emmanuel Coupvent des Graviers The ten most recent improvements. New records are also published as an Atom feed. Best known values Truncated to 8 decimals; ▲ marks entries proven optimal. nsquaretriangleconvex 3 0.50000000 1.00000000 1.00000000 4 0.50000000 0.33333333 0.50000000 5 0.19245008 0.17157287 0.27639320 6 0.12500000 0.12500000 0.16666666 7 0.08385900 0.09722222 0.11111111 8 0.07237642 0.06778921 0.08000013 9 0.05487599 0.05484693 0.06406475 10 0.04653741 0.04337674 0.05199307 11 0.03703703 0.03652988 0.04255319 12 0.03259885 0.03100478 0.03921568 13 0.02701991 0.02655652 0.03093720 14 0.02430397 0.02377577 0.02783558 15 0.02121054 0.02109076 0.02456405 16 0.02052785 0.01835554 0.02227287 17 0.01726167 0.01624232 0.01905733 18 0.01552605 0.01489434 0.01823889 19 0.01394786 0.01348189 0.01593109 20 0.01293819 0.01260939 0.01446420 21 0.01147376 0.01139597 0.01319108 22 0.01067285 0.01026612 0.01225496 23 0.00975728 0.00926978 0.01108248 24 0.00908900 0.00898515 0.01071264 25 0.00823006 0.00780263 0.00931610 26 0.00757759 0.00747156 0.00871478 27 0.00710240 0.00702932 0.00802912 28 0.00690052 0.00669434 0.00771884 29 0.00627593 0.00612369 0.00728911 30 0.00603852 0.00595740 0.00716018 31 0.00583706 0.00516128 0.00630967 32 0.00581090 0.00480075 0.00585071 33 0.00507134 0.00472821 0.00547013 34 0.00483489 0.00443091 0.00520621 35 0.00443228 0.00403268 0.00477380 36 0.00418488 — —