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Routine record updates appear automatically in the live feed. This page covers major milestones: new problem releases, optimality proofs that close an instance, verified advances beyond published literature, and platform rule updates.

  1. SITE UPDATEThis week’s challenge expands to n=50P96 adds n=21–50 to this week’s standings: 48 competitive instances in total, with the same deadline.
  2. PROOFP95: n = 5–7 proved optimalProof by zzzcy #308 adopted. With arbitrary rotations, the minimum sides are 11, 14 and 17. The three instances close with historical records preserved.
  3. NEW RELEASEWeek 10: consecutive L shapes in a squareStarted October 31 at 9pm Pacific (November 1 at noon Beijing). Now expanded to 48 competitive instances, n = 3–50, with free rotation.
  4. PROOFP94: n = 1–8 proved optimalProof by zzzcy #308 adopted. Both continuous and nine-decimal optima verified; historical records preserved.
  5. AHEAD OF PUBLISHED41 new breakthroughs verifiedAcross P12, P16, P66, P82, P94 and P95. Six external reference records were also updated.
  6. PROOFP16 spread points in a triangle: n = 5, 6, 7 proved optimalYinfeng Xu's 1996 paper in Acta Mathematicae Applicatae Sinica gives the exact maximum minimum distance of 2 ≤ n ≤ 7 points in the isosceles right triangle: 4 − 2√3 for n = 5, 1/2 for n = 6 and (4 + √2)(√(1 + 2√2) − √2)/7 for n = 7. Erich Friedman's table also lists them as proved. NUE_13 pointed this out; the site cites the proof, the three instances close, and their record holders are unchanged.
  7. PROOFP11 n = 2–4 and P51 n = 2 proved optimalTwo discussion proofs by NUE_13 are adopted. P11: the optimal radii of 2, 3 and 4 equal circles in the 3-4-5 right triangle are 5/28, 25/164 and 5/36; the site completed the steps of the original argument that were not rigorous, and the n = 4 bound reuses HwaterB's three-piece covering from P19. P51: two lights in the square leave a minimum intensity of at most 24/5, attained by the symmetric layout Friedman published (the proof was generated with DeepSeek, as disclosed). All four are continuous optima, and no nine-decimal grid optimum is claimed; the instances close with their record holders unchanged. NUE_13 receives two +2 proof awards.
  8. NEW RELEASEFour families extended: 368 new instancesP16 spread points in a triangle, P12 circles in a rectangle and P13 graduated circles in a square now run to n = 100; P82 circles in regular polygons runs to n = 100 in the pentagon and hexagon and to n = 48 in the octagon. The new instances cite the values and authors published in Packomania's tables, and the site supplies only a basic starting layout, so every one is open against an external record. P12 n = 7–12 also gain the external benchmark they lacked, and the existing P13 and P16 instances now credit the authors the tables name. Instances past n = 30 (n = 40 for P82) are listed in each family's table and accept JSON submissions checked by the same exact verifier.
  9. PROOFP60, P61, P63, P65, P67: 17 instances proved optimalzzzcy emailed computer-assisted proofs for five problem families (written with AI assistance): P67 n = 3–9, P63 n = 8, 13, 21, and 34, four P65 instances, P60 d = 5 n = 16, and P61 d = 3 n = 5. The site read only the static certificate data without running attached scripts, and replayed every certificate with its own independent checkers. All 17 instances close as proven optimal with their record holders unchanged.
  10. PROOFP72 smallest tetrahedron in a cube: n = 5 proved optimal, 1/6Emailed by NUE_13 (generated with DeepSeek, as disclosed): if one point lies inside the tetrahedron formed by the other four, the minimum volume is at most 1/12; otherwise three crossing tetrahedra partition the convex hull, and since five points in the unit cube span volume at most 1/2, the minimum tetrahedron volume is at most 1/6. Every step was verified and two intermediate details were completed by the site. This is the first P72 instance with a proven optimum; the record holder remains unchanged.
  11. PROOFThree days, 25 instances proved optimal: twelve in P93, four in P05, nine from the discussionsSummary of the three proof batches adopted between 3 and 5 October: zzzcy's computer-assisted proof for P93, The-Anh Vu-Le's Lean 4 formal proofs for P05, and four mathematical proofs from the discussions and mailbox. Each entry links to its complete proof.
  12. NEW RELEASEP87 equal discs covering a disc now runs to n = 100Adds 75 new instances for n = 26–100. Published tables stop at n = 25, so these instances have no external benchmark and start from standard site baselines. Instances up to n = 40 support the interactive graphical editor; instances above 40 are listed in the family table and accept JSON submissions checked by the same exact covering verifier.
  13. PROOFP93 sum of radii in a triangle: n = 1–11 and 15 proved optimalzzzcy emailed a computer-assisted proof (written with AI assistance) establishing that the current site records for these twelve instances achieve the maximum possible score on the nine-decimal grid. Reading only the static certificate data, the site re-enumerated all contact triangulations and replayed every certificate with its own independent checker. Those twelve instances close with their record holders unchanged. The same proof package also determines the continuous optima for n = 21, 28, 36, and 45; because their nine-decimal grid scores can still be improved, those four instances remain open.
  14. PROOFP05 squares in a circle: n = 3, 5, 6, 7 proved optimalThe-Anh Vu-Le formalised in Lean 4 with Mathlib the minimum enclosing circle for 1–7 unit squares. Citing his proof repository, the site marks n = 3, 5, 6, and 7 as proven optimal (n = 4 was already archived) while preserving historical records. External benchmarks from Friedman's tables for P05 n = 36–40 and P90 n = 27, 33–40 have also been added.
  15. SITE UPDATEScoring v3: records on frozen problems no longer scoreFrozen problems (P07, P15, P27) no longer accept submissions, which meant their records could not change hands and their Site #1 points had effectively become permanent. Those records retain their history, attribution, and profile display, but no longer count toward total contribution points; the problem catalogue now includes a Frozen Problems section explaining each case. Three players are affected for a total of 7 points. Thanks to NUE_13 for pointing this out.
  16. PROOFDiscussion and email proofs adopted: nine instances solvedP89 n=4, 8, 16, 18, P91 n=12, 27, P90 n=1, P82 (m=6, n=3), and P17 n=6 are archived as proven optimal with their historical records retained. Two new proof contributions by NUE_13 and one each by hadsajnc and LittleChasa receive proof awards; reused arguments receive attribution without duplicate points.
  17. AHEAD OF PUBLISHEDTwo new constructions pass the published records: P92 n = 50 and P61 (d = 4, n = 11)Tmrg raised the sum of radii of 50 circles in a unit disc to 6.538814808, about 0.00067 above Joost de Winter's 6.53814+ (added to Friedman's table on October 1), using an arrangement distinct from both the published figure and earlier certified site layouts. zzzcy lowered the coherence of 11 complex projective lines in C^4 to 0.425147038, clearing the eight-decimal interval of the public Game of Sloanes table with an inner-product profile distinct from earlier certified constructions. Both results were verified in exact arithmetic and certified after review.
  18. PROOFThree P89 and P91 instances proved optimalAdopted the diameter lower-bound proof by 😰 #106: the minimum enclosing square side is 1 for P89 n=1, 2 and 2 for P91 n=1.
  19. SITE UPDATESite-wide discussions are open!
  20. PROOFDiscussion proofs adopted: seven instances solvedP19 n=7, P16 n=4, P31 n=4, and P90 n=2, 8, 18, 32 close as proven optimal, preserving historical records and grid-precision notes. Four distinct proof contributions by HwaterB, XSsMC, and NUE_13 each receive a proof award; remaining unproved conjectures stay open.
  21. SITE UPDATESeven P51 targets refreshed; external P08 sources addedUpdated external benchmarks for P51 n=28–32, 34, 35 from Friedman’s table, and added external author credits and source precision intervals for P08 via r=2/s. Historical records and attribution remain intact; world-leader bonuses follow current leadership under the active scoring rules.
  22. NEW RELEASETans, dominoes and L's in a square: three new families, 104 instancesP89–P91 bring the polyomino rows of Friedman's packing tables—tans, dominoes, and L-trominoes in a square—minimising the enclosing square's side length. All 84 published instances carry their source benchmarks: 50 regular grid layouts and 31 reconstructed diagrams are verified and displayed, while three instances (tans n = 15, L-trominoes n = 22 and 28) start from baseline grids with their external targets preserved. Twenty additional instances beyond the published tables (dominoes n = 33–40 and L-trominoes n = 29–40) open from baseline grids. Pieces support exact right-angle turns combined with arbitrary rotations.
  23. SITE UPDATEScoring update: site #1 earns 1, world #1 adds 3The former permanent breakthrough award now follows current world leadership. An internal record takeover transfers the holder's 1+3 points; a strictly better external record removes only the +3 world-leader bonus. New holders still require review for world #1 certification. Historical reviews and attribution remain; proof points, special points, and settled weekly honours are unchanged. External adoption of a site result or differences within published precision intervals do not count as a loss of leadership.
  24. PROOFP17 n=5 solved continuously: √6 − √2Adopted XSsMC #250's two-square reduction combined with the classical three-point theorem and an exact attaining construction, awarding one permanent +2 proof award. Historical records and nine-decimal grid notes are preserved.
  25. SITE UPDATEP51: 17 external targets refreshed; historical credit retainedUpdated external benchmarks and author credits across 17 instances from Erich Friedman's table. Earlier verified reconstructions remain as starting layouts. Historical reviews and attribution are preserved, and active points follow the current-leadership rules.
  26. NEW RELEASESix classic packing and covering families: 274 new instancesP82–P87 add six regular packing and covering families using the interactive editor with snapping, automatic sizing, and standard literature units. 268 instances display verified reference layouts, and six start from baseline layouts with their external targets preserved. Six continuously solved P82 instances are included for reference only.
  27. PROOFP17 n=4 solved continuously: three strips give √13/3Adopted HwaterB #40's three-strip pigeonhole proof together with an attaining staggered construction, crediting one permanent +2 proof award. Historical records and nine-decimal grid notes are preserved.
  28. PROOFP21 n=6 solved: five squares give the exact bound √2Adopted HwaterB #40's five-square pigeonhole proof, with the existing integer-coordinate construction attaining √2 exactly. Historical records are preserved and one permanent +2 proof award is credited.
  29. PROOFP58 n=5,6: classical optima restored, continuous cases archivedUsing affine invariance, the classical analytical optima for the triangle Heilbronn problem at n = 5 and 6 are restored and archived as continuously solved, with the site's finite-decimal records preserved.
  30. NEW RELEASEP05 tilted squares in a circle extends to n = 40, with all 21 Friedman configurations rebuiltAdds 26 new instances for n = 15–40. Benchmarks for n = 15–35 are taken from Erich Friedman's squares-in-circles survey, with each configuration reconstructed from the published figure, checked by the exact verifier, and credited to its original discoverer. Instances n = 36–40 start from baseline layouts.
  31. SITE UPDATEP56: twelve sub-problems take the site's own constructions; proven optima now show only their closed formReference layouts at n = 11, 13, 15, 16, 19, 21, 25, 26, 28, 32, 36, and 40 have been updated with the site's offline search and staggered-row constructions (credited to MinMax Arena), while preserving prior submissions in the history. Proven-optimal instances now display their exact closed-form expressions (such as 2/√3), with decimal approximations derived directly.
  32. PROOFP56 lattice paper: continuous optima proved for seven casesPublished a finite-point lower bound, a classification of triangular-lattice equality cases for n ≥ 10, and exact optimisation of staggered-row configurations. Instances n = 7, 8, 14, 20, 22, 23, and 30 are archived as proven optimal for the continuous problem with historical records and credit preserved; n = 33 and 35 remain open.
  33. SITE UPDATEScoring is now public, including adopted proof ideasContribution points are now broken down into record, proof, breakthrough, and special-contribution points. Adopted, reviewed proof ideas earn permanent awards without repeat points for follow-up additions or rewrites. Player profiles display all four categories, including retroactive credit for previously published proofs.
  34. PROOFHwaterB and MinMax Arena jointly publish the P20 four- and five-point proofA partition into three 60° sectors establishes that the continuous optimal minimum distance for four and five points in a semicircle is 1. The four-point optimum is attained exactly on the grid; the unique five-point optimum involves √3/2, so the nine-decimal record is retained as a grid approximation rather than being labelled optimal.
  35. PROOFHwaterB and MinMax Arena jointly publish the P21 nine-point proofUsing an eight-region convex partition, HwaterB #40 proves that the nine-point optimal minimum distance in a cross is √5/2, establishing the optimality of their existing record and resolving the conjecture raised in NUE_13 #31's submission.
  36. SITE UPDATEBreakthrough records now use one review and certification systemThe new Breakthrough Review page separates numerical leads over external benchmarks into Pending and Verified. Pending constructions pass the arena verifier but earn no breakthrough points; certification is granted only after the external source, metric, published precision, and complete construction are reviewed, with one award per player and sub-problem.
  37. NEW RELEASEThree packing families expand to 800 verifiable instancesP01 and P02 now run through n = 300, and P06 through n = 200, importing Packomania's public constructions and benchmark values across all 800 instances. Instances above n = 30 are organised in compact family tables with full per-instance JSON submission and exact verification.
  38. SITE UPDATEDiscussion is now open to every signed-in userStarting threads and replying no longer require holding a record. Any signed-in user can join problem discussions; public attribution, rate limits (five new threads and thirty replies per hour), and automated review remain in effect.
  39. NEW RELEASEP78 joins the covering collection: equilateral triangles covering a discOpens 12 instances (n = 7–18), the first curved-boundary container in the covering series. The verifier checks coverage of the unit disc by equilateral triangles using exact arithmetic in Q(√3); all 12 benchmark values from Erich Friedman's table are converted and cited, with every pictured layout reconstructed and verified.
  40. PROOFA submission from NUE_13 led to proofs for P21 n = 4 and n = 5The constructions and exact optima proposed by NUE_13 are confirmed; the site completed the proof using a four-arm distance inequality and a five-unit-square partition so that no symmetry assumption is required. The accompanying n = 9 conjecture was subsequently proved by HwaterB #40 on September 5.
  41. NEW RELEASEThe covering collection launches with five familiesP73–P77 open 115 independently challengeable instances, spanning equal-disc coverings of polygons and freely rotated equilateral triangles covering a square or triangle. Published constructions with diagrams have been reconstructed and verified, and the remaining instances start from clearly labelled baseline layouts.
  42. PROOFMinMax Arena proved P08 optimal at n = 4A diameter decomposition of the feasible centre region reduces four-circle packings in an L-tromino to a single branch and yields the exact optimal radius; Cphysics's nine-decimal construction reaches the highest value representable on the verifier grid.
  43. PROOFUser #40 proved P19 optimal at n = 4Because the L-tromino is covered by three unit squares each of diameter √2, the pigeonhole principle bounds the four-point minimum distance by √2; Dilses #32's construction attains equality.
  44. PROOFUser #46 proved P10 optimal for n = 3, 4 and 5Whenever the radius exceeds 1/2, every circle centre is forced into the central unit square whose diameter is strictly less than 2r, simultaneously establishing that the optimal radius for n = 3, 4, and 5 is 1/2.
  45. AHEAD OF PUBLISHEDP51 n = 5 passed the published bestCphysics reached a minimum illumination of 22.237716, clearing Friedman's then-published 22.213+ interval. Historical news: the external table was updated to 22.252+ in September 2026; the problem page reflects the current benchmark while historical contributions remain.