Discoveries
8 results
In October 2026 I ran a series of computer searches with an AI assistant on open problems whose best known answers are kept in public tables. They found new records, solved four of the problems exactly and turned up a counterexample to a conjecture published in 2025. Every result comes with a file anyone can check.
Published on Zenodo
Meeting schedules (covering designs)
- The puzzle
A group of people meets in smaller groups of a fixed size, and every pair of people must sit together in at least one meeting; harder versions ask the same of every three or four people. The fewest meetings that can manage this is called a covering number. For most group sizes nobody knows it, so a public table records the best schedules found so far.
- What I found
With 68 people meeting 6 at a time, the best known schedule needed 160 meetings. I found one with 159, and no schedule can use fewer, so this case is now solved. The same happened with 63 people in groups of 11 (42 meetings down to 41), 76 people in groups of 9 (86 down to 85) and 86 people in groups of 7 (186 down to 185). In all, 27 cases got shorter schedules than the table listed.
- How long it stood
The four solved cases had stood since 2008 and 2009, and the other 23 records dated from 2009 to March 2026. The table itself has been kept for about 30 years, first as the La Jolla Covering Repository and now at coveringrepository.com.
- Why it matters
The table is a standard reference in combinatorics, the mathematics of arrangements, and designs of this kind are used to plan experiments and to build lottery systems. Most of the improvements are small, one to three meetings fewer; the four solved cases are final.
Details
Covering numbers C(v,k,t): 27 new upper bounds, four of which meet the Schönheim lower bound and are exact. The archive's verifier checks every design exhaustively; the previous values are those listed on coveringrepository.com on 26 October 2026.
| New bound | Previous |
|---|---|
| C(63,11,2) = 41 | 42 |
| C(68,6,2) = 159 | 160 |
| C(76,9,2) = 85 | 86 |
| C(86,7,2) = 185 | 186 |
| C(41,6,3) ≤ 573 | 574 |
| C(46,8,4) ≤ 3553 | 3555 |
| C(47,8,4) ≤ 3985 | 3986 |
| C(49,7,3) ≤ 652 | 654 |
| C(49,7,4) ≤ 8802 | 8804 |
| C(52,7,3) ≤ 774 | 775 |
| C(58,10,2) ≤ 42 | 43 |
| C(61,8,3) ≤ 833 | 834 |
| C(70,8,3) ≤ 1238 | 1239 |
| C(75,24,4) ≤ 208 | 209 |
| C(76,23,4) ≤ 293 | 294 |
| C(77,8,3) ≤ 1672 | 1675 |
| C(77,24,4) ≤ 236 | 237 |
| C(83,23,4) ≤ 448 | 449 |
| C(84,14,2) ≤ 45 | 46 |
| C(86,9,3) ≤ 1543 | 1544 |
| C(89,8,3) ≤ 2610 | 2611 |
| C(92,7,3) ≤ 4442 | 4443 |
| C(93,7,3) ≤ 4660 | 4662 |
| C(93,24,4) ≤ 436 | 437 |
| C(95,7,3) ≤ 4787 | 4788 |
| C(96,7,3) ≤ 5040 | 5042 |
| C(99,7,3) ≤ 5418 | 5420 |
Preprint on Zenodo
Networks (graph theory)
A 2025 conjecture, disproved
- The puzzle
In 2025 four mathematicians published a conjecture in the Electronic Journal of Combinatorics: a certain inequality about networks should hold for every network. They proved it in some cases and checked every network with up to 9 nodes by computer.
- What I found
A network with 14 nodes, built from the smallest finite geometry, the Fano plane, where the inequality fails, and whole infinite families of networks where it fails as well. So the conjecture is false.
- How long it stood
The conjecture was published in 2025.
- Why it matters
A counterexample settles the question: the conjecture cannot be proved as stated. The note is a preprint and has not been peer reviewed yet.
Details
Conjecture 20 of Chen, Guo, Li and Wang (Electron. J. Combin. 32(4) (2025) #P4.37), on the k-th Laplacian eigenvalue of a graph and its complement, fails for the split graph of the Fano plane with n = 14 and k = 6, and for the split graphs of infinitely many symmetric designs. The record holds exact proofs and a verification script.
Published on Zenodo
Larger symmetric networks
- The puzzle
Connect as many computers as possible when each may have a fixed number of cables, so that any message reaches any other computer in at most two hops. Public tables list the largest such networks known, with a separate table for networks that have a strong kind of symmetry.
- What I found
For five cable counts I found symmetric networks larger than that table lists, for example 468 computers with 27 cables each, where the table had 392.
- How long it stood
The table of symmetric networks was last updated in 2022.
- Why it matters
The question comes from designing computer and communication networks. For three of the five cable counts, larger networks without this symmetry are already known, so these are records for the symmetric kind only.
Details
Five Cayley graphs of diameter 2 whose orders exceed the Cayley-graph table of the degree-diameter problem (combinatoricswiki); they do not exceed the general-graph table.
| New bound | Previous |
|---|---|
| d = 18: n = 210 | 200 |
| d = 19: n = 240 | 200 |
| d = 20: n = 264 | 210 |
| d = 22: n = 320 | 288 |
| d = 27: n = 468 | 392 |
Preprint on Zenodo
Corrections to a second 2025 paper
- The puzzle
A second 2025 paper, in the journal Filomat, proposed two related conjectures about networks, again an inequality that should hold for every network.
- What I found
As printed, both statements are slightly wrong. One misses an exception, the ring of five nodes, and a step in the proof of a lemma fails for networks made of separate pieces. The note gives these small counterexamples and a corrected version of the lemma.
- How long it stood
The paper was published in 2025. The main question, for networks in one piece, is still open.
- Why it matters
It is a small correction to the record: it fixes how the conjectures are stated and leaves the main question open. The note is a preprint and has not been peer reviewed yet.
Details
W.-J. Li and J.-M. Guo, Filomat 39:13 (2025) 4551-4564: Conjecture 1.2 fails for the 5-cycle, an exception lost when it is derived from Conjecture 1.1, and Conjecture 1.1 fails for disconnected graphs such as C5 + K1, because the proof of Lemma 2.1 assumes that every component satisfies the inequality. The record holds exact computations and a corrected Lemma 2.1.
Published on Zenodo
Codes (coding theory)
- The puzzle
Choose as few strings of 20 zeros and ones as possible so that every such string differs from one of the chosen ones in at most 3 positions. A set like this is called a covering code, and the smallest possible size is unknown.
- What I found
A set of 1917 strings that does the job. The published tables listed 2048 as the best known.
- How long it stood
The tables had listed 2048 since at least 1999, and the main one has not changed since 2009.
- Why it matters
Covering codes are a classic topic in coding theory, the mathematics behind sending and storing data reliably, and tables like these are how the field keeps score. The gap to the best known lower bound, 892, stays wide.
Details
Binary covering code K(20,3) ≤ 1917, against 2048 in Kéri's table (unchanged since 2009, lower bound 892) and in Litsyn's table (last modified 1999), where 2048 comes from a linear code. The archive's script checks all 2^20 words exactly; an independent check in C agrees.
| New bound | Previous |
|---|---|
| K(20,3) ≤ 1917 | 2048 |
Published on Zenodo
Geometry
Fewer crossings
- The puzzle
Place a given number of points on a sheet and join every pair with a straight line. The goal is to place them so that as few lines as possible cross. A public table at TU Graz records the best placements found so far.
- What I found
For 13 values between 55 and 100 points I found placements with fewer crossings than the table lists, from 1 to 60 fewer. With 100 points: 1,463,417 crossings, where the table had 1,463,457.
- How long it stood
The table was last updated in May 2015, and I found no published improvement on these values since then.
- Why it matters
The question is a classic of discrete geometry and close to how circuit and network diagrams are laid out. The Graz group keeps unpublished point sets for many sizes, so these improve the public table; whether they beat the group's private records is unknown.
Details
Rectilinear crossing numbers: point sets in general position for n = 55, 64, 66, 72, 74, 88, 93, 95, 96, 97, 98, 99 and 100 with fewer crossings than Oswin Aichholzer's table (TU Graz, last modified 26 May 2015). The archive's script checks general position and counts crossings exactly in integer arithmetic.
| New bound | Previous |
|---|---|
| n = 55: 125201 | 125207 |
| n = 64: 234446 | 234447 |
| n = 66: 266150 | 266151 |
| n = 72: 380907 | 380925 |
| n = 74: 426392 | 426398 |
| n = 88: 867885 | 867887 |
| n = 93: 1087897 | 1087899 |
| n = 95: 1186885 | 1186887 |
| n = 96: 1238638 | 1238662 |
| n = 97: 1292296 | 1292312 |
| n = 98: 1347551 | 1347559 |
| n = 99: 1404532 | 1404592 |
| n = 100: 1463417 | 1463457 |
Published on Zenodo
Packing coins in a triangle
- The puzzle
Fit a given number of identical coins, without overlapping, into a right-angled triangle with two equal sides, and make the coins as large as possible.
- What I found
For five numbers of coins, 164, 182, 193, 255 and 284, I found arrangements with larger coins than any public record I could find. The coins grow by at most 0.06 percent.
- How long it stood
For three of them the record came from the Packomania table, last updated in March 2011; for the other two it was set by Moki&Julio in 2026.
- Why it matters
Packomania's tables are where packing records are compared, and questions like this one are close relatives of practical ones, such as cutting material with little waste.
Details
Packings of equal circles in the isosceles right triangle with legs 1, each radius certified in exact decimal arithmetic. They beat Packomania's records for all five and Moki&Julio's for the two that Moki&Julio list; eleven more packings from the same search are left out because Moki&Julio match or beat them.
| New bound | Previous |
|---|---|
| r164 ≥ 0.0282699699 | 0.0282698456 |
| r182 ≥ 0.0268918240 | 0.0268917976 |
| r193 ≥ 0.0261481472 | 0.0261431538 |
| r255 ≥ 0.0228693810 | 0.0228636249 |
| r284 ≥ 0.0217147355 | 0.0217014202 |
Submitted, under review
Terence Tao's list of open problems
- The puzzle
Terence Tao, a mathematician who won the Fields Medal in 2006, keeps a public list of mathematical constants whose exact values are unknown. For each one it records the best bounds proved so far, and researchers, some of them working with AI systems, keep improving them.
- What I found
Better bounds for two of these constants, built on work that two other contributors, echul-seal and V. Patankar, posted earlier the same day. Each comes with a file anyone can check exactly.
- How long it stood
Both constants joined the list in October 2026, and their bounds have already been improved more than once.
- Why it matters
The step is small, in the fifth and sixth decimal places. The submission is under review and has not been accepted yet.
Details
Exact step-function certificates for the constants 1c and 1d in Terence Tao's repository of optimization constants, refining the certificates in pull requests #269 by echul-seal and #270 by V. Patankar.
| New bound | Previous |
|---|---|
| C1d ≤ 1.4556234448 | 1.455682936505816 |
| C1c ≥ 0.4103454241 | 0.4103432812 |
Method
Searches and verification were carried out with Claude (Anthropic) under my direction.
Updated