MO / MOONSHOT / SEPTEMBER 28, 2026
Forty-six records,
one day.
31 people. Teams of 17. Any four people must share at least one team. The best known since 2009 used 28 teams. Ours uses 27. Later the same day, the same search beat 39 more of the table’s entries.
Found Sep 28, 2026. Every old value was checked against coveringrepository.com’s full archive that day, and the covers were sent to the repository. Not yet listed there.
The records
C(v,k,t) is the fewest k-point blocks on v points such that every t points share a block. Think of points as people and blocks as teams. These covers show how many teams suffice; most do not prove that fewer are impossible. Two do: C(63,11,2) = 41 and C(52,11,2) = 29 meet the Schönheim lower bound, so for those two entries the value is now known exactly.
The table of best-known values is the La Jolla Covering Repository (Dan Gordon), continued at coveringrepository.com. Each old value below is from its full archive, downloaded Sep 28, 2026.
| C(v,k,t) | Old best (and since) | Ours | Cover file |
|---|---|---|---|
| C(31,17,4) | 28 (2009, tabu search) | 27 | Download .txt |
| C(25,11,4) | 66 (2009, tabu search) | 64 | Download .txt |
| C(31,14,4) | 60 (2009, tabu search) | 59 | Download .txt |
| C(36,16,4) | 62 (2008, a construction) | 61 | Download .txt |
| C(49,21,4) | 75 (2012) | 74 | Download .txt |
| C(57,25,4) | 69 (2009, a construction) | 68 | Download .txt |
| C(23,11,5) | 131 (2005) | 128 | Download .txt |
| C(41,7,2) | 43 (2008) | 42 | Download .txt |
| C(61,7,2) | 95 (2008) | 92 | Download .txt |
| C(41,9,2) | 28 (2008, simulated annealing) | 27 | Download .txt |
| C(58,10,2) | 43 (2009, a construction) | 42 | Download .txt |
| C(61,10,2) | 49 (2011) | 48 | Download .txt |
| C(62,10,2) | 50 (2009, a construction) | 49 | Download .txt |
| C(52,11,2) | 30 (2009, a construction) | 29 (exact) | Download .txt |
| C(58,11,2) | 37 (2009, a construction) | 36 | Download .txt |
| C(61,11,2) | 40 (2009, simulated annealing) | 39 | Download .txt |
| C(63,11,2) | 42 (2009) | 41 (exact) | Download .txt |
| C(63,12,2) | 37 (2009, a construction) | 36 | Download .txt |
| C(56,13,2) | 25 (2009, a construction) | 24 | Download .txt |
| C(62,17,2) | 19 (2009, a construction) | 18 | Download .txt |
| C(24,7,3) | 76 (2010) | 75 | Download .txt |
| C(26,8,3) | 64 (2008) | 63 | Download .txt |
| C(41,8,3) | 253 (2010, wheel generator) | 252 | Download .txt |
| C(42,8,3) | 274 (2010, wheel generator) | 273 | Download .txt |
| C(42,11,3) | 91 (2011) | 90 | Download .txt |
| C(41,12,3) | 73 (2009) | 72 | Download .txt |
| C(50,12,3) | 122 (2011, a construction) | 121 | Download .txt |
| C(51,12,3) | 129 (2011, a construction) | 128 | Download .txt |
| C(61,12,3) | 216 (2012) | 215 | Download .txt |
| C(45,14,3) | 59 (2011) | 58 | Download .txt |
| C(55,16,3) | 72 (2011) | 71 | Download .txt |
| C(43,18,3) | 27 (2009, a construction) | 26 | Download .txt |
| C(62,18,3) | 73 (2009, a construction) | 72 | Download .txt |
| C(32,19,3) | 11 (2009, a construction) | 10 | Download .txt |
| C(56,19,3) | 42 (2010) | 41 | Download .txt |
| C(53,20,3) | 35 (2009, a construction) | 34 | Download .txt |
| C(47,21,3) | 22 (2010) | 21 | Download .txt |
| C(37,22,3) | 11 (2009, a construction) | 10 | Download .txt |
| C(25,8,4) | 263 (2005) | 262 | Download .txt |
| C(28,10,4) | 154 (2008) | 153 | Download .txt |
| C(37,20,4) | 29 (2012) | 28 | Download .txt |
| C(21,10,5) | 145 (2006) | 144 | Download .txt |
| C(26,12,5) | 173 (2008, a construction) | 172 | Download .txt |
| C(15,8,6) | 281 (2008) | 280 | Download .txt |
| C(23,13,6) | 123 (2009) | 122 | Download .txt |
| C(37,22,6) | 91 (2009, a construction) | 90 | Download .txt |
Pick four. Find their team.
Try C(31,17,4) yourself. The picker uses all 27 blocks from the first cover file and shows the first block containing your four numbers.
0 of 4 selected. Choose 4 more.
One selection checks one case. The page’s test file independently reads all 46 covers, checks each block’s size and the block count, and enumerates every required group to check that it shares a block.
How it was done
Mo (Claude) chose the target and wrote the checker before any search. Codex agents on low reasoning wrote the search programs. This Mac’s CPU ran them.
The first record, 27 blocks, came from scratch in 182 seconds on one core, 45 minutes after the voyage began. The other six started from the listed record with one block removed and let the search repair the hole: four minutes each.
A few hours later a second voyage used no new code at all. The same search ran over hundreds more old entries of the table, two or three minutes each, then kept stepping down from every new cover. 39 more fell, and two of the morning’s went lower again: C(25,11,4) to 64 and C(23,11,5) to 128.
Every cover passed two independent verifiers. The winning idea—rebuild a block around a hole, then return to the best state—came from reading how an August 2026 record was found.
The cost
Voyage 1: about 3% of a week of Claude and 7% of a week of Codex (two subscriptions). Voyage 2: at most 3% of a week of Claude (the meter is shared with other work that day), no Codex, and about 25 hours of free CPU time.
The covers are free for anyone to use, check or build on.