Beat a record
On these problems a better construction is the whole result, and a computer checks it: upload the file, and if it
is valid it is verified — no reviewer, no waiting for opinions. Write the search program yourself or have a
chatbot write it; it runs on your computer.
- Pick a problem below and read what beats the record.
- Run a search (the prompt under each problem gets you a starting program).
- Submit the result on the problem's Submit page with the file attached. The machine check usually runs
within an hour; a verified result is credited to you, and a curator confirms when it beats a published record.
Records found here
- C(61,25,3) ≤ 27 (previous best 28, listed since 2009), shown here on
20 Oct 2026
by @caleb, checked by machine. Hossein Farahani independently found a different 27-block covering two days later.
Approachable 2 verified results so far
Beats the record: A (v,k,t) covering design with fewer blocks than coveringrepository.com lists for some v, k, t.
Open (records table). Best known upper bounds on C(v,k,t) are listed at coveringrepository.com; the La Jolla Covering Repository is archived (frozen March 2026).
Thousands of table entries, many found long ago with modest computing. Local search (simulated annealing, tabu search with block swaps) still improves entries, especially for larger v and k. Upload the block file named v.k.t.txt, one block per line.
File format: One block per line: k distinct integers from 1..v separated by spaces. Either start with header lines `v:`, `k:`, `t:`, or use the La Jolla Covering Repository format: no header, file named `v.k.t.txt` (e.g. `53.20.3.txt`) or a statement naming C(v,k,t).
Hard 3 verified results so far
Beats the record: A sorting network on n ≥ 13 channels with fewer comparators than the best known (45, 51, 56 and 60 for n = 13–16).
Open for n ≥ 13. Minimum sizes are proven for n ≤ 12; the best known sizes for n = 13–16 are 45, 51, 56 and 60 comparators.
Best known networks come from clever prefixes plus exhaustive or SAT-based completion. A checker tests every 0/1 input, so a valid network is verified at once.
File format: Header `n: <channels>`, then one comparator per line as `i,j` (0-based, i < j). Several comparators per line separated by spaces also work.
Hard
Beats the record: More unit vectors at pairwise angles of at least 60° than the best known kissing configuration in some dimension from 10 to 31.
Open. 2025–2026 brought new lower bounds in several dimensions (e.g. τ₁₁ ≥ 604, τ₁₂ ≥ 841, τ₁₉ ≥ 11,948); a September 2026 preprint claims small further gains in dimensions 25–31.
Recent records came from modifying lattice and code constructions and from AI-guided search (2025–2026). Exact integer or rational coordinates make the check exact.
File format: Header `d: <dimension>`, then one vector per line: d numbers (integers, fractions a/b or decimals) separated by spaces.
Hard
Beats the record: A cap set in F₃⁸ (no three points on a line) with more than 512 points.
Open. The best known cap in F₃⁸ has 512 points (FunSearch, Romera-Paredes et al., Nature 2023/24); no larger construction has been reported since.
512 was found by FunSearch (2023/24) with program search over priority functions. Products and extensions of good caps in lower dimensions are the usual starting point.
File format: Header `n: <dimension>`, then one point per line as a string of n digits from {0,1,2}.
Very hard
Beats the record: A cap set in F₃⁷ with more than 236 points.
Open. 236 ≤ r₃(7) ≤ 291 (lower bound: Edel's construction; a 2022 preprint claims ≤ 288). Exact values are known only for n ≤ 6.
236 (Edel) has stood for a long time; the same methods as in dimension 8 apply.
File format: Header `n: <dimension>`, then one point per line as a string of n digits from {0,1,2}.
Very hard 1 verified result so far
Beats the record: 41 or more unit vectors in R⁵ at pairwise angles of at least 60° (the best known is 40).
Open. 40 ≤ τ₅ ≤ 44 (lower bound from the D5 lattice; upper bound Mittelmann–Vallentin 2010).
40 comes from the D5 lattice and has not been beaten; the upper bound is 44.
File format: Header `d: <dimension>`, then one vector per line: d numbers (integers, fractions a/b or decimals) separated by spaces.
Very hard
Beats the record: A Hadamard matrix of an order 4k above 2000 for which none is known.
The Hadamard conjecture is open. Matrices for all 12 previously open orders below 2000 (668, 716, …, 1964) were announced on 12 August 2026 (Alpöge, Voinov, Reynolds-Haertle, with Claude), in encoded form and without a paper yet; the smallest open order is now above 2000.
Constructions combine Williamson-type and Goethals–Seidel arrays with computer search. Matrices are large; the checker verifies orthogonality exactly.
File format: n rows of n entries, each row a string of '+' and '-' (or space-separated 1 / -1).
Very hard
Beats the record: A red/blue colouring of the edges of K₃₆ with no red K₄ and no blue K₆ (would show R(4,6) ≥ 37).
Open. 36 ≤ R(4,6) ≤ 40 (lower bound: Exoo 2012; upper bound: Angeltveit–McKay, listed in Radziszowski's survey).
The lower bound 36 dates from 2012; circulant and Cayley colourings are the usual search space.
File format: Header `s: <red clique size>`, `t: <blue clique size>`, then the n×n 0/1 adjacency matrix of the red graph, one row per line.
Very hard 1 verified result so far
Beats the record: A 2-colouring of K₄₃ with no monochromatic K₅ (would show R(5,5) ≥ 44).
Open. 43 ≤ R(5,5) ≤ 46 (lower bound: Exoo 1989; upper bound: Angeltveit–McKay, arXiv 2024, J. Graph Theory 2026).
Widely believed not to exist (R(5,5) is conjectured to be 43); a valid colouring would be a sensation.
File format: Header `s: <red clique size>`, `t: <blue clique size>`, then the n×n 0/1 adjacency matrix of the red graph, one row per line.
Very hard
Beats the record: A bilinear algorithm for 4×4 matrix multiplication with fewer than 48 multiplications over Q (or fewer than 47 over GF(2)).
Open. The best ranks are 47 over GF(2) (AlphaTensor 2022) and 48 in characteristic ≠ 2 (AlphaEvolve 2025 over C; Dumas–Pernet–Sedoglavic over Q).
Records from AlphaTensor (2022) and AlphaEvolve (2025); flip-graph search is the main human-designed method.
File format: JSON {"m":3,"n":3,"p":3,"field":"Q","u":[r matrices m×n],"v":[r matrices n×p],"w":[r matrices m×p]}. Entries are integers, "a/b" fractions or decimals. field may be "Q" or "Zp" for a prime p.
Very hard 1 verified result so far
Beats the record: A bilinear algorithm for 3×3 matrix multiplication with 22 or fewer multiplications.
Open. The rank is at most 23 (Laderman 1976) over every field; lower bounds are 19 in general (Bläser 2003) and 21 over F₂ (2026 preprints).
Laderman's 23 (1976) has resisted all searches, including recent AI-driven ones.
File format: JSON {"m":3,"n":3,"p":3,"field":"Q","u":[r matrices m×n],"v":[r matrices n×p],"w":[r matrices m×p]}. Entries are integers, "a/b" fractions or decimals. field may be "Q" or "Zp" for a prime p.
Other problems are checked by Lean proofs or by review: see all problems or try a task with your chatbot.
