Four problems that didn't give

In October the lab spent compute on four open problems and solved none of them: a 17-input sorting network with 70 comparators, 4x4 matrix multiplication with 47 multiplications, a perfect 1-factorisation of the complete graph on 64 vertices, and Conway's 99-graph. Claude ran the searches and wrote the code, and Codex refereed part of the matrix theory. The sorting and matrix sessions started on October 18, when I asked for new utility sessions and said to try for the 47. Two nights later I told the lab to take on historic problems if it thought it should, and it started the other two. On October 21 I cut token spending and parked all of it except one exact search on Conway's graph.

Sorting 17 inputs with 70 comparators

A sorting network is a fixed list of compare-and-swap steps on n wires, and it sorts everything if it sorts all 2^n inputs of 0s and 1s, which for 17 wires is 131,072. The record at 17 is 71 comparators. Valsalam and Miikkulainen found the first 71 (JMLR, 2013), and Bert Dobbelaere's SorterHunter later matched it at depth 12. The lower bound is 60 (Dobbelaere's list shows 63, from a 1972 two-value bound of Van Voorhis's that it adopted in 2025), so nothing proved rules out 70, and we found no claim of one as of October 18.

Claude's main search is a parallel-tempering walk over 70-comparator lists that are allowed not to sort, penalised for what they get wrong. The searches at 70 made about 38.2 billion proposals in all. At 71 the same machinery finds a record-size network once per 99 million proposals (the plain walk: once per 410 million), and it turned up 29 new 71s, distinct from each other and from the three public ones.

At 70 the penalty never reached zero. The closest are seven networks, each confirmed by a checker, that sort all but one or two of the 131,072 inputs; about one run in three of the strongest search ended at a near-miss like that. Claude then asked a SAT solver whether windows of comparators in the near-misses could be refilled, at the same size, so that the network sorts. It never found a refill. In the last run, of eight windows of 14 or 16 comparators, one came back UNSAT with DRAT and LRAT proofs that both checked, one was UNSAT from the solver alone, and six stayed open. Those results cover only those windows and say nothing about whether a 70 exists.

Multiplying 4x4 matrices with 47 multiplications

Over any ring where 2 is invertible, the best known way to multiply 4x4 matrices uses 48 multiplications. AlphaEvolve found one in 2025 with coefficients in the Gaussian rationals Q(i), and Dumas, Pernet and Sedoglavic gave a version with rational coefficients (arXiv:2506.13242). The known 47s all live in characteristic 2. AlphaTensor found one in 2022, and Kauers and Moosbauer found more than 100,000 and report that none of them can be lifted from Z/2 to Z/4. Over the complex numbers, 47 is open.

The only from-scratch search that passed calibration, meaning it could find the known 48 from random starts, used symmetry, the same idea as in the constant-weight codes post. Claude fixed a group of order 16 inside the symmetry group of the known 48 and searched only decompositions built from whole orbits. That finds a 48 in 5 to 8 percent of random restarts (63 of 1,023). For 47, Claude deleted one orbit of 8 terms and inserted orbits of size 4, 2 and 1 (orbit sizes are powers of 2, so an odd total needs a fixed term). A control that put the deleted orbit's kind back recovered the 48 in 160 of 160 runs. The real attempt, 10,000 runs over 500 instances, hit nothing, and its best squared residual, 5.15, didn't improve when the candidate list grew.

Earlier rounds found no merges among terms of the known 48s (441 symmetry classes), and a SAT search over F3 couldn't even rediscover a symmetric 48. A plan to make new characteristic-2 47s and test them for lifting stalled when Claude's flip-graph engine never got below 49. Codex refereed Claude's theory toward a conjecture about the known 48s: it confirmed four identities, corrected a normalisation error and a reduction Claude had stated as an equivalence, and gave counterexamples to two consequences, which Claude withdrew. The conjecture itself is unproved.

A perfect 1-factorisation of K_64

A 1-factorisation of K_n, for even n, splits its edges into n-1 perfect matchings, and it is perfect if every two matchings together form a single Hamiltonian cycle. Kotzig conjectured in 1963 that every such K_n has one. K_64 has been the smallest open case since Pike found one for K_56 (arXiv:1810.08734) with a search of about 1.06 million worker-hours.

Claude's two checkers both pass Pike's K_56, Maciosowski's two K_92s and all 3,155 perfect 1-factorisations of K_16 in Gill and Wanless's catalogue, and both reject corrupted controls. The best search was an exact-cover depth-first search over even starters, small seeds whose translates make up the whole factorisation. It found 1,071 perfect 1-factorisations from K_16 to K_46, all passing both checkers, at 5.3 per core-hour at K_42, 2.0 at K_44 and 1.0 at K_46. A cost model that matches those rates to within a factor of 2 puts one K_64 at about 440,000 core-hours (plausible range 150,000 to 1.5 million), about 40 times a 10,000 core-hour budget. The whole effort used 10.5 CPU-hours.

The multiplier symmetry Maciosowski used at 92 is no use at 64: Claude showed that no multiplier reduces the conditions there, which agrees with a lemma in Maciosowski's paper. One structured family of starters was predicted to be 4,200 times richer in perfect ones than random merges at K_64, but nearly all of that sat on a single base path, the "zigzag". None of its 32,768 even starters is perfect; the best satisfies 26 of the 31 conditions. Exact repairs around that near-miss found nothing, and annealing over base paths plateaued. K_64 stays parked unless a new structural idea turns up or someone runs the existing search on a cluster.

Conway's 99-graph

Conway's 99-graph would have 99 vertices, every edge in exactly one triangle, and exactly two common neighbours for every two non-adjacent vertices: a strongly regular graph srg(99,14,1,2). Nobody knows whether it exists. The question dates to 1969, and Conway later made it one of his $1,000 problems, paying for a graph or for a proof that none exists. He died in 2020, and Claude found no one who has taken over the offer.

The best reported partial graph, from Thakkar and Severini (arXiv:2608.11211), satisfies 3,501 of 4,950 constraints (70.73%), one per vertex degree and one per pair of vertices. An evolutionary program-search system wrote their search, and the graph isn't public. Claude rebuilt that 2-switch search three ways. The best reached 3,124 (63.11%), confirmed by two scorers, and every variant stopped improving within ten minutes. With one core for at most an hour per run, we can't tell whether their number needs much more compute or a detail we don't have.

By the published results, the automorphism group of a 99-graph would be trivial, of order 2, or of order 3 with no fixed points, and in that last case Ishida's 2026 identity forces exactly six invariant triangles. Claude built and validated an enumerator for that case. It splits the search into 163,943 blocks, one for each essentially different way to fill the six triangles' rows of a 33-by-33 orbit matrix. At the time of writing, on October 21, 20,903 blocks (12.7%) had been searched to the end, with no orbit matrix in any of them, including all 6,940 blocks in which exactly one pair of triangles is joined. That is a computation, not a certificate. The rest, mostly blocks with no two triangles joined, was estimated at 90 to 120 core-hours with this code, and one low-priority process was working through those. Eigenvalue counting alone also allows 13, 20 or 27 triangles; Ishida's identity rules those out, and checking them without it hasn't started.