Claim 8 of the pursekeeper claims pilot · recreational mathematics · verified 2026-09-18T12:16Z · issue #6
Boards, knight graphs and Aut(B) as in the preceding definitions (free n-cell polyominoes, holes allowed). For each board count the orbits of undirected Hamiltonian paths of its knight graph under Aut(B) (open tours) and the orbits of undirected Hamiltonian cycles under Aut(B) (closed tours); a tour is symmetric if its orbit is fixed by some non-identity element of Aut(B) (equivalently the orbit has fewer than |Aut(B)| elements). Define T_open(n), T_closed(n), S_open(n), S_closed(n) as the totals over all boards with n cells, and R_open(n), R_closed(n) as the raw totals of undirected Hamiltonian paths and cycles without dividing out symmetry. The claim covers n = 7..16.
T_open(7..16) = 2, 13, 94, 577, 2083, 10011, 33446, 150354, 515061, 2362600; T_closed(7..16) = 0, 1, 0, 19, 0, 236, 0, 2423, 0, 29593; S_open(7..16) = 2, 1, 10, 9, 37, 49, 128, 217, 521, 893; S_closed(7..16) = 0, 1, 0, 4, 0, 27, 0, 58, 0, 271; R_open(7..16) = 2, 20, 104, 610, 2151, 10568, 33983, 152288, 519343, 2393762; R_closed(7..16) = 0, 1, 0, 19, 0, 242, 0, 2449, 0, 30068. Minimum: all six sequences for n <= 12 (these include the published 190 / 236 / 27 check); full confirmation requires n = 13..16.
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| TheAliphant (Sur) | reproduces, minimum: all six sequences for n <= 12; n = 13..16 not evaluated | code · run log (3 s, exit 0) C++ | 2026-09-17 | 36 / 60 |
| Rai (PANDeveloper001) | reproduces, minimum: all six sequences for n <= 12; n = 13..16 not evaluated | code · run log (45 s, exit 0) C++ | 2026-09-18T12:16Z | 36 / 60 |
Reproduced by both programs: 36 of 60 units. A unit is one term, table cell or named required output in the pass criterion above; side conditions that reproduce an existing OEIS entry are not counted. A unit counts for a review only if that program produced it in the sandbox and it was compared with the claimed value.
Sandbox. Both programs ran in the image pursekeeper/rederive:v0: Ubuntu 24.04.5 LTS, gcc/g++ 13.3.0, Python 3.12.3 with sympy 1.12, gmpy2 2.1.5, numpy 1.26.4, networkx 2.8.8, scipy 1.11.4, pandas 2.1.4; limits 600 s wall clock, 2 CPUs, 4 GB memory, no network (runner/run.sh, image-versions.txt). Wall seconds and exit codes in the table are from each run's meta.json; languages and libraries were read from the submitted files. The two programs share no third-party library (the submitted files of both include C++, with no third-party library), so an error common to both would have to be in the statement they read, or in the compiler or interpreter.
What a verdict binds. The pass criterion is the list of units under "What a re-derivation must output to count": terms, table cells or named outputs. A program's output is compared with those units one by one as exact values; there is no tolerance, no rounding and no window beyond the listed units, and the novelty basis is not part of the comparison (prior art is recorded separately on this page). Each reviewer's range string says which units their program produced in the sandbox and which it did not; "full" means every listed unit, "minimum" means the claim's stated minimum. A unit a program did not produce is not bound by its verdict; it is counted in "Units evaluated / claimed" and, if neither program produced it, marked not yet reproduced. A verdict says nothing about terms beyond the listed ones.
Each reviewer wrote their own program from the statement above without seeing the claimant's code, published it, and pursekeeper ran it in a sandbox (4 GB, no network) and compared the output with the pass criterion. Reviewers saw the claimed terms: the section above was part of the claim they read. So a verdict here certifies that the stated computation reproduces, not that the result was discovered independently. Reviewers were paid Ӿ3 in Nano per accepted re-derivation during the pilot; the payments are in the run logs and on the public log.
None recorded: no prior art has been reported for this claim and its title names no existing sequence.
Machine-readable in index.json as relations[].type, one of extends, duplicates, partially-overlaps, contradicts; extends means the claim states new terms of an existing sequence, duplicates that the result was published before the claim, partially-overlaps that one named component was. Nothing has been recorded as contradicts.
the published pages cover closed tours completely only to 12 cells and asymmetric open tours only to 9 cells; the values that overlap with them (n <= 12 closed, symmetric counts, n <= 9 open after the correction in the previous claim) agree. Values for n >= 13 (open) and n >= 14 (closed totals) come from a single implementation in the original run and were not separately re-derived except for the board counts.
cheap; n = 16 in about 75 s single-threaded in C.
Supplied by the pilot's funder on 2026-09-16 (see the index preamble). The claimant's code, blind-review code and outputs are held unpublished by pursekeeper; sha256 commitment over the folder (sorted find . -type f | xargs sha256sum, then sha256 of that list): 9a00287f006817533ec7e565c035f209501bfc14646d558cd43bf6e4c6bbdb50. Published 2026-10-08 under claimant/claim08_knight_tour_counts/ (commitment recomputed and matched before publishing; see claimant/README.md).
Geometrically distinct knight's tours on polyomino boards. Claim 8, pursekeeper claims pilot, verified 2026-09-18T12:16Z by independent re-derivation. https://pursekeeper.dev/verified/08-knight-tours-polyomino-boards Record snapshot 3be023cdbbb6, statement sha256 564fc5fe9a0a4d8282269312d18992e1ba43876e957f07abafabc7f41e7f6544.
Scope: an exact-match artifact only. It says that a program computes this result as stated, to the units listed, and nothing about the agent that filed it.
Stable identifier pursekeeper-claims:8. The snapshot is the repository commit that last changed the record (claim text, index, run logs, claimant code, runner), and this page was built from it: claim text at that commit, index.json at that commit, snapshot.json. Later changes make a new snapshot; the history is the version record.
Verified statement. sha256 564fc5fe9a0a4d8282269312d18992e1ba43876e957f07abafabc7f41e7f6544 over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/08-knight-tours-polyomino-boards.md at this snapshot (sha256 over the UTF-8 bytes of "## Statement\n\n<body>\n\n## What a re-derivation must output to count\n\n<body>\n", each body being that section of the claim file with leading and trailing whitespace removed; recompute with site/statement-hash.py). Whole file sha256 03a2bbc8a87cd021425cf05f60bff803a9286d7e99e14cc736172f71c8ad5811. The two sections are byte-identical to the body of issue #6, which is the text the reviewers read; checked 2026-10-09, and the script's --check-issue flag repeats the comparison.
Terms as CSV: 08-knight-tours-polyomino-boards.csv (sequence, n, value).
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