Rebuild the community's strict 460 boards exactly, and in doing so discover the move ordinary solvers can't make: paying two mismatches at a single cell.
Reproduceseeded — reproduces with the given seed·reruns the search (See below)·Budget: not logged (exploratory run, not the standardized single-core bench)
Complexity
Time
DFS with an exact 14-cell tail; 8 threads, 5 s restart cadence, per-witness budget
Space
bounded by the DFS frontier plus the fixed frame and the witness schedule
Backtracking search is exponential in the worst case; the exact tail and the witness schedule tame it to a targeted replay rather than a blind hunt.
Hardware & run
Native runCPU only
Cores
8
RAM
16 GiB
GPUs
0
CPU
Apple M1
Machine
MacBook (Apple M1, 8 cores)
Budget
not logged (exploratory run, not the standardized single-core bench)
REPLAY closes the study with a
different kind of learning. Every experiment before it mined a statistic from the
whole corpus; REPLAY learns from a single board, and it learns the one thing a
statistic cannot show, the exact move a record used that our search could not make.
The public fully-clued boards this project was measured against reach 460 (the
community record has since advanced to 464), but this project's own break-allowing
search stalled at 457 or 458 no matter what. REPLAY set out to reproduce those 460
boards exactly, piece for piece, to learn what they were doing that our search could
not. The answer turned out to be a single overlooked move.
A break-allowing search normally lets a cell carry at most one mismatch as
it's placed. REPLAY relaxes that to allow two at certain cells, and reorders
how candidate placements are ranked so that the moves a known good board
actually used outrank cheaper-looking ones. With those two changes it can
walk the same path the witness board took.
Played back this way, the community 460 boards rebuild exactly, every piece
in place, and the score checks out. The reproduction is the proof that the
missing ingredient was real.
▶Interactive: the double-break cellsExplore →
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▶Interactive: replay the break scheduleExplore →
Find the double-break cells on a real board
Pick a record board. We score it live, find every mismatched edge, and flag the cells that carry two mismatches at once. A solver that allows at most one break per cell — almost every solver — literally cannot place these cells, so it stalls a few points below. That is the move REPLAY had to add to walk the community's boards exactly.
463 / 480
24 broken cells
9 double-break cells
double-break cell (two mismatches)the rest of the board is perfectly matched
The double-break cells are computed here from the board's own edges (the same matched-edge rule the engine and the Bucas leaderboard use) — not hand-placed. Open the board in the viewer to inspect every edge. On the community's strict-460 boards there are 4–5 such cells; REPLAY's prior-over-cost ordering plus a cost-2 break operator is what makes them reachable.
Both community strict-460 boards replay exactly to 460. The discovery: each
contains four or five cells that pay two mismatches at once. A search that
allows only one mismatch per cell literally cannot reach those boards, which
is exactly why the project's earlier runs saturated at 457 to 458. Allowing
the double break lifts the strict ladder to 460.
It's a clean explanation of a long-standing plateau, and a caution: a
reasonable-looking rule (one break per cell) silently fenced off the very
boards we were chasing.
The replay is a depth-first search run in a deliberately constrained mode.
Prior-over-cost ordering. Ordinary break-tolerant DFS ranks candidate
placements by immediate mismatch cost, cheapest first. REPLAY flips the
priority to prior-over-cost: candidates the witness board actually used are
ranked ahead of cheaper-looking ones, so the search is pulled down the known
good path instead of wandering off it. This is the --prior-over-cost flag
driving off the witness board's own schedule.
Exact tail. The last 14 cells are solved exactly (--exact-tail 14)
rather than heuristically, so the endgame that ordinary runs fumble is closed
deterministically.
The relaxation that mattered. The per-cell mismatch budget is raised from
one to two. That single change is what admits the witness boards at all.
Run on a fixed 460 frame, 8 threads, a 5-second restart cadence and a per-run
budget, both community strict-460 witnesses rebuild piece-for-piece and the
score checks out, the reproduction is the proof that the missing ingredient
was the double break.
The finding. Each witness contains four or five cells that pay two
mismatches at once. A search capped at one break per cell cannot represent
those boards, which is precisely why the project's earlier break-tolerant runs
saturated at 457–458. It is a search-completeness result dressed as a record
attempt: the wall was in the move set, not the compute.
This one is seeded and closer to deterministic than the stochastic builders:
the DFS replay off a fixed frame and witness schedule rebuilds the 460 boards
reliably. The targets it reconstructs are the community's own strict-460
records, so the boards themselves are on the record
timeline; what this experiment adds is the replay that rebuilds them. The
replay needs two inputs beyond the puzzle, a border frame and the community
witness board it reconstructs; a runnable backing directory that ships both is
planned.
Does allowing two breaks per cell open a path to 461 and beyond, or just to
the known 460s? Are there boards needing a triple break? And can the
double-break cells be predicted from a partial board rather than discovered
by replay?