Reproducestochastic — won't reproduce exactly; the board is verifiable·re-verifies a stored board·Budget: not logged (exploratory run, not the standardized single-core bench)
Pipeline
1
beam producernot yet written up
Two fast top-half beams build from empty with all pieces free and no frame pinned
2
exact tail
Admissible-estimate pruning drops doomed partials, then an exact finisher solves the bottom rows, border emerging last
Complexity
Time
four stages: two fast top-half beams, a band grow, and an exact bottom finisher
Space
banks the surviving partials between stages; pruned at each handoff by an admissible estimate
Frame-free construction removes the usual border pin, so the search space is larger up front; the admissible cutoff at each stage handoff is what keeps it tractable.
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)
Almost every solver starts by locking down the border, because the border is
the most constrained part and pinning it shrinks the search. STAGED is an
experiment in refusing that crutch. It builds the board in stages from the
top down with all 256 pieces free, and never commits to a border until the
very end, when the border simply falls out of what remains. The question it
answers: can you reach a strong board without ever anchoring to a frame?
The build runs in four stages. The first two fill the top half of the board
with a fast plain search, banking the partial boards that survive. At the
handoff, a cheap admissible estimate throws out any partial that clearly
can't be finished well, so later stages only work on promising starts.
The third stage grows the next band of rows from those survivors. The fourth
is an exact finisher on the bottom rows that minimizes mismatches, and it
chooses the bottom border last, against whatever pieces are
still unused. So the frame is not designed up front; it emerges as a
consequence of everything above it.
STAGED reaches 436 of 480 from scratch, with an emergent border and all five
official clues respected, built end to end with no frame to lean on. That's
well below the records, and that gap is the finding: it measures how
much the usual frame-first anchor is worth, and it showed the frame-free
machinery works at full scale.
Along the way it pinned down the anatomy of the very best boards: they are a
perfect block of most of the board plus a thin band of mismatches
concentrated in a few top rows. That shape is what later builders aim to
reproduce on purpose.
Four stages, each handing survivors to the next through an admissible filter.
Top-half beams (stages 1–2). Fill the top half with a fast plain search,
banking the partial boards that survive. All 256 pieces are free, no frame
is pinned.
Admissible handoff. At each stage boundary, a cheap admissible estimate
(an optimistic upper bound on the best possible finish) discards any partial
that provably can't be completed well. Because the estimate never
under-counts the achievable score, discarding is safe, it only removes
partials that cannot win.
Band grow (stage 3). Extend the surviving partials down the next band of
rows.
Exact finisher (stage 4). Solve the bottom rows exactly, minimizing
mismatches, and choose the border last from whatever pieces remain. The
frame is not designed up front; it falls out of everything above it.
The result is 436 from scratch with an emergent border, all five clues
respected, well below the records, and that gap is the measurement: it prices
what the usual frame-first anchor is worth. The by-product mattered more than
the score: STAGED pinned the anatomy of the best boards, a large perfect block
plus a thin band of mismatches in a few top rows, the target shape later
builders aim at deliberately.
Stochastic (the top-half beams use randomized tie-breaking), so a re-run won't
reproduce the exact board; the committed 436 board is the artifact of record and
is checkable in the viewer. The engine is the shared beam
producer, run in
stages so the border emerges last rather than being fixed first. It needs no
corpus or seed board, so a runnable backing directory for the full staged
pipeline is planned alongside the other from-scratch builders.
Can a generator that deliberately spends its mismatches in the top rows, to
keep the rest perfect, reach the 450s frame-free? How much of the
436-to-record gap is the missing frame anchor versus the harder endgame? And
does a learned finish-quality estimate pick better survivors than the cheap
admissible one?