Reproduceseeded — reproduces with the given seed·reruns the search·Budget: baseline arm 45 s × 8 frames; CAS arm deterministic per-shell beam over 20 frames (the source's per-shell MIP took 25 to 45 s per frame)
Pipeline
1
frame input
A seeded border DFS enumerates distinct perfect 60/60 frames of the official puzzle
2
beam producernot yet written up
Rings are solved outside-in, each as one assignment over the remaining pieces (width-1024 per-shell beam; the source runs used a per-shell MIP)
Complexity
Time
seconds to find each 60/60 frame; the source per-shell MIP took 25 to 45 s per frame, the baseline arm 45 s per frame
Space
one ring solved at a time; outer rings are frozen once placed
Freezing each ring is what makes the shells cheap to solve, and also what starves the inner ones.
Hardware & run
Native runCPU only
0.10core-hours
Cores
8
RAM
16 GiB
GPUs
0
CPU
Apple M1
Machine
MacBook (Apple M1, 8 cores)
Budget
baseline arm 45 s × 8 frames; CAS arm deterministic per-shell beam over 20 frames (the source's per-shell MIP took 25 to 45 s per frame)
Start
seeded frame DFS; the shell beam is deterministic given a frame
Re-runs everything from scratch: frame enumeration, the 20-frame CAS arm and the 8-frame baseline arm, regenerating the committed results file (per-frame scores, deltas and board URLs).
CAS (concentric annular solving) builds the board the way an onion grows:
lay a perfect border ring, then solve inward, shell by shell, until the four
centre cells close the board. I asked whether that outside-in schedule could
be a record path. It came back with a clean no and a useful yes: from any
perfect frame the pipeline plateaus in a narrow 429 to 437 band, and from
that exact starting state it beats a destroy-and-repair continuation on
every single frame.
The 16×16 board is 8 concentric rings of 60, 52, 44, 36, 28, 20, 12 and 4
cells. CAS fixes the outermost ring first as a perfect 60/60 frame: all 60
border cells filled with border pieces, every rim edge grey, all 60
adjacencies inside the ring matched. It then solves each inner ring in turn
as one assignment problem over the pieces still unused, maximising matched
edges against the ring outside it plus the matches inside the ring itself.
Once a ring is placed it is frozen; the search never revisits it.
The original runs solved each shell with a MIP,
25 to 45 seconds per frame. The committed reproduction swaps that for a
deterministic width-1024 per-shell beam, which lands in the same band, and
adds a comparison arm: pin the same frame, leave the interior empty, and
hand that state to a
destroy-and-repair local search
instead.
Twenty enumerated perfect frames, twenty scores between 429 and 437 matched
edges out of 480 (mean 432.8). The source audit, on its own 20 frames, sat
at 430 to 436 with mean 432.4. The band and the mean reproduce to within one
point on each edge, and the frame does not matter: every 60/60 frame lands
on the same plateau. The best reproduction board scores 437; it is checkable
edge by edge in the viewer.
▶Expected vs measured (20 CAS frames, 8 baseline frames)Explore →
The second half is the head-to-head. Started from the same pinned frame with
an empty interior, the local-search continuation loses every single time. In
the source measurement (full alns_only, 60 seconds per frame) it reached 385
to 398 and lost all 8 overlapping comparisons by 34 to 49 points. The
reproduction's lighter proxy baseline does better in absolute terms (411 to
419) and still loses every frame, by 13 to 23 points. The gap's magnitude is
not comparable across the two setups; its direction and its every-frame
consistency are.
The plateau is not a tuning problem, and it is not an objective gap either:
all 480 edges are in the CAS objective. The mechanism is piece starvation.
Greedy outer shells lock in pieces by how well they fit outward, and by
rings 5 to 7 the remaining pool simply no longer contains pieces whose
colour profiles fit the inward-facing constraints. It is the ring-scale
version of piece theft: a scarce piece spent
early, rings before the cell that needed it.
The source volumes tried the obvious rescues, and both confirm the
diagnosis. A CAS-ALNS hybrid scored 418, worse than CAS alone; an ALNS
refine pass on top of a finished CAS board reached 437 to 439, a lift of a
few points that stays far below what other pipelines in this notebook reach
from scratch on the same hardware. For where the community bar sits, see the
records page.
CAS is two things at once, and both halves are the finding. As a record path
it is refuted as a greedy schedule: the outside-in order starves its own
endgame, from any frame. As a sub-problem tool it is the strongest option I
have measured for one specific state, a perfect frame plus an empty
interior. Destroy-and-repair operators expect a full board to mutate; handed
an empty interior they must first build one, badly, while CAS is exactly the
constructive tool for that state. Which algorithm is right depends on the
state you start it from; composition matters as much as the components.
CLOISTER is
the sibling experiment that keeps the pinned frame but drops the
ring-by-ring schedule.
Three caveats bound the claim. The reproduction pins the five official clue
pieces; the source pages do not record whether the original runs did. All
five clues sit in the interior, so the frame arm is unaffected, and pinning
can only make the reproduction slightly conservative, the safe direction for
a plateau claim. The baseline arm is a light proxy, so only the direction
and the every-frame consistency of the gap carry over, not its size. And the
numbers are properties of the official 16×16 piece set: the mechanism lives
in rings 5 to 7, which an 8×8 board (4 rings) never reaches, so small-board
tests can neither confirm nor refute the claim.
just research-cas-annular re-runs the whole thing from scratch: it
enumerates the frames by seeded border DFS, runs the 20-frame CAS arm and
the 8-frame baseline arm, and regenerates the committed results file with
per-frame scores, deltas and a viewer URL for every board.