This is the niche-picking tool. If you want to choose where to spend effort,
read down the columns: pick the wall you find most interesting, and the table
hands you every method that has attacked it and the exact score where it
stopped. For a broader survey by method family, see the
map of every known approach; for the angles
still open, the open problems board; and for the
approaches that provably do not move the score, the
dead ends.
Every cell of the map below is grounded in this project's published work. The
four walls themselves are corroborated by the published literature; the
experiment results are this project's own work, each written up on its own page; they
are not claimed to be externally verified.
Every one of them is a version of the same statement: there is nothing local
to prune on.
- No forced moves. Every interior cell
keeps 73–137 legal pieces, so the branching factor never collapses.
- The hardness peak. With ≈17 interior
colours the puzzle sits at the phase transition: about one expected solution,
the worst place to search.
- The area law. Genuinely-distinct
partial boards collapse past ~80 cells, but no local score can see that
global fact.
- Rigidity. Records are locally frozen;
the step to a better board is one giant, indivisible swap with no gradient
to follow.
Columns are the walls; rows are the methods, best score first. A filled mark
(●) means the method is fundamentally working against that wall. The community
ceiling on this puzzle is 470; the full solution is 480.
Almost every method ends up against rigidity, the wall that says the great
boards are isolated islands. The build-from-scratch and corpus methods (PRIOR,
KEYRING, PALIMPSEST, GAUNTLET) try to reach a new island by steering
construction with learned signal; they top out at 458–463 and a couple of them
do reach genuinely new families, but none crosses to the ceiling. The
concentrate and exact methods (LADDER, BANDSAW) instead attack the search
itself (the high branching factor and the unsearchable peak) and pay for it
at the endgame. The anchor methods (CLOISTER, MIDDEN) localise the damage but
hit the area-law wall in the interior. No single wall is the whole story, and
no method gets through all four.
The ceiling is never arbitrary. For each method, its own write-up records the
exact reason the score stopped climbing, quoted here in one line.
- PALIMPSEST (463/480). Reached
463, the project best, by reading the corpus to find which shared choices
are traps and steering a 15-basin sweep around them. Forcing the search to
avoid the traps directly made boards worse: the value was in where to look,
not a hard rule.
- PRIOR (460/480). Plateaus at 460:
the learned position prior gets a from-scratch build into the 460 class
fast, but the corpus signal alone is not enough to leave it.
- KEYRING (460/480). Three learned
signals voting (position, adjacency, 2×2 patch) reached 460 in a corner
arrangement no board had cracked before, a new family, but the patch
signal is marginal and the polish still caps at 460.
- REPLAY (460/480). Replays the
community's strict-460 boards exactly and reveals the move ordinary search
misses: 4–5 cells that take two mismatches at once, unreachable for a search
that allows at most one.
- GAUNTLET (458/480). Running the
beam in nine scan directions opened a brand-new 458 family (scan order is a
stronger diversity axis than the random seed), but a second round topped
out at 457 with no 461: the new family saturates like the others.
- CLOISTER (453/480). As a
standalone interior solver it confirms a real rim-compatibility bonus, but
that bonus cannot be retrofitted after the fact (the same rigidity as the
full board), so it settles in the low 450s.
- MIDDEN (452/480). Choosing where
(not when) the board may break extends the perfect run from 153 to 167–174
cells, but the dispersed geometry still fails at the endgame: nothing
absorbs the last damage.
- LADDER (451/480). Floods cheap
probes and promotes the deepest, reaching a 451 strict board with no record
to copy, the first escape from the universal 444–450 band, but the supply
of perfect openings runs out and the rungs all converge to one ceiling.
- LODESTONE (451/480). A
scarce-demand prior used only as a tiebreaker lifts the from-scratch median
by two (449→451) and tightens the variance, but any larger weight collapses
it: scarcity is a real but weak signal, and it never touches the basin
ceiling.
- MOSAIC (448/480). Composes exact
4×4 block solutions with soft seams, reaching 448 from scratch, but the
shortfall lands almost entirely in the last three corner blocks, where the
piece pool runs thin: the same piece-theft, now a single bright spot.
- BANDSAW (437/480). Solves an
endgame band to proven optimality, and in doing so measures the exactness
wall: the search tree grows about twentyfold per extra allowed mismatch, on
both sides, so meeting in the middle stops paying at full size.
- STAGED (436/480). Builds the whole
board with no pre-set frame and an emergent border, reaching 436, well
below the records. That gap is the finding: it measures exactly how
much the usual frame-first anchor is worth.
Read down the table and the lesson of the whole project is visible at a
glance: the methods that move the score change the shape of the search, whether
a scan order, a learned prior, or a confined region, never its raw speed. And
every one of them stops at a wall that is global, not local. The ten edges
from 470 to 480 are not a polishing problem; they are on the far side of all
four walls at once.
A separate 2026 campaign by William Millilaw reached the same conclusion from
the diversity side. Sweeping the full method list, he found that almost
everything (adaptive local search from scratch, snake placement, the top of a
trained generator's distribution) keeps rediscovering the same handful of
basins, and only parallel tempering reliably produced genuinely new ones,
boards far apart from the known set. Even that plateaus quickly. His reading is
the one this table keeps making: the bottleneck is not the score a method
reaches but the number of distinct basins it can find, and no method in the
standard arsenal finds enough of them.
Each saturation line is distilled from the project's lab notebook (one
entry per experiment). The four walls are corroborated by the published
literature: the 17-colour phase transition by Ansótegui, Béjar, Fernández
& Mateu, "How Hard is a Commercial Puzzle: the Eternity II Challenge";
the missing gradient and deep basins by the Eternity II local-search
literature.