# Which wall stops which method > The research section has two halves: the structural walls that make Eternity II hard, and the algorithms built to climb them. This page is the bridge: each method lined up against the wall it actually attacks, and the score where that wall stopped it. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/walls-and-methods/ - Updated: 2026-07-21 - Topics: structure - Source: Ansótegui, Béjar, Fernández & Mateu, How Hard is a Commercial Puzzle: the Eternity II Challenge — https://repositori.udl.cat/server/api/core/bitstreams/0b6533fe-54e5-4070-85fe-80f7d35837d8/content --- 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](/research/build/approaches-map); for the angles still open, the [open problems](/research/open-problems) board; and for the approaches that provably do not move the score, the [dead ends](/research/build/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. ## The four walls, in one line each Every one of them is a version of the same statement: there is nothing local to prune on. - **[No forced moves](/research/why/no-forced-moves)**. Every interior cell keeps 73–137 legal pieces, so the branching factor never collapses. - **[The hardness peak](/research/why/phase-transition)**. With ≈17 interior colours the puzzle sits at the phase transition: about one expected solution, the worst place to search. - **[The area law](/research/why/entropy-area-law)**. Genuinely-distinct partial boards collapse past ~80 cells, but no local score can see that global fact. - **[Rigidity](/research/why/rigidity-wall)**. Records are locally frozen; the step to a better board is one giant, indivisible swap with no gradient to follow. ## The map 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. | Method | Best | [Forced moves](/research/why/no-forced-moves) | [Hardness peak](/research/why/phase-transition) | [Area law](/research/why/entropy-area-law) | [Rigidity](/research/why/rigidity-wall) | New basin | | --- | --- | :-: | :-: | :-: | :-: | --- | | [PALIMPSEST](/research/lab/experiments/raphael-anjou/learning/palimpsest) · from the corpus | 463/480 | | | | ● | no | | [PRIOR](/research/lab/experiments/raphael-anjou/learning/prior) · from scratch | 460/480 | | ● | | ● | no | | [KEYRING](/research/lab/experiments/raphael-anjou/learning/keyring) · from scratch | 460/480 | | | | ● | new family | | [REPLAY](/research/lab/experiments/raphael-anjou/learning/replay) · decode & replay | 460/480 | | | | ● | no | | [GAUNTLET](/research/lab/experiments/raphael-anjou/pipelines/gauntlet) · from scratch | 458/480 | | | | ● | new family | | [CLOISTER](/research/lab/experiments/raphael-anjou/pipelines/cloister) · anchor & constrain | 453/480 | | | ● | ● | no | | [MIDDEN](/research/lab/experiments/raphael-anjou/pipelines/midden) · anchor & constrain | 452/480 | | | ● | ● | no | | [LADDER](/research/lab/experiments/raphael-anjou/pipelines/ladder) · concentrate effort | 451/480 | ● | ● | | | new family | | [LODESTONE](/research/lab/experiments/raphael-anjou/learning/lodestone) · from scratch | 451/480 | ● | | | | no | | [MOSAIC](/research/lab/experiments/raphael-anjou/pipelines/mosaic) · solve a piece exactly | 448/480 | ● | ● | | | no | | [BANDSAW](/research/lab/experiments/raphael-anjou/meet-in-the-middle/bandsaw) · solve a piece exactly | 437/480 | ● | ● | | | no | | [STAGED](/research/lab/experiments/raphael-anjou/pipelines/staged) · from scratch | 436/480 | ● | | ● | | no | ### How to read it 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. ## Where each one stopped, and why 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](/research/lab/experiments/raphael-anjou/learning/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](/research/lab/experiments/raphael-anjou/learning/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](/research/lab/experiments/raphael-anjou/learning/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](/research/lab/experiments/raphael-anjou/learning/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](/research/lab/experiments/raphael-anjou/pipelines/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](/research/lab/experiments/raphael-anjou/pipelines/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](/research/lab/experiments/raphael-anjou/pipelines/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](/research/lab/experiments/raphael-anjou/pipelines/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](/research/lab/experiments/raphael-anjou/learning/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](/research/lab/experiments/raphael-anjou/pipelines/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](/research/lab/experiments/raphael-anjou/meet-in-the-middle/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](/research/lab/experiments/raphael-anjou/pipelines/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. ## The shape of the gap 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. > **Note** > > 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. ## Related - [Why a faster computer doesn't help](https://eternity2.dev/research/why/prune-vs-speed) — The single most important idea in hard combinatorial search: shrinking the space you search beats searching it faster, by an exponential margin. Eternity II is engineered so you can barely shrink it at all. - [The rigidity wall](https://eternity2.dev/research/why/rigidity-wall) — Every record board we have is frozen in place. You cannot nudge your way from a great board to a perfect one, and we can prove it. - [Experiments](https://eternity2.dev/research/lab/experiments) — The lab's named search experiments, one section per researcher. Each is a real run against Eternity II with its idea, its best board, and the questions it left open. Raphaël Anjou's notebook is here in full; the notebook is open to anyone else's. - [No forced moves](https://eternity2.dev/research/why/no-forced-moves) — The usual way to crack a logic puzzle is to find a spot where only one piece fits, place it, and repeat. That lever doesn't exist here: every interior piece has between 73 and 137 possible neighbours, and not one is ever pinned to a single option. - [A map of every known approach](https://eternity2.dev/research/build/approaches-map) — The survey the community keeps asking for and never finds: every family of attack tried on Eternity II, what each one actually reached, where it walls out, and a link to the deep page. One organising insight runs through all of them. - [Dead ends](https://eternity2.dev/research/build/dead-ends) — Approaches we tried that look promising and don't move the needle on Eternity II, written down with what we found so you can spend your time elsewhere. - [Open problems](https://eternity2.dev/research/open-problems) — The open frontier of Eternity II in one place: every angle still worth a run, the wall it attacks, what has been tried and where it stopped, and whether it is a newcomer-tractable target or a hard, well-mapped one.