Measurements

Joshua Blackwood's record backtracker, decoded through Jef Bucas's notes (a colour quota schedule and a late-game mismatch allowance, tuned near-optimally), then built and run on my M1: left as published it flies to 248 of 256 pieces ignoring the clues; pin the five official clues and the same engine stalls near 45.

Peter McGavin's own C backtracker, the community's fastest: a 2007 optimization recipe compounded for two decades through generated code, lookup tables and counter tricks, then built on my M1 and pointed at the real 256-piece puzzle, where single core it drives past 200 of 256 pieces at ~109M placements/s.

Fifteen solvers, ours and our implementations of the community's two record backtrackers, each run once on ten corner-pinned variants of the official puzzle, single core, 60 seconds per run. The finding: node count is not score.

One constraint-propagation engine, run under a dozen ordering and propagator presets, on the same ten corner-pinned variants as the leaderboard. A study of what each knob buys, kept off the headline board because the best preset reaches less than half a contender's score.

The three comparisons at the heart of the DFS study, worked through: fill order (row-major wins, a bad order is catastrophic), heuristics (MRV rescues border-first but costs throughput; more propagation did not help), and breaks (they smash the depth wall; the break schedule is the lever, not the per-cell cap).

The results, worked through: on these boards the five clue-shaped hints never help a backtracker, they range from a mild cost to a catastrophe, and the fill order decides how much damage they do; the scores are bimodal, not a smooth gradient; and the hint-count question is confounded by a free pinned-seam floor.

The five comparisons at the heart of the repair study, worked through: blind random destroy wins while every conflict-targeting operator loses; construction sets the floor; simulated annealing is the strongest acceptance rule; and the clever refinements (exact refill, restarts) buy nothing at this budget.

One question, asked carefully: among depth-first backtrackers for Eternity II, what does each fill order, each heuristic, and the break mechanism actually buy? A family of from-scratch backtrackers, each one change apart, run on the same ten corner-pinned variants, single core, sixty seconds.

Give a backtracker five correct pieces for free, in the puzzle's own clue geometry. It turns out not to help, and depending on the fill order it can hurt badly, because a pinned piece is a hard constraint a fixed fill order must satisfy on arrival. A family of fill paths, run on the same hinted boards, single core, measured against no hints at all.

The sibling of the DFS study, for the other way people attack Eternity II: destroy part of a board, rebuild it, keep the change if it helps. One question, asked carefully. What does each decision in that loop buy: which region to destroy, how to rebuild it, when to keep a move, when to restart, and what board to start from?

A safe, portable Rust depth-first backtracker, specialised at runtime by emitting and compiling per-puzzle Rust, taken from 43 to 123 million search-nodes per second on one core. Measured fairly against Peter McGavin's C on the same machine: a tie on hard, deep boards like the real Eternity II, and about 44% of its speed on easy ones. Every rung searches the identical tree; the whole gain is code, not algorithm.

Measure the official 256 pieces with no solver in sight and every structural door is shut: no rotation-symmetric piece, 5 twin pairs in 32,640 pairings, a 307-of-480 matching cap if nothing rotates, colour budgets that pair to exactly 480 with zero slack, and a 17+5 palette sitting at the one-expected-solution point.

Eternity II uses 22 colors. They split 17 interior to 5 frame-only, and that count of around 17 lands near where this kind of puzzle is hardest to solve (the transition is a band, not a single integer).

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 near-perfect board doesn't scatter its few errors evenly. It packs them into one band of five rows and leaves all the rest flawless. Which band is decided by the direction the search filled the board, and you can see the mirror on the real record boards.

On a 16×16 puzzle built like Eternity II, eighteen hints scattered across the board solve it in minutes, while piling hints into contiguous rows needs a hundred just to bring the search down to tens of billions of placements. Position, not count, is the lever, and it points straight at the endgame.

Five of Eternity II's 22 colors appear only along the border ring, each on exactly 24 edges, never once in the interior. A structural split that shapes how every solver treats the frame.