# The board as a codeword > Read a full board as a codeword whose 480 internal joints are parity-like checks, and the matched-edge score becomes 480 minus the number of failed checks. It is a clean lens with one load-bearing identity underneath it, and it is worth being exact about what the coding view buys and what it only renames. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/permutation-code-wall/ - Updated: 2026-07-22 - Topics: structure - Reproduce: `cd research/topics/permutation-code-wall/compute && cargo run --release > ../results/permutation_code_wall.json` - Source: Eternity II piece set and interactive board viewer (bucas.name) — https://e2.bucas.name/ - Source: The board-as-codeword lens: article, checker source and results JSON (GitHub) — https://github.com/raphael-anjou/eternity2/tree/main/research/topics/permutation-code-wall --- There is a way of looking at a completed board that borrows its whole vocabulary from error-correcting codes, and it is genuinely illuminating once you know exactly what it is claiming. Lay the 16 by 16 grid out, treat each tile as a symbol, and treat every place two neighbouring tiles meet as a *check* that either passes (the two facing edges are the same colour) or fails. There are 480 such internal joints. The list of which checks fail is a *syndrome*, and a full solution is the one board whose syndrome is empty. Score a board on the [matched-edges convention](/research/records) used throughout this wiki (count the internal joints that match, out of 480), and the arithmetic falls out on its own: the score is 480 minus the number of failed checks. That single line, score equals 480 minus the syndrome weight, is the whole engine of the coding lens. Everything else it says about the puzzle (minimum distance, decoders, stopping sets) is built on it. So it is worth pinning down what that identity actually is, and separating the part of this picture that is a real, countable fact about the official set from the part that is a change of clothes. ## What the lens renames, and what it measures The plain split, stated directly because it is the point. The identity itself renames rather than reveals. "Syndrome weight" is the count of unmatched interior edges; the scorer already calls that count *breaks* and already defines it as the top score minus the achieved score. So "score equals 480 minus the syndrome weight" is not a discovery about Eternity II. It is the definition of the score written in coding language. That is not a criticism: a good renaming can make a structure legible. It just means the identity buys you a viewpoint, not a number. Underneath the vocabulary, though, sit three facts that are real properties of the published 256-piece design, each exactly countable, each checkable against the official instance: - **480 checks.** A 16 by 16 grid has exactly 480 internal joints. This is $2WH - W - H$ with $W = H = 16$, it is the length of the scorer's own adjacency list, and it is the kit's maximum possible score. Three independent routes to the same 480. - **A permutation code.** The 256 tiles are all distinct up to rotation, so a legal board uses each of the 256 pieces exactly once. In coding terms the codeword is not a free string of symbols; it is a permutation, and that constraint is far stronger than the joint checks alone. - **A border code.** Exactly five colours (labelled 1 through 5) appear only on the frame, never on an interior tile, and the grey frame colour sits on zero interior half-edges. The rim is a separate little code stacked on the joints. These are what give the lens something concrete to stand on. The [parity theorem](/research/why/parity-defect-floor), read in this language, is a statement that this code has no weight-one syndrome: you cannot fail exactly one check on a legal full board, so 479 is an empty rung and the defect floor is 2. The [area law](/research/why/entropy-area-law) and [piece theft](/research/why/piece-theft) are, in this language, statements about the permutation code: it is the use-each-piece-once rule, not the colour matching, that carries the difficulty. ## The identity, checked bit for bit Because the identity is load-bearing, the reproduction confirms it directly through the kit's canonical scorer, the same rim-excluding scorer the site and every engine use, so a failed check here means the same thing a break means everywhere else on this wiki. The check starts from a genuine solution: a solved, generated 16 by 16 board that scores 480 out of 480 with an empty syndrome. It then injects an exactly-known number of broken checks by corrupting one facing half-edge at a time to a colour no piece carries, accepting a corruption only when it drops the score by exactly one. That per-step guard is what makes the injected count a known quantity rather than an inferred one: $k$ breaks means exactly $k$ failed checks, each verified on its own. Then it reads back the score, the break count, and an independent recount of the syndrome, and confirms all three agree. Run across five solved boards and eight break counts (forty rows in all, on the matched-edges convention, out of 480), every row obeys the identity: | Injected breaks $k$ | Score | Breaks | Independent syndrome recount | 480 minus $k$ | | ------------------: | ----: | -----: | ---------------------------: | ------------: | | 0 | 480 | 0 | 0 | 480 | | 1 | 479 | 1 | 1 | 479 | | 2 | 478 | 2 | 2 | 478 | | 5 | 475 | 5 | 5 | 475 | | 10 | 470 | 10 | 10 | 470 | | 29 | 451 | 29 | 29 | 451 | | 60 | 420 | 60 | 60 | 420 | | 120 | 360 | 120 | 120 | 360 | Every row satisfies score equals 480 minus $k$, breaks equals $k$, and the independent syndrome recount equals $k$, on all five seeds. The run is deterministic and finishes in under a second on Apple Silicon; the committed results JSON is byte-stable on rerun. The row at $k$ equals 29 is there on purpose: it reads score 451, which is exactly the score-and-break pairing the paper's from-scratch champion bottom-band board carries (29 breaks, 451 matched edges). The identity 480 minus 29 equals 451 holds against the score-and-break pairing the paper reports for that champion board. (That 451 is a from-scratch notebook figure, well below the community's best full boards of 470 and 464 on the [records page](/research/records); the point here is only that the coding arithmetic lands on it.) ## The structural facts, on the official instance | Fact | Expected | Measured | | --- | --- | --- | | Check count | 480 | 480 three ways: formula $2WH - W - H$, enumerated adjacency list, kit maximum score | | Permutation code | 256 tiles, each used once | 256 pieces, all distinct up to rotation | | Border code, frame colours | five, confined to the rim | 5 frame colours (1 through 5), 17 interior colours | | Border code, grey | outward rim only | 0 grey half-edges on any interior piece | All four are recomputed from the instance data rather than assumed, and all four match. ## What the error-correction view does and does not buy What it buys is a clean mental model and a place to put the two hard constraints. The joint checks are a weak code on their own: locally, a near-perfect board has plenty of low-mismatch colour patterns nearby, which is why the [mismatches cluster](/research/why/mismatch-geometry) into a single band rather than spreading. The strength lives in the permutation code layered on top, the requirement that the symbols be a genuine rearrangement of all 256 pieces. Naming that layer, and separating it from the colour checks, is a useful way to say where the difficulty sits: not in matching colours, but in matching them while every piece is spent exactly once. What it does not buy is a new number. The central identity is the score's own definition in a different alphabet, and the vocabulary of decoders and minimum distance describes the same landscape the [rigidity](/research/why/rigidity-wall) and [area-law](/research/why/entropy-area-law) pages measure directly, without adding a measurement of its own. The stronger claim a coding reading wants to make, that the permutation code's minimum distance is what pins a specific record board in place, is a board-specific measurement that lives on one private champion board the public kit does not ship, so it is neither confirmed nor denied here. What reproduces exactly is the scaffold: 480 checks, a 256-piece permutation code, a five-colour border code, and the one identity that holds the whole picture up. ## Related - [Why 479 is impossible](https://eternity2.dev/research/why/parity-defect-floor) — A counting argument on the official piece set forbids a score of exactly 479/480: every color's half-edges come in even numbers, and a single broken joint would leave two odd counts. The floor below perfect is 478, and at most 76 one-move near misses can surround any solution. - [Where the mismatches live](https://eternity2.dev/research/why/mismatch-geometry) — 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. - [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. - [Entropy and the area law](https://eternity2.dev/research/why/entropy-area-law) — Eternity II has two rules: edges must match, and each piece is used once. The first is generous. All the hardness lives in the second. - [Piece theft, where solvers die](https://eternity2.dev/research/why/piece-theft) — A solver fills a few rows for free, then hits a wall in the middle of the board. Here's the mechanism: a scarce piece spent in the wrong place, rows ago. - [Records & solvers](https://eternity2.dev/research/records) — Eternity II has never been solved, but nearly two decades of community effort have pushed the best board to 470/480. Who holds what, how they did it, and why some headline "480" boards are not actually the real puzzle.