# Why 479 is impossible > 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. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/parity-defect-floor/ - Updated: 2026-07-22 - Topics: structure, search-space - Source: David Eddy opens the Parity thread: the number of edges of each type is even, and the last-piece consequence (msg 332, June 2007) — https://groups.io/g/eternity2/message/332 - Source: Christophe Weibel's worked example of a final piece that fails to match its hole (msg 335) — https://groups.io/g/eternity2/message/335 - Source: Brendan Owen's verdict on parity in E2: correct by construction, useful only near the end of a search (msg 346) — https://groups.io/g/eternity2/message/346 - Source: Parity defect floor: article, checker source and census results (GitHub) — https://github.com/raphael-anjou/eternity2/tree/main/research/topics/parity-defect-floor --- Line up full boards by score, counting matched internal joints out of 480 (the matched-edges convention used throughout this wiki), and the ladder looks continuous: rung after rung, each occupied by some board somebody has built. It has exactly one hole. No legal full placement of the 256 pieces scores 479. That is a theorem about the published piece set, provable by counting alone, and this page walks the whole argument: the evenness that forces it, the defect floor of 2 that follows, and the census that bounds how many one-move near misses can surround any solution. It is also one entry in [the theorem sweep](/research/why/theorem-sweep), the survey of what can be proven from the bag before any search runs. ## Count the half-edges 256 tiles with four edges each carry 1024 half-edges. The gray frame color accounts for 64 of them, exactly the number of outward-facing slots on the rim (16 per side). The remaining 960 carry the 22 colors, and the census is strikingly even: | Color group | Half-edges | | ---------------------------------- | ---------: | | Gray frame edge | 64 | | Five frame-join colors (1 to 5) | 24 each | | Five interior colors (6 to 10) | 48 each | | Twelve interior colors (11 to 22) | 50 each | Every colored count is even. The colored total, 960, is exactly twice the number of internal joints ($2 \times 16 \times 15 = 480$): in a frame-legal full placement, where all 64 gray half-edges face outward, the colored half-edges exactly fill the 960 internal joint slots with nothing left over. None of this depends on where the pieces go. These are properties of the bag. There is a pleasing shortcut hiding here: the evenness could have been predicted without examining a single piece. The designers guarantee a solution exists, and a solved board pairs every colored half-edge with a same-colored partner, so each color's count is twice its joint count, which is even by definition. The census merely confirms on the real set what solvability already promised. ## The argument, in four lines Fix any full placement with a legal frame (every rim-facing edge gray) and fix a color $c$. Each of the 480 internal joints shows two half-edges. Some joints show $c$ on both sides, some on exactly one. Counting $c$'s half-edges: $$h(c) = 2 \cdot \#\{\text{joints showing } c \text{ twice}\} + \#\{\text{joints showing } c \text{ once}\}.$$ Since $h(c)$ is even, the number of joints showing $c$ exactly once is even too. This holds for every color simultaneously, in every full placement, solved or broken. Now suppose a placement scores 479. It has exactly one broken joint, and a broken joint shows two different colors, say $a$ and $b$ (if both sides agreed it would be a match; gray cannot appear, since all 64 gray half-edges face outward). Every other joint is matched, showing its color twice. So exactly one joint shows $a$ exactly once. One is odd. That contradicts the evenness of $h(a)$, and the placement cannot exist. ## The floor below perfect is 478 Write the defect of a board as $480$ minus its score. Parity forbids defect 1 and forbids nothing else: at defect 2 the odd counts can absorb each other in pairs, so the argument goes silent. Every legal full placement is therefore either a solution or misses at least two joints. One more censused fact closes the remaining loophole: no Eternity II tile is fixed by any rotation (0 of 256), so a board cannot score 480 while differing from a solution only by a tile spun in place. Score 480 means solution. Anything else means 478 or below. For today's leaderboards the missing rung is academic: the community's best full boards stand at 470 under the center-clue-only convention and 464 with all five clues placed ([the records page](/research/records) keeps the full ladder). But it changes what "almost solved" can ever mean. There is no 479 to pass through on the way to 480. The last step of the climb is 478 to 480, two joints at once, and any method that improves boards one joint at a time is structurally unable to take it. ## The 478 shell is sparse A solution exists; the puzzle was constructed from one. What sits right next to it at 478? A board one move away from a solution must come from a move that breaks exactly two joints, and only two kinds of single move can do that. Both are countable in the bag: - **Twin swaps.** Two interior tiles that agree, in some orientations, on three of their four edges. Placed so the agreeing edges line up, exchanging them disturbs one edge of each: two joints. The set contains exactly **50** such near-twin pairs. - **In-place turns.** A tile whose repeated colors let a half-turn or a quarter-turn keep two of its four edge colors in position, so turning it where it stands breaks exactly the other two joints. The set contains **23** half-turn tiles and **3** quarter-turn tiles of this kind. Adding them up, any solution has at most $50 + 23 + 3 = 76$ defect-2 neighbours reachable by a single swap or rotation. This is an upper bound from the bag: whether a given twin pair actually lines up inside a specific solution depends on that solution, so the realized count for the designed solution is open. The ceiling stands regardless. Around the very top of the ladder, near misses are sparse, a few dozen boards at most, while rungs further down are populated astronomically. It is the same scarcity that [forbidden patterns](/research/why/forbidden-patterns) show at the 2×2 scale, read at the summit instead. ## The frame pays its bill in advance The census has a second story to tell. Colors 1 to 5 appear on zero interior tiles; their $5 \times 24 = 120$ half-edges live entirely on the side edges of the 60 rim tiles. The rim ring has exactly 60 joints, so in any solution the five frame-join colors saturate them exactly, 12 joints per color, with no slack at all. And because a rim tile's orientation is forced (gray outward), each of the 56 edge tiles shows one fixed color to the interior wherever it lands. Summed over the bag, the interior receives a fixed demand vector over colors 6 to 22, namely (4, 5, 3, 3, 1, 1, 2, 3, 4, 6, 4, 2, 3, 6, 4, 3, 2), 56 inward edges in all. Whatever frame you build, the interior's boundary bill is the same 17 numbers, known before any search starts. ## What the community saw in June 2007 The evenness was spotted before the puzzle even shipped. In June 2007 David Eddy opened a mailing-list thread titled Parity with the observation that "the number of edges of each type is even", and drew the last-piece consequence: if the final piece has four different edges, the final hole must show the same four colors in some order, which he put at a 1 in 6 chance of matching ([msg 332](https://groups.io/g/eternity2/message/332)). Christophe Weibel supplied the worked example showing the last piece genuinely can fail to match its hole ([msg 335](https://groups.io/g/eternity2/message/335)), and Brendan Owen, who had studied the design, closed the thread with the practical verdict: parity is correct for E2 by construction, and in a search it only ever helps near the very end ([msg 346](https://groups.io/g/eternity2/message/346)). All of that is true, and the thread stopped there. Pushed one step further, the same observation contains the theorem above: the evenness does more than make the last piece chancy, it deletes the 479 rung outright, sets the defect floor at 2, and caps the one-move shell around every solution at 76. ## Check it yourself Every number on this page reduces to a one-pass census of the official set: the half-edge counts and their evenness, the 960 exact-fit identity, the zero rotationally symmetric tiles, the 4/56/196 corner/edge/interior split, the 50/23/3 defect-2 generators, the frame saturation, and the inward demand vector. The checker recomputes each value beside its expected one and emits a pass flag per claim; all 16 checks pass and the output is byte-stable across runs. The [article, checker source and results are on GitHub](https://github.com/raphael-anjou/eternity2/tree/main/research/topics/parity-defect-floor), and the reproduction block on this page points at the same topic. One caveat on scope: the parity argument covers full 256-piece placements with a legal gray frame. Partial boards, and boards that break the frame rule, are outside it. ## Related - [The theorem sweep: thirteen structural laws](https://eternity2.dev/research/why/theorem-sweep) — One research arc, thirteen families of structural theorems: ring purity, the 479 parity floor, the 470 wall as a phase boundary, flux invariants, the entropy area law, and the impossibility results that price every standard shortcut. This page is the map. - [Forbidden patterns](https://eternity2.dev/research/why/forbidden-patterns) — Almost every small patch of pieces you could build is impossible. For a 2×2 square, 99.72% of the ways to place four pieces can never be made to match. - [Designed to be unsolvable: the recipe](https://eternity2.dev/research/why/design-recipe) — Eternity II follows a recipe for the hardest possible edge-matching puzzle: compact shape, no symmetric or duplicate pieces, split palettes, flat frequencies, one expected solution. The community reverse-engineered every ingredient in the launch year. - [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.