# Parity arguments > Count anything on an edge-matching board twice, once from each side, and the totals must agree, giving impossibility proofs for the price of one pass. The 479 story shows both the power and the trap: a clean parity claim, true for every interior move, defeated through the sixty border edges nobody scores. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/build/analysis/parity-arguments/ - Updated: 2026-07-02 - Topics: structure, search-space - Source: kubzpa's parity claim that a 479 score is impossible (August 2007) — https://groups.io/g/eternity2/message/1640 - Source: psykowally's 478 construction: rotate an interior piece with equal opposite edges (msg 1642) — https://groups.io/g/eternity2/message/1642 - Source: Verhaard's refutation: 479 through a flipped border piece (January 2009) — https://groups.io/g/eternity2/message/6317 - Source: angwin_uk & mjqxxxx on border edge-type balance, 12 per side (August 2007) — https://groups.io/g/eternity2/message/2073 - Source: The 2011 parity mega-thread: checkerboard and rotation-balanced piece sets — https://groups.io/g/eternity2/message/8898 - Source: John Gilbert's negative result: balanced sets stall backtrackers faster (msg 8913) — https://groups.io/g/eternity2/message/8913 - Source: Nick's pen-and-paper near-balanced set, and Jamison's global-sum verification (msgs 8977/8978) — https://groups.io/g/eternity2/message/8977 --- A parity argument is double-entry bookkeeping applied to a game board. Every edge inside the puzzle has two sides, so anything counted over the whole board (colour occurrences, matched edges, orientation sums) gets counted twice, once from each side, and the two ledgers must agree. A state where they disagree is not merely unpromising; it is impossible, and no search is needed to prove it. On a puzzle where [no move is ever forced](/research/why/no-forced-moves) and lookahead is expensive, an invariant that costs one pass over the board and never lies is worth taking seriously, as long as you remember which direction it points. ## The 479 story, in three acts The community's best parity tale starts two weeks after launch. In August 2007, kubzpa argued that a placement with *exactly* 479 matched edges, one single mismatch, cannot exist ([msg 1640](https://groups.io/g/eternity2/message/1640)). The intuition is a parity flip: disturb any piece and the edges it touches change state together, so mismatches should come in pairs. psykowally immediately supplied the constructive companion for 478: take a solved board and rotate an interior piece whose opposite edges carry equal colours by 180°: exactly two edges break ([msg 1642](https://groups.io/g/eternity2/message/1642)). The argument is correct for every interior move. It fails at the frame. In January 2009 Louis Verhaard pointed out the leak: the 60 outward-facing grey edges are *unscored*, so a border piece whose two ring-facing sides share a colour can be flipped end-for-end, breaking exactly one scored edge (the seam edge behind it) while the change on the grey side costs nothing ([msg 6317](https://groups.io/g/eternity2/message/6317)). One mismatch, score 479, parity defeated through the edges the scoring convention ignores. This project checked the claim against the official piece set: 14 border pieces qualify (computed), so any full solution implies a 479. That is the version of the story that [known facts](/research/build/known-facts) now carries. Verhaard added a bureaucratic punchline: the prize entry form recorded piece numbers but not rotations, so Tomy's scorer would have read such a board as a 480. The lesson generalises. A parity argument is only as strong as the boundary conditions it accounts for, and Eternity II's scoring rules puncture the boundary in sixty places. ## Flip it yourself The three acts fit on one small board. Below is a genuinely solved, engine-generated framed 8×8: every scored edge matched, and an outward grey rim the score ignores, exactly like the real puzzle's 60 grey edges (32 of them at this size). Every claim in the story above is a single click here. > **[Figure]** Interactive: the parity argument — interactive: ParityFlipLab. Rendered on the canonical page (link above); not shown in this markdown export. ## Step by step 1. **Start solved.** All 112 scored edges match, this board's stand-in for 480/480. The grey band is the rim the scoring convention never reads. 2. **Act one: click any unmarked interior piece.** A 180° turn swaps the up/down edges together and the left/right edges together, so scored edges break in pairs per axis: 0, 2 or 4 at a time, never an odd count. Try as many as you like; no interior click will ever produce exactly one mismatch. That is kubzpa's argument, and for interior moves it is airtight. 3. **Act two: click a sky-ringed interior piece.** One opposite pair equal, the other not: exactly two scored edges break, and the badge reads a 478-class board, psykowally's constructive companion. 4. **Act three: click an emerald-ringed border piece.** Its two ring-facing sides share a colour, so the 180° flip leaves both lateral edges matched. Only the seam edge behind it breaks (*one* scored edge) while the outward change parks on the grey rim (flashed amber), where no scorer ever looks. One mismatch. 479. Verhaard's refutation, in one click. 5. **Audit the boundary.** The counter panel tells you how many pieces of this draw qualify for each move; on the official set, 14 border pieces qualify (computed), so any full solution implies a 479. The proof was correct everywhere the scoring looked; the leak is precisely the edges it exempted. ## What it costs A parity or balance check is a single pass over the scored edges: $$ O(\text{edges}) \;=\; O(480) \ \text{on the full board}, \qquad O(56) \ \text{for NS-1's seam}, $$ with a constant so small it is effectively free: 480 edge reads are microseconds, against search steps counted in billions. That asymmetry of price is why such checks compose with everything: NS-1 after the border closes rejects 10–28% of deep dead-ends for 56 reads, the cheapest pruning this project knows. But the asymmetry of *information* runs the other way, and it never softens: a violated invariant is a proof of impossibility, a satisfied one proves nothing at all. One pass over the edges buys a certificate that only ever says no. It is worth exactly its price, provided nobody mistakes it for guidance. ## Colour sums and multiset balance The second family of counting arguments tallies colours rather than mismatches. Already in August 2007, angwin_uk observed that the border is built balanced: five border edge types, twelve of each on either side of every border piece's grey edge ([msg 2073](https://groups.io/g/eternity2/message/2073)). mjqxxxx sharpened the point: border pieces sit in a fixed orientation, so each type must split *equally* into left-facing and right-facing edges, a strictly stronger condition than even counts ([msg 2098](https://groups.io/g/eternity2/message/2098)). Follow that thought inward and you reach the seam. In any complete solution, the multiset of colours the border ring presents to the interior must equal the multiset the interior presents back: every colour handed in is handed out. That is Hopfer's NS-1 condition, formalised in 2022 and treated at length on [the border balance page](/research/why/border-balance): a genuine necessary condition, cheap to check, and blind to everything that happens interior-to-interior. Same mathematics, two uses. In 2007 the balance served to [estimate how many border solutions exist](/research/build/analysis/solution-counting); in 2022 it was turned around into a pruning certificate. ## The 2011 expedition: balance is abundant and buys nothing After the contest ended, the list spent the summer of 2011 pushing parity as far as it would go. Juraj Pivovarov posed the *oriented set* problem: split the 256 pieces into checkerboard piles A and B so that every colour's directional edge counts balance, because knowing either the orientations or the pile assignment of a solution would make the rest easy ([msg 8898](https://groups.io/g/eternity2/message/8898)). Peter McGavin reduced the condition to per-colour sums, left equals right and top equals bottom ([msg 8906](https://groups.io/g/eternity2/message/8906)). Juraj then counted the qualifying checkerboard rotation-balanced sets: his first estimate of roughly $4.5 \times 10^{485}$ ([msg 8929](https://groups.io/g/eternity2/message/8929)) drew a "something must be wrong" from Michael Field ([msg 8930](https://groups.io/g/eternity2/message/8930)), and the corrected count settled around $3 \times 10^{147}$ ([msg 8931](https://groups.io/g/eternity2/message/8931)). All the while, he framed the search for even one as a hard instance of PARTITION. Two results ended the expedition, both worth keeping. John Gilbert ran the experiment: balanced checkerboard sets can be found one at a time, but feeding the balance to a backtracker as a constraint makes it stall *faster*: each placement now draws on half the candidate pieces, and the restriction costs more than it prunes ([msg 8913](https://groups.io/g/eternity2/message/8913)). And Nick, working with pen and paper while bored on a train ([msg 8960](https://groups.io/g/eternity2/message/8960)), drove a full 256-piece checkerboard-plus-rotation assignment to within a single edge-flip of balance ([msg 8977](https://groups.io/g/eternity2/message/8977)); Jason Jamison verified the global sums under Nick's coding (all tops equal all bottoms at 2,809, all lefts equal all rights at 2,881) and reported that the near-balanced set still stalled his backtracker about 19 pieces out of a corner ([msg 8978](https://groups.io/g/eternity2/message/8978)). Balance is real, abundant, and buys the search nothing. ## What parity buys a solver author Three things, none of them a solution. **Cheap necessary conditions.** A parity or balance check costs one pass and composes with anything: a backtracker, a [local search](/research/build/local-search/local-search-alns), a sanity check on someone else's claimed board. The NS-1 figures above show the going rate. **Certificate asymmetry.** Every argument on this page points one way: a violated invariant says *definitely broken*, a satisfied one never says *definitely fine*. Swap two border pieces and NS-1 stays at zero; balance a piece set perfectly and the backtracker stalls anyway. Parity prunes; it does not guide. **A boundary-conditions reflex.** The 479 proof was correct everywhere the prover looked, and wrong because the scoring rules created sixty edges he did not have to look at. Before trusting any counting argument on this puzzle, including this project's own, audit what the frame, the scoring convention, and the unscored grey are quietly exempting. On Eternity II, the exceptions live at the border, and the border is where the arguments go to die. ## Related - [The border balance](https://eternity2.dev/research/why/border-balance) — A solved board hides a simple bookkeeping law: every colour the border hands to the interior, the interior hands straight back. Break it and you know instantly the board is wrong; obeying it, though, guarantees nothing. - [Known facts & numbers](https://eternity2.dev/research/build/known-facts) — The numbers every Eternity II researcher ends up re-deriving, collected in one place with their provenance: the puzzle definition, the clue placements, scoring conventions, the record table, search-space sizes and the structural counts.