# Ring purity: the border is a closed sub-puzzle with zero slack > Five of the 22 colors never touch the 196 interior pieces. The piece list forces every valid solution to spend all 120 frame half-edges on the border ring: a self-contained sub-puzzle at exactly zero slack (120 = 120), an Eulerian circuit on five vertices, coupled to the interior through just 56 inward edges. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/ring-purity/ - Updated: 2026-07-22 - Topics: structure - Source: Ring purity: reproduction topic with the article, the committed checker and the results JSON (GitHub) — https://github.com/raphael-anjou/eternity2/tree/main/research/topics/ring-purity - Source: Brendan Owen digitises the piece set: 24 half-edges of each of the 5 border-join colours, 48 or 50 of each interior colour (groups.io msg 1054, July 2007) — https://groups.io/g/eternity2/message/1054 - Source: Brendan Owen, Design the hardest puzzle: the deliberate 17+5 palette split (groups.io msg 1947, August 2007) — https://groups.io/g/eternity2/message/1947 --- Five of the 22 edge colors never appear on any of the 196 interior pieces. In the standard numbering they are colors 1 to 5: the frame colors. Each occurs on exactly 24 half-edges, all of them on the 60 border pieces, for a total supply of 120. Ring purity is the statement that the piece data leaves this supply no choice at all: every valid solution spends the entire 120 on the 120 ring-facing edge slots of the border ring, with zero slack. The border is a self-contained sub-puzzle, and its only contact with the interior runs through 56 inward-facing edges.
120 = 120
ring demand vs frame supply, zero slack
5 × 24
frame colors × half-edges each
56
inward edges coupling ring to interior
## Three facts about the piece list The proof rests on three properties you can verify by scanning the 256 pieces once. No search, no sampling. First, frame colors occur only on border pieces. The checker does not trust the labels: it derives the frame set from the data, as the colors with zero occurrences on interior pieces, and finds exactly five. Second, every one of the 56 edge pieces carries exactly two frame-colored slots and one non-frame slot among its three non-zero slots, and the non-frame slot sits opposite the gray slot. This holds 56 out of 56 times. Third, both non-zero slots of every corner piece are frame-colored, and the two gray slots are adjacent. This holds 4 out of 4 times. For the record, the rest of the palette is as flat as the frame part: of the 17 interior colors, 5 occur 48 times and 12 occur 50 times. Brendan Owen measured exactly these counts the week he digitised his set ([msg 1054](https://groups.io/g/eternity2/message/1054)), and the strict separation of the two palettes was a design decision ([msg 1947](https://groups.io/g/eternity2/message/1947)); the [design-recipe page](/research/why/design-recipe) tells that story. ## The forcing argument An edge piece sits on the rim with its gray slot facing outward, so the slot opposite the gray one faces the interior. Its inward neighbour is an interior piece, which by the first fact has no frame color to offer. The inward slot therefore cannot be frame-colored. By the second fact the edge piece owns exactly one non-frame slot, and it sits exactly opposite the gray slot, which is the inward position. So the two frame slots are forced onto the two ring-facing positions, along the rim. Corners contribute both their non-zero slots to the ring by the third fact. Now count. Demand: $56 \times 2 + 4 \times 2 = 120$ ring-facing slots. Supply: $5 \times 24 = 120$ frame half-edges. The two numbers match exactly. Every frame half-edge is consumed on the ring, none is left over, and no non-frame color ever appears on a ring joint. That is the zero-slack part: the frame-color budget is spent to the last half-edge. ## The ring is an Eulerian circuit The saturation has a clean graph-theoretic reading. Take a multigraph on 5 vertices, one per frame color, and one edge per border piece, joining its two ring-facing colors. A valid arrangement of the border ring is exactly an Eulerian circuit of this multigraph: walking the ring reads off a closed trail that uses every piece-edge once, and conversely. On the real instance the multigraph has 60 edges, every degree equal to 24, and it is connected, so an Eulerian circuit exists, as it must, since a full solution exists. Of the 60 edges, 14 are self-loops (pieces showing the same frame color on both ring slots), and the edges span all 15 possible color pairs, the 10 unordered pairs plus the 5 loops. ## What the law is worth Two strength measures come with the theorem, one exact and one sampled. The exact one is first-step branching. Fix any placed border piece; the next piece along the ring must match the exposed frame color. Of the border pieces, only those incident to that color qualify: depending on the color, 20, 21, 21, 22 or 22 of them, mean 21.2, against 59 candidates for an unconstrained ordering. Color matching alone cuts the first branching factor by 2.78x. The sampled one is a sequential importance sampling estimate over 1000 seeded greedy ring builds. The mean log10 path probability of a completed build is -27.30 (sd 0.64) on seed 1 and -27.39 (sd 0.62) on seed 2, against a uniform baseline of $\log_{10}(1/59!) = -80.14$. Read that as: the color-matching law concentrates probability mass by roughly 53 orders of magnitude relative to a uniform ordering of the border pieces, and still leaves a residual rarity near $10^{-27}$. A purely greedy color-matching walk accordingly almost never finishes a ring: 8.7% completion on seed 1, 9.3% (seed 2, run for batch consistency; the committed JSON records seed 1) on seed 2, with the source run at 9.4%. All 87 completed seed-1 builds closed into circuits, open ends matching, which is the Eulerian structure showing up in the sample. ## What stays open The exact count of undirected Eulerian circuits of the real ring multigraph is open. A single-orientation shortcut through the BEST theorem was attempted during the original investigation, caught as an undercount by a K5 sanity check, and withdrawn; no circuit count is claimed anywhere. The greedy completion measure also ignores the geometric corner/edge alternation of the physical ring: it measures the color-matching law in isolation, not the full ring constraint. ## Where it sits, and how to check it The theorem splits the puzzle along the rim. Inside the ring, the 56 edge pieces each show one color from the 17-color interior palette, and those 56 inward edges are the border's entire interface with the 196-piece interior; the bookkeeping across that seam is the subject of the [border-balance page](/research/why/border-balance). The visible face of the same split, the five rare colors quarantined at the rim, is on the [rare-color-geography page](/research/why/rare-color-geography). And this page is the detailed account of one law in the [theorem sweep](/research/why/theorem-sweep), alongside the instance's other exact structure. Every number above is recomputed by the committed checker in the reproduction topic linked under sources: a single Rust program that loads the official instance, derives the frame colors from the data, verifies each clause of the theorem exhaustively, builds the ring multigraph, runs the two seeded Monte Carlo measures, and emits one JSON file (committed as `results/ring_purity.json`) holding every figure quoted here. The deterministic clauses reproduce byte for byte; the sampled ones agree within sampling error across independent seeds. ## 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. - [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. - [The rare colors live on the frame](https://eternity2.dev/research/why/rare-color-geography) — 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. - [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.