# Tuned to the hardness peak > Eternity II uses 22 colors. They split 17 interior to 5 frame-only, and that 17 is exactly where this kind of puzzle is hardest to solve. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/phase-transition/ - Updated: 2026-07-02 - Topics: structure - Reproduce: `just research-phase-transition` - Source: Brendan Owen, Design the hardest puzzle: the 17+5 derivation (eternity2 mailing list, August 2007) — https://groups.io/g/eternity2/message/1947 - Source: Ansótegui, Béjar, Fernández & Mateu, On the hardness of solving edge matching puzzles as SAT or CSP problems (Constraints, Springer 2013) — https://link.springer.com/article/10.1007/s10601-012-9128-9 - Source: Ansótegui, Béjar, Fernández & Mateu, How Hard is a Commercial Puzzle: the Eternity II Challenge — https://repositori.udl.cat/server/api/core/bitstreams/0b6533fe-54e5-4070-85fe-80f7d35837d8/content --- Hard search problems have a difficulty knob. Loosen it and there are many solutions, so a search trips over one fast. Tighten it and there are none, which is often easy to prove. In between sits a narrow band where solutions are scarce but real, and that's where search blows up. People call it a phase transition, like water freezing. For edge-matching puzzles the knob is the number of colors. Too few and pieces fit together countless ways; too many and they barely fit at all. The [published analysis](https://link.springer.com/article/10.1007/s10601-012-9128-9) puts the peak at around 17 interior colors, the setting where a puzzle this size has about one solution. Eternity II uses 17. The community had that number within weeks of launch: in August 2007, Brendan Owen derived $I = (196! \cdot 4^{196})^{1/392} \approx 17.14$ interior colours from the "about one expected solution" criterion, exactly the published parameters, years before the academic analyses ([Design the hardest puzzle, msg 1947](https://groups.io/g/eternity2/message/1947)). ## The difficulty wall > **[Figure]** Interactive: the colour-count difficulty peak — interactive: PhaseTransitionLab. Rendered on the canonical page (link above); not shown in this markdown export. ## Or measure it yourself The chart above is precomputed. This one isn't: the engine solves fresh puzzles live, one per colour count, and draws the peak from real runs in your browser. > **[Figure]** Interactive: solve across the difficulty peak live — interactive: PhaseTransitionLiveLab. Rendered on the canonical page (link above); not shown in this markdown export. ## The transition, measured on real pieces The colour-count argument says *where* the peak is. In March 2008 Brendan Owen went and watched a phase transition happen, directly, on the actual pieces. He took a thin 2-by-L interior rectangle and tiled it with random sets of E2's interior pieces, four hundred random sets at each length, and recorded how many could be completed with no mismatch ([msg 4909](https://groups.io/g/eternity2/message/4909)). > **[Interactive: RectangleTransitionChart]** Rendered on the canonical page (link above); not shown in this markdown export. The shape is the phase transition in miniature. A 2-by-1 or 2-by-2 strip is so short that random pieces often just fit: 58% solvable at length 1. Then it dies completely. From length 4 through 15, not one random set out of four hundred tiles the rectangle at any length: solutions are so scarce they effectively do not exist. And then they come back. At length 16 one set in four hundred works, by 19 it is 8.5%, by 21 it is 57%, and by 22 nearly nine in ten. The region where solutions are vanishingly rare but not yet impossible is exactly the hard band, and it is not a story or a model here, it is a count. It is also why a 14-cell-wide interior is so punishing: it sits in the steep part of that climb, where a solution exists but almost no random arrangement is one. ## The split, straight from the pieces Sorting the official set's colors by where they appear shows the design plainly.
5 frame-only colors
{data.frameOnlyColors.map((c) => (
> **[Interactive: MotifSwatch]** Rendered on the canonical page (link above); not shown in this markdown export. '{colorToLetter(c)}'
))}
These appear only on border and corner pieces, never in the interior. They are the rare colors, kept to the edge.
17 interior colors
{data.interiorColors.map((c) => (
> **[Interactive: MotifSwatch]** Rendered on the canonical page (link above); not shown in this markdown export. '{colorToLetter(c)}'
))}
The palette of the inside of the board, where almost all the matching happens.
## The set in numbers | | Count | | ----------------- | ----: | | Corner pieces | 4 | | Edge pieces | 56 | | Interior pieces | 196 | | Interior colors | 17 | | Frame-only colors | 5 | ## Why it matters This is the clearest single sign that Eternity II was made hard on purpose. Board size, piece count and color split all aim at the same target: a puzzle with about one solution, placed at the worst possible spot for any search to find it. The difficulty was chosen, the way a good exam is neither trivial nor impossible. How do we know the peak is real and not just a story? Two ways meet here. The published analysis [derives it](https://repositori.udl.cat/server/api/core/bitstreams/0b6533fe-54e5-4070-85fe-80f7d35837d8/content): for framed edge-matching puzzles, the color count where you'd expect about one solution falls near 17, and that is the hardest setting to search. And you can watch a piece of it yourself in the demo above: build real puzzles, count the work, and see it explode when colors are scarce. The impact is concrete. It means the gap to a solution isn't a tuning problem you can grind away with a faster machine; the puzzle was placed where search is worst on purpose, so beating it needs a genuinely better idea, not just more effort. See the difficulty measured live on the [Algorithms page](/algorithms). ## Related - [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. - [Why a faster computer doesn't help](https://eternity2.dev/research/why/prune-vs-speed) — The single most important idea in hard combinatorial search: shrinking the space you search beats searching it faster, by an exponential margin. Eternity II is engineered so you can barely shrink it at all. - [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. - [No forced moves](https://eternity2.dev/research/why/no-forced-moves) — 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.