# The rare colors live on the frame

> 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.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/rare-color-geography/
- Updated: 2026-07-01
- Topics: structure
- Reproduce: `just research-rare-color-geography`
- Source: Brendan Owen derives the 17+5 colour design as the hardest possible puzzle (groups.io message 1947) — https://groups.io/g/eternity2/message/1947

---
Bar,
  BarChart,
  CartesianGrid,
  Legend,
  ResponsiveContainer,
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The 22 colors don't play equal roles. Sort them by where their edges sit and
they split into two clean classes: five rare colors fenced off to the border,
and seventeen common colors that do almost all their work inside the board.

## Where the rare colors are allowed

Pick one of the five rare colors and watch the board: all 24 of its edges light
up around the frame ring, and the 14×14 interior stays completely blank. Switch
to a common color and the interior fills instead. That empty middle, for every
rare color, is the whole finding.

> **[Figure]** interactive: RareColorRing. Rendered on the canonical page (link above); not shown in this markdown export.

## The five rare colors
> **[Interactive: RareSwatches]** Rendered on the canonical page (link above); not shown in this markdown export.
<div className="mx-auto grid max-w-md grid-cols-3 gap-3 text-center">
  <div className="rounded-lg border p-3">
    <div className="text-2xl font-bold tabular-nums">{data.edgesPerRareColor}</div>
    <div className="text-xs text-muted-foreground">edges each</div>
  </div>
  <div className="rounded-lg border p-3">
    <div className="text-2xl font-bold tabular-nums">{data.rareEdgesTotal}</div>
    <div className="text-xs text-muted-foreground">rare edges, all on the frame</div>
  </div>
  <div className="rounded-lg border p-3">
    <div className="text-2xl font-bold tabular-nums text-emerald-600">0</div>
    <div className="text-xs text-muted-foreground">rare edges in the interior</div>
  </div>
</div>

## Frame vs interior, color by color

> **[Figure]** interactive: FrameInteriorChart. Rendered on the canonical page (link above); not shown in this markdown export.

> **[Figure]** Interactive: the rare-colour border geography — interactive: RareColorLab. Rendered on the canonical page (link above); not shown in this markdown export.

## Why it matters

The split itself is structural, not a defensive trick. Because the outer rim
is solid grey, every border piece keeps its grey edge facing out, so its
coloured edges only ever meet other border edges or the interior: the border
and interior colours live in separate pools no matter what. The five border
colours read as "rare" simply because there are far fewer border edges to
colour, and they could be relabelled to any five values (even reusing interior
ones) without changing the puzzle at all. (See the [design note on
why](/research/why#engineered-to-resist-cleverness), with thanks to Vasily V.
on the groups.io list for the correction.)

What is real, and what matters for solving, is the consequence: the 14×14
interior is left to seventeen common colours with no rare, highly-constraining
signal anywhere in it, which is a large part of why interior search has so
little to grip. The border, where the colour pool is small and matches are
tight, is correspondingly the part every strong board gets essentially perfect.
Eternity II's colour counts were balanced to remove the statistical handholds
that sank Eternity I; this geography is where you feel the result.

See [why the interior gives no forced moves](/research/why/no-forced-moves).

## 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.
- [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.
- [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.
- [CLOISTER](https://eternity2.dev/research/lab/experiments/raphael-anjou/pipelines/cloister) — Fix a perfect border, then search the interior with the border's edges treated as hard constraints from the very first cell.