# Known facts & numbers

> 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.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/build/known-facts/
- Updated: 2026-07-21
- Topics: structure
- Source: Wikipedia, "Eternity II puzzle" (the prize, the deadline, Verhaard's 467 and the 10^557 estimate) — https://en.wikipedia.org/wiki/Eternity_II_puzzle
- Source: 469: “EternityII Solver” thread (the Blackwood & McGavin board), eternity2@groups.io, 2020-09 — https://groups.io/g/eternity2/message/10045
- Source: 470: “470 Found” (Joshua Blackwood), eternity2@groups.io, 2021-03 — https://groups.io/g/eternity2/message/10117
- Source: 460 with all five clues: “Highest points (of 480) with using all 5 (!) hints?” thread, eternity2@groups.io, 2023-03 — https://groups.io/g/eternity2/message/11074
- Source: 464 with all five clues (Benjamin Riotte): “Record of Eternity2 with 5 hints?” thread, eternity2@groups.io, 2026-07 — https://groups.io/g/eternity2/message/11919
- Source: Brendan Owen's expected-solution-count estimates (groups.io message 5197) — https://groups.io/g/eternity2/message/5197
- Source: Louis Verhaard, algorithm and score details for the 467 (shortestpath.se) — https://www.shortestpath.se/eii/eii_details.html
- Source: e2.bucas.name (Jef Bucas), the community board viewer; every record board re-scores there — https://e2.bucas.name

---
Every conversation about Eternity II eventually stops while someone re-derives
the same handful of numbers. This page is that handful, written down once.
Facts marked **computed** are re-derived exactly from the official piece set,
the same data our [reference numbers](/research/reference) and
[Why is it hard?](/research/why) pages are checked against, and anyone can
reproduce them. Record facts come from the primary mailing-list messages;
encyclopedic facts from Wikipedia.

## The puzzle by the numbers

| Fact | Value | Provenance |
| --- | --- | --- |
| Board | 16×16 = 256 cells, framed | official |
| Pieces | 256, all distinct | official; computed |
| Corner pieces (2 grey sides) | 4 | computed |
| Edge pieces (1 grey side) | 56 | computed |
| Interior pieces (0 grey sides) | 196 | computed |
| Colours (grey border excluded) | 22 | computed |
| of which frame-only colours | 5 | computed |
| of which interior-palette colours | 17 | computed |
| Internal edges (adjacent cell pairs) | 480 = 2 · 16 · 15 | computed |
| Pinned clue placements | 5 | official |
| Prize | $2,000,000, unclaimed; deadline noon, 31 Dec 2010 | Wikipedia |

The grey outer-border colour is not one of the 22: it never appears on an
internal edge. The 5 + 17 colour split is not an accident: 17 interior
colours is [exactly the hardness peak](/research/why/phase-transition) for a
puzzle of this size.

## The five clues

The rules sheet pins one piece; Tomy sold four smaller
[clue puzzles](/research/build/clue-puzzles) whose
solutions each revealed one more placement, and the remaining placements were
published later. Squares use the board notation of
[e2.bucas.name](https://e2.bucas.name): rows A–P from the top, columns 1–16
from the left. Piece numbers are 1–256 in the official list order, and the
rotation is clockwise from the piece's stored orientation.

| Square | Piece | Rotation | Which clue |
| --- | --- | --- | --- |
| I8 | 139 | 0° | mandatory (in the rulebook) |
| C3 | 208 | 90° | clue puzzle |
| C14 | 255 | 90° | clue puzzle |
| N3 | 181 | 90° | clue puzzle |
| N14 | 249 | 180° | clue puzzle |

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

A board that respects all five is called **strict-canonical**; most record
boards respect only the mandatory centre clue ("canonical"). That distinction
splits the record table in two; see below.

The clues also settle a counting question: with only the mandatory piece
placed, [complex theory](/research/why/complex-theory) expects ≈ 14,702 full
solutions; with all five, ≈ 1. The 5-clue puzzle has, by design, a single
expected solution.

## Scoring conventions

- **Score = matched internal edges, out of 480.** The 60 outward-facing grey
  edges around the frame are not counted. This is the convention of the prize
  scrutiny, of every record in the table below, and of this wiki.
- **"Breaks" = 480 − score.** Solver authors often report mismatch counts
  instead: Blackwood's 468 was announced to the mailing list as "12 breaks".
- **479 is possible, but only through the border.** A launch-summer
  [parity argument](/research/build/analysis/parity-arguments) claimed exactly
  one mismatch cannot exist
  ([kubzpa, 2007](https://groups.io/g/eternity2/message/1640)), and it holds
  for interior moves (rotating an interior piece with equal opposite edges
  breaks two, giving 478,
  [msg 1642](https://groups.io/g/eternity2/message/1642)). But the parity
  leaks through the 60 *unscored* outward border edges: flip a border piece
  whose two ring-facing sides share a colour and exactly one scored edge
  breaks ([Verhaard, 2009](https://groups.io/g/eternity2/message/6317)).
  The official set has 14 such border pieces (computed), so any full
  solution implies a 479.
- **Check the piece set before comparing scores.** Several headline "480"
  boards use [variant piece sets](/research/build/variants) (Clue-1/Clue-2
  designs, mixed sets, the unframed TopCoder variant). They are different
  puzzles; see [Records & solvers](/research/records).

The 480 internal edges split into three families (computed):

| Edge family | Count |
| --- | --- |
| Border ↔ border (around the 60-piece frame ring) | 60 |
| Border ↔ interior (the seam, 4 × 14) | 56 |
| Interior ↔ interior (inside the 14×14 core) | 364 |
| **Total** | **480** |

## The record table

The full timeline, with board previews and per-record references, lives at
[Records & solvers](/research/records). The numbers themselves:

| Score | Who | When | Track |
| --- | --- | --- | --- |
| 467 | Louis Verhaard's team | Sept 2008 | canonical; won the $10,000 partial prize |
| 468 | Joshua Blackwood | Aug 2020 | canonical |
| 469 | Peter McGavin (Blackwood's solver) | Sept 2020 | canonical |
| 469 | various (~7 boards, independent finds plus one single-swap of McGavin's) | Nov 2020 | canonical |
| **470** | **Joshua Blackwood** | **Mar 2021** | **canonical, the ceiling since** |
| 470 | Jef Bucas | Dec 2024 | canonical (tie of the 470) |
| 460 | Bruno Gauthier | Mar 2023 | strict-canonical; best board respecting the four optional clues, 2023 until beaten Jul 2026 |
| **464** | **Benjamin Riotte** | **Jul 2026** | **strict-canonical record; beat Gauthier's 460, all five clues respected** |
| "480" | various | 2023 | variant piece sets, not the canonical puzzle |

The 469s were long quoted as the ceiling, but board-level verification shows
the 468/469/470 boards are all the same starter-only regime (the contest
rules pinned only the starter piece), so the 470 is the record on the same
puzzle, not an easier variant. A full canonical solution (480/480) has never
been reported by anyone, in any public source. The gap of 10 edges has stood
since March 2021. Board previews and the per-record references for every row
are at [Records & solvers](/research/records).

**Open targets.** The full 16×16 is not the only unsolved board with a name.
The nearest named one is Brendan Owen's
[10×10 set_2](/research/build/benchmarks#brendans-1010s-one-fell-one-is-still-standing):
its twin set_1 fell after about 180 core-years, and set_2 is still open with its
difficulty already priced by theory. The
[open problems board](/research/open-problems) collects it alongside the rest of
the live frontier.

## How big is the search

| Count | Value | Provenance |
| --- | --- | --- |
| All placements, no constraints: 256! · 4²⁵⁶ | ≈ 1.15 × 10⁶⁶¹ | computed |
| Corners in corners, edges on the border (orientation forced), clue pinned: 4! · 56! · 195! · 4¹⁹⁵ | ≈ 1.115 × 10⁵⁵⁷ | computed; matches the widely quoted Wikipedia figure |
| Widest point of the search tree (complex-theory estimate, row scan) | ≈ 10⁴⁵ ways to extend, around depth 50–200 | Owen / McGavin, see [complex theory](/research/why/complex-theory) |
| Expected full solutions, 1 clue / 5 clues | ≈ 14,702 / ≈ 1 | same |

The 10⁵⁵⁷ figure is worth being able to re-derive rather than cite: the 4
corner pieces go to the 4 corners (orientation forced by the two grey sides),
the 56 edge pieces to the 56 border slots (orientation forced), and the
remaining 195 free interior pieces (196 minus the pinned clue) each take one
of 4 rotations.

## Symmetry facts

All computed from the official piece set:

- **No piece is rotation-symmetric.** Every one of the 256 pieces has 4
  distinct orientations, so rotation never shrinks a piece's candidate set.
- **No two pieces are identical, even up to rotation.** There are no
  interchangeable duplicates to exploit.
- **Five pairs share a colour multiset without being the same piece**: pieces
  3 & 4, 6 & 15, 8 & 52, 110 & 111, 172 & 182 (1-based official numbering)
  carry the same four colours in a different cyclic order.
- **The clues break all board symmetries.** A 16×16 board has no centre cell,
  so every rotation or reflection moves the mandatory clue square I8: no
  non-trivial symmetry maps the clued puzzle to itself, and symmetry-breaking
  offers no further search-space reduction.

## Colours, edge by edge

The 256 pieces carry 1,024 edge stickers; 64 are grey (2 per corner piece,
1 per edge piece), leaving 960 coloured edge instances. Their census
(computed; see [rare-colour geography](/research/why/rare-color-geography)):

| Colour group | Colours | Edge instances each | Total |
| --- | --- | --- | --- |
| Frame-only ("rare") | 5 | 24 (all on frame pieces, none interior) | 120 |
| Interior palette, low | 5 | 48 | 240 |
| Interior palette, high | 12 | 50 | 600 |
| **All** | **22** | | **960** |

Two consequences researchers keep rediscovering:

- **Every per-colour count is even, and the halves sum to exactly 480.**
  A matched internal edge consumes two instances of one colour, so a perfect
  solution needs $\sum_c N_c / 2 = 480$ matches, which is exactly the number
  of internal edges. There is **zero slack**: in a full solution every single
  edge instance of every colour finds its partner.
- **The five rare colours live only on the frame** (24 instances each), which
  makes the frame ring a nearly self-contained sub-puzzle; the geometry is
  worked through in [rare-colour geography](/research/why/rare-color-geography).

## Local structure

Exact, exhaustive counts over the official interior pieces (computed; details
and live demos on the linked pages):

| Fact | Value | See |
| --- | --- | --- |
| Random 2-cell interior placements that are infeasible | 14,890 / 38,220 ≈ 39% | [forbidden patterns](/research/why/forbidden-patterns) |
| Random L-tromino placements infeasible | ≈ 83.3% | same |
| Random 2×2 interior placements infeasible | 1,427,039,544 / 1,431,033,240 ≈ 99.72% | same |
| Interior pieces with a forced neighbour | 0: every piece has 73–137 possible right-neighbours | [no forced moves](/research/why/no-forced-moves) |
| Border↔interior colour balance in any full solution | deficit $\Delta = 0$ (a necessary condition, Hopfer 2022) | [the border balance](/research/why/border-balance) |

## Reading the provenance

- **computed**: re-derived exactly from the official 256-piece set; the
  relevant pages carry `just research-…` reproduction commands and the raw
  results live in the [repository](https://github.com/raphael-anjou/eternity2).
- **groups.io**: the record scores (468, 469, 470, the strict 464) exist only
  in mailing-list messages and on the community board viewer; the exact
  announcement messages are linked from [Records & solvers](/research/records),
  and each board can be re-scored at [e2.bucas.name](https://e2.bucas.name).
- **Wikipedia**: covers only the encyclopedic layer (the prize, the deadline,
  and Verhaard's 467).

## Related

- [Reference numbers](https://eternity2.dev/research/reference) — Exact counts of how many valid ways a small block can be filled at a given position of the official Eternity II board, under increasingly constrained rules: known-good numbers to check your solver's edge-matching and constraint code against.
- [Records & solvers](https://eternity2.dev/research/records) — Eternity II has never been solved, but fifteen years of community effort have pushed the best board to 470/480. Who holds what, how they did it, and why some headline "480" boards are not actually the real puzzle.
- [Tuned to the hardness peak](https://eternity2.dev/research/why/phase-transition) — 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.