# Solution counting: measuring what you cannot find

> Nobody has ever seen a full Eternity II solution, yet the community knows, to within a factor of two, how many exist. This page is the story and the craft of that number: exact censuses on small boards, the expectation formula and its twenty-year convergence on 14,702, and the culled-search estimates that were trusted only when four independent runs agreed.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/build/analysis/solution-counting/
- Updated: 2026-07-02
- Topics: structure
- Source: Brendan Owen's launch-day estimate: 5,930 with the hint, ~4.65M without, reconciling the 'roughly 5 million' Monckton had given Dave Clark (msg 987, 2007-07) — https://groups.io/g/eternity2/message/987
- Source: The closed-form expected-solutions product posted to the list (msg 3385, 2007-11) — https://groups.io/g/eternity2/message/3385
- Source: kubzpa's paper 'E2 has 15 millions of solutions' (msg 3497, 2007-12; v2 after peer review, msg 3583) — https://groups.io/g/eternity2/message/3497
- Source: Owen's four independent culled border searches converge on ≈4.05×10³⁷ (msg 2696, 2007-09; first figure msg 1225) — https://groups.io/g/eternity2/message/2696
- Source: McGavin: 1.1527×10⁷ solutions without the starter vs 14,702 with it, and nodes-per-solution unchanged (msg 8924, 2011) — https://groups.io/g/eternity2/message/8924
- Source: McGavin's canonical statement: 14,702 with the starter, 4×10⁻⁸ with all five clues (msg 11193, 2024) — https://groups.io/g/eternity2/message/11193
- Source: Clue #3 counted exactly: 2,195,647,488 solutions (msg 8168, 2010) — https://groups.io/g/eternity2/message/8168
- Source: Both 9×9 benchmarks exhausted: 2 and 3 solutions against a predicted 3.2 (msg 8793, 2011; corrections 8801–8803) — https://groups.io/g/eternity2/message/8793

---
Nobody has ever exhibited a full Eternity II solution. The community
nevertheless agrees on how many exist, to within a claimed factor of two:
about **14,702** with the mandatory starter piece, and about **4×10⁻⁸**,
effectively exactly one, with all five clues placed
([msg 11193](https://groups.io/g/eternity2/message/11193)). This page is
about how a group of people learned to count something none of them could
find, and why they bothered.

The *why* is not curiosity. A puzzle with 14,702 solutions and a puzzle with
1 are different beasts in every way that matters. **Design**: the puzzle's
parameters were tuned so the expected count sits near one; the community
reverse-engineered the designer's knob within weeks of the 2007 announcement,
when Alan O'Donnell wrote the count as a function of the number of colours
$B$ and found it crosses $1$ at $B \approx 14.67$
([msg 94](https://groups.io/g/eternity2/message/94)); the real puzzle's
[17 interior colours](/research/why/phase-transition) sit just past that
knife edge. **Difficulty**: solution count divided into search-tree size
(nodes per solution) is the real price of the hunt, and it behaves
counter-intuitively: Peter McGavin computed that dropping the starter
constraint multiplies the solutions by 784 while leaving nodes-per-solution
essentially unchanged, $9.2766\times10^{42}$ against $9.2751\times10^{42}$
([msg 8924](https://groups.io/g/eternity2/message/8924)). More solutions do
not mean an easier puzzle; they mean a proportionally bigger haystack.
**Verification**: exact counts on small regions are the community's
correctness currency: the way two solvers prove they read the same pieces
without ever sharing them (see
[the benchmark culture](/research/build/benchmarks)).

One more reason the count matters: the solutions are not variations on a
theme. Brendan Owen and others stressed that distinct solutions are
"generally a complete restructure of the entire puzzle", not local swaps of
a few pieces ([msgs 7364/7365](https://groups.io/g/eternity2/message/7365)),
so 14,702 is a count of genuinely different boards, scattered through the
search space.

## Three ways to count

Twenty years of list traffic sort every counting result into three regimes,
each with its own guarantees, its own price, and its own failure modes.

| Regime | What it gives | What it costs | Flagship result |
| --- | --- | --- | --- |
| Exact enumeration | The true count | The entire search tree | Clue #3: exactly 2,195,647,488 |
| First-moment expectation | An average over puzzles like this one | Pencil-and-paper arithmetic | 14,702 with the starter |
| Sampled / culled search | An estimate with informal error bars | A fixed node budget per run | Border ring ≈ 4.05×10³⁷ |

### Exact enumeration: counting by exhausting

Where the tree is small enough to walk entirely, counting is just searching
without stopping at the first success. The community used this from the
first weeks as a *verification protocol*: tile the top-left 3×3 corner with
all 256 pieces and you must find exactly **2,633,221** solutions
([msg 2229](https://groups.io/g/eternity2/message/2229)), a number safe to
publish under the piece-set copyright, and impossible to reproduce with even
one piece mistyped. The 5×5 corners followed in 2009, cross-verified by
three independent programs (top-left: 1,596,901,885,652 partials with all
five hints, [msg 7103](https://groups.io/g/eternity2/message/7103),
[7105](https://groups.io/g/eternity2/message/7105)). Matching the consensus
got you into the community's half-joking "Right Numbers Club"; the whole
story is on the [benchmarks page](/research/build/benchmarks).

Two exact results stand above the rest. In 2010, apal1969 exhausted the
official Clue #3 puzzle (a real, commercial Tomy instance) and found
exactly **2,195,647,488** solutions
([msg 8168](https://groups.io/g/eternity2/message/8168)): the only
commercially sold instance of the family whose solution count is known
exactly rather than estimated. And in 2011, both of Brendan Owen's 9×9
benchmark puzzles were exhausted, at about $1.9\times10^{14}$ and
$1.45\times10^{14}$ nodes and three weeks each on a dual-core Opteron.
They yielded **2** and **3** solutions;
[complex theory](/research/why/complex-theory) had predicted 3.2 for the
first ([msg 8793](https://groups.io/g/eternity2/message/8793)). Even the counting
had to be counted carefully: the first report claimed one solution each,
and it took McGavin's independently shuffled backtrackers to surface the
missing boards ([msgs 8801–8803](https://groups.io/g/eternity2/message/8801)).
Exhaustive does not mean bug-free; only replication is.

### The first moment: counting by expectation

For the full 16×16 board, exhaustion is off the table, so the community's
main tool has always been the *first-moment* (expected-value) calculation:
multiply the number of ways to arrange the pieces by the probability that
every internal edge matches, treating edge colours as independent draws.
The first versions appeared within **days** of the January 2007
announcement, months before anyone had pieces
([msg 38](https://groups.io/g/eternity2/message/38),
[94](https://groups.io/g/eternity2/message/94)). On launch day, with the
real piece statistics in hand, Owen computed ≈ **5,930** solutions with the
mandatory hint and ≈ **4.65 million** without it, then used the split to
audit the marketing: Christopher Monckton had given eternity2.net's Dave
Clark a best estimate of "roughly 5 million" solutions, so Owen concluded
Monckton's mathematicians had simply forgotten the hint constraint
([msg 987](https://groups.io/g/eternity2/message/987)). David Eddy added
the same day that
[parity corrections](/research/build/analysis/parity-arguments) (every
colour's edge count must be even) contribute a factor of roughly $2^{16}$
that brings the estimate
families into agreement ([msg 992](https://groups.io/g/eternity2/message/992)).

Convergence took years, and it was not monotone. The closed-form product
was posted in November 2007
([msg 3385](https://groups.io/g/eternity2/message/3385)); mjqxxxx had
already begun a fully rigorous counting framework that July
([msg 1221](https://groups.io/g/eternity2/message/1221)); kubzpa's December
2007 paper argued for ~**15 million** solutions with the starter. It got
genuine peer review on the list: the list's fixes produced a corrected
second version ([msg 3497](https://groups.io/g/eternity2/message/3497),
[3583](https://groups.io/g/eternity2/message/3583)), and mjqxxxx then caught
a Monte-Carlo inconsistency in v2
([msg 3589](https://groups.io/g/eternity2/message/3589)) that kubzpa traced
to a biased shuffle, revising his estimate again
([msg 3591](https://groups.io/g/eternity2/message/3591)). A "20,000 solutions"
figure circulated for years before being traced to the archived French
official site ([msg 8515](https://groups.io/g/eternity2/message/8515)). The
number that survived is complex theory's: jagbrain derived **14,702** from
an independent closed-form Markov model in 2008, matching Owen's iterative
method exactly ([msg 5758](https://groups.io/g/eternity2/message/5758));
McGavin published the same figure in 2011
([msg 8924](https://groups.io/g/eternity2/message/8924)) and restated it as
the canonical answer in 2024: 14,702 with the starter, "probably accurate
to within a factor of 2", and $4\times10^{-8}$ with all five clues, "very
strongly suggesting… a single, unique solution"
([msg 11193](https://groups.io/g/eternity2/message/11193)). The machinery
behind those numbers, the depth-by-depth expectation and what it predicts
about the search tree, lives on the
[complex theory page](/research/why/complex-theory); this page only needs
its output.

One caveat the community itself raised belongs here. E2 was *generated from
a solution*, and picking a puzzle by picking a solution oversamples
solution-rich piece sets, so the count conditioned on the generation
process should run *higher* than the plain expectation, by an amount the
thread tried and failed to pin down
([msgs 6892/6894](https://groups.io/g/eternity2/message/6892)). Expectation
formulas price a random puzzle with E2's statistics; E2 is not quite a
random such puzzle.

### Sampled and culled search: counting by surviving

Between exact and expected sits the third regime: run a real search, but
randomly cull it to a fixed budget, and scale the survivors back up. The
archive's methodological gem is Owen's September 2007 census of the border
ring. He ran **four separate searches** of the 60-piece frame, each
randomly culling the tree while keeping about 10 million active nodes per
depth. Each of the four independently estimated **≈ 4.05×10³⁷** border
solutions ([msg 2696](https://groups.io/g/eternity2/message/2696), first
posted at [msg 1225](https://groups.io/g/eternity2/message/1225)). The
figure disagreed with eternity2.net's published estimate by thirteen orders
of magnitude, and the replication is exactly why the community sided with
Owen. A single culled run can be silently biased by anything; four
independent runs agreeing is an error bar you can see. That habit
(replication as error control, not authority) is the same one that later
governed the corner censuses and the 9×9 recounts.

## Step by step: the expectation formula on a tiny board

The first-moment calculation is worth doing once by hand. Take a miniature
E2: a framed 3×3 board (4 corner pieces, 4 edge pieces, 1 interior piece)
with a single palette of $c$ colours on every internal edge.

**Step 1: count the arrangements.** Corners go to corner cells with their
orientation forced by the two grey sides; edge pieces likewise; the single
interior piece keeps its 4 rotations:

$$
A \;=\; 4! \times 4! \times 4 \;=\; 2304 .
$$

**Step 2: count the constraints.** The board has $2\cdot3\cdot2 = 12$
internal edges. If colours were independent uniform draws, each edge would
match with probability $1/c$.

**Step 3: multiply.**

$$
\mathbb{E}[S] \;=\; \frac{2304}{c^{12}}
\quad\Longrightarrow\quad
c=2:\ 0.56, \qquad c=3:\ 0.0043 .
$$

That is the entire designer's dilemma in one line: at two colours the tiny
puzzle expects about half a solution, at three it is almost surely
unsolvable. A fraction of a colour swings the expectation across the
"exactly one" line. Scale the same product up to the real board, with $4!$
corner arrangements, $56!$ edge arrangements, $195!\cdot4^{195}$ interior
arrangements (the starter is pinned), 60 frame-ring edges at $1/5$ and
$56+364=420$ remaining edges at $1/17$, and you get the form posted to the
list in November 2007 ([msg 3385](https://groups.io/g/eternity2/message/3385)):

$$
\mathbb{E}[S] \;=\; 4!\;56!\;195!\;4^{195}
\left(\tfrac{1}{5}\right)^{60}\left(\tfrac{1}{17}\right)^{420}
\;\approx\; 0.02 .
$$

There is no hiding the gap: **0.02 is not 14,702**. The naive product is
nearly six orders of magnitude too small, and the community knew why from
the start. Treating every edge as an independent $1/17$ coin flip prices
the last piece like the first, when in reality the real piece set pairs up
perfectly (every colour has an even count, with
[zero slack](/research/build/known-facts)), so the match probabilities in
the endgame are conditioned far above $1/17$ by everything already placed.
Those parity and depletion corrections were flagged in the very message
that posted the formula, and Eddy's $\sim 2^{16}$ even-count factor
([msg 992](https://groups.io/g/eternity2/message/992)) is the largest of
them. [Complex theory](/research/why/complex-theory) is precisely this
calculation done properly, tracking the surviving piece and colour counts
depth by depth instead of assuming a fixed probability, and *that* is the
version that lands at 14,702 and matches exhaustive counts on small boards
to within its claimed factor of two. The structure of the formula
(arrangements × match probability) is right; where you take the probability
from is the whole game.

## What it costs

The three regimes are really three points on a cost–knowledge curve.

**Exact enumeration costs the tree itself.** Counting is exhaustive search
that refuses to stop: each 9×9 census cost on the order of $10^{14}$ nodes
and three weeks of 2011 hardware, and the price grows
with the search space, not with the answer. For the full puzzle the tree is
$\sim 10^{45}$ wide at its plateau; the exact count of E2's solutions will
not be computed by anyone, ever, by this route.

**Expectation costs almost nothing, and that is its trap.** The
first-moment formula is $O(1)$ arithmetic (complex theory's refined version
is a few hundred multiplications, one per cell), which is why it existed
before the puzzle shipped. But a first moment carries no information about
*variance*: it tells you the average count over puzzles with E2's
statistics, not whether the mass sits in typical instances or in rare
solution-rich freaks, and not whether the partial boards being counted at
each depth are genuinely distinct. The
[entropy and area-law results](/research/why/entropy-area-law) show exactly
where that blindness bites: past ~80 cells, distinctness collapses in a way
no independence model can see. Use expectations to compare orders and set
expectations, never as bounds.

**Sampling buys error bars only through repetition.** A culled search costs
one node budget per run and gives an estimate of unknown bias; Owen paid
four budgets for the border ring and bought the only kind of confidence
this regime offers. The rule the community settled on is the one worth
carrying out of this page: a sampled count you have seen once is an
anecdote; the same count from independent runs, seeds, and authors is a
measurement.

## Related

- [Complex theory: counting the search before you run it](https://eternity2.dev/research/why/complex-theory) — Brendan Owen's complex theory estimates how wide the search tree is at every depth, and even how many solutions exist at all. Many in the community consider it the single most important thing to understand about Eternity II.
- [The community's benchmarks](https://eternity2.dev/research/build/benchmarks) — How a community forbidden from sharing the pieces built a shared test culture anyway: derived-count verification protocols, the Txibilis and beginner suites, node-count duels, full enumerations, and the one benchmark that is still standing open today.
- [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.