# Reference numbers

> 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.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/reference/
- Updated: 2026-07-11
- Topics: backtracking
- Reproduce: `just research-subgrid`
- Source: sylvogel's published subgrid counts (groups.io message 11879) — https://groups.io/g/eternity2/message/11879
- Source: This project's reproducible Rust generator + committed results (GitHub) — https://github.com/raphael-anjou/eternity2/tree/main/research/topics/subgrid-placement-counts

---
How many distinct, edge-matching ways can you fill a small block of the
official board? These counts are the ground truth to test a solver against: if
your edge-matching and constraint code disagrees with the numbers below on a
2×2 corner, the bug is in your code, not the table.

The upright figures here are computed from the official piece set by this
project's reproducible Rust generator (linked below), so anyone can rerun them.
The handful of italic figures are counts too large to enumerate exactly in
seconds (tens of billions to tens of trillions of fillings); those are
**sylvogel's published values**, reproduced here for completeness and credited
in the sources. Everything else is recomputed here from scratch.
> **[Interactive: ReferenceTableView]** Rendered on the canonical page (link above); not shown in this markdown export.

## Related

- [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.
- [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.