# Learned move-ordering

> When several pieces fit, which do you place next? Instead of one rule of thumb, carry several signals learned from strong boards and let them vote. Voting keeps the search from over-trusting any one hunch and marching into the same dead end.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/build/learning/learned-value-ordering/
- Updated: 2026-07-16
- Topics: construction, learning
- Source: Beam search (this project's concept page): the construction whose value-ordering is learned — https://github.com/raphael-anjou/eternity2/blob/main/web/content/research/build/construct/beam-search.mdx

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Building a board one piece at a time, the hard decision is which piece to place
next when several would fit equally well. A single rule of thumb, cheapest cost
first, most-constrained cell first, tends to march the search into the same dead
end every run. A [corpus prior](/research/build/learning/corpus-priors) improves
on that with one learned signal. This technique improves on it again by carrying
*several* learned signals at once and letting them vote, so no single hunch owns
the decision.

## Several signals, one vote

From the corpus of strong boards you can learn more than one thing at a time, and
the signals are cheap counts over the same board set:

- **Piece-in-position.** How often a piece appears in a given cell across good
  boards, the positional prior of the previous page.
- **Piece-pair adjacency.** How often two particular pieces end up touching. A
  neighbourliness the positional count does not see.
- **The 2×2 patch prior.** For every little four-cell patch, a Laplace-smoothed
  log-odds ratio of how often it appears in high-scoring boards versus low-scoring
  ones. A patch common in strong boards and rare in weak ones scores positive; a
  consensus-trap patch scores negative. This is the sharpest of the three, because
  it is the only one that contrasts good against bad rather than merely counting
  good.

At each step the beam scores every candidate placement by its matched-edge gain
plus a weighted sum of the three signals, and keeps the top few. A little injected
randomness stops the parallel beams collapsing onto one path.

> **[Figure]** Interactive: the three-signal placement vote — interactive: KeyringDiagram. Rendered on the canonical page (link above); not shown in this markdown export.

This is the [KEYRING](/research/lab/experiments/raphael-anjou/learning/keyring)
experiment, a keyring of hunches rather than one. It reached 460 in a corner
family no earlier search had cracked, which is the real payoff: because the strong
boards are known to sit in isolated basins, a signal diverse enough to reach a
*new* family is worth more than one that re-derives a known board.

## Why voting beats one signal

A single learned signal has the same failure mode as a single hand-written rule:
wherever it is wrong, it is wrong every time, and it drags every beam the same way.
Three signals that were learned from different views of the corpus, position,
adjacency, and patch quality, disagree in different places, and their disagreement
is exactly the randomness that keeps the beams exploring different regions. The
patch prior carries the most weight because it alone encodes *bad*: it can push the
search away from a placement, not just toward one, which is the seed of the idea
that grows into [anti-pattern mining](/research/build/learning/anti-pattern-mining).

The same caveat as every learned signal applies. The three priors are mined from a
corpus that is itself sub-perfect, so they encode the community's ceiling as much
as its wisdom, and the vote reaches the top of the search's range reliably without
lifting that range. Where that boundary sits is
[its own page](/research/build/learning/when-learning-collapses).

## Related

- [Corpus priors](https://eternity2.dev/research/build/learning/corpus-priors) — The simplest way to learn from strong boards: count where each piece tends to sit, or how often it serves a scarce demand, and use that count as a gentle tiebreak in construction. It has to stay a tiebreak; the moment it becomes part of the objective it collapses the search.
- [Anti-pattern mining](https://eternity2.dev/research/build/learning/anti-pattern-mining) — The subtle idea in learning from strong boards: not every agreement between them is good. Some shared placements are real structure; some are a shared trap that caps every search just short of the top. Separate the two, and attack the trap.
- [KEYRING](https://eternity2.dev/research/lab/experiments/raphael-anjou/learning/keyring) — Build a board from scratch, ranking each next piece by three signals learned from past strong boards. Reached 460 in a board family no earlier search had cracked.
- [Beam search](https://eternity2.dev/research/build/construct/beam-search) — Keep the K most promising partial boards alive at once and grow them cell by cell. Beam search is the workhorse behind this project's from-scratch builders, and a clean illustration of why breadth alone stalls in the deep interior.
- [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.