# 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. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/build/learning/corpus-priors/ - Updated: 2026-07-16 - Topics: construction, learning - Source: Beam search (this project's concept page): the bounded-width construction a prior biases — https://github.com/raphael-anjou/eternity2/blob/main/web/content/research/build/construct/beam-search.mdx --- The simplest thing you can learn from a pile of strong boards is a count. For each cell of the board, how often does each piece sit there across the good boards? That single tally is a *prior*: a gentle statistical preference for what tends to belong where. It copies no board in particular; it reads the whole crowd of them at once and distils a bias. ## The count matrix Take every board in the corpus that scores above a threshold, and tally a matrix $M[c][p]$: how many strong boards place piece $p$ in cell $c$. Normalised per cell, $M[c][\cdot]$ is a distribution over pieces for that position, the learned preference. It costs one linear pass over the corpus to build, and nothing at search time beyond a lookup. > **[Figure]** Interactive: the position-prior heatmap — interactive: PriorDiagram. Rendered on the canonical page (link above); not shown in this markdown export. The prior is only ever used to break ties. A [beam search](/research/build/construct/beam-search) extends its partial boards cell by cell; when two candidate placements match the same number of edges, the prior tips the choice toward the piece that is more typical of strong boards in that spot. The matched-edge count still decides; the prior only speaks when the count is silent. This is the [PRIOR](/research/lab/experiments/raphael-anjou/learning/prior) experiment, which builds a competitive board from an empty grid with no board to anchor to, and reaches 460 that way. Three sharpenings of the same count are worth naming, because each trades signal strength against noise: - **Rotation-aware.** Split each piece by its four rotations, so the prior prefers not just the right piece but the right orientation. The matrix grows from $256 \times 256$ to $1024 \times 256$; the signal is finer but thinner per cell. - **Per corner-family.** Strong boards fall into families by their four corner pieces. A prior built from a single family's boards captures structure the pooled matrix averages away, which is what let PRIOR reach a fresh basin rather than the crowded common one. - **Sharper threshold.** Rebuild the prior from only the very best boards. The ceiling of the construction rises, at the cost of a thinner, noisier signal learned from fewer examples. ## A prior on demand, not position The count does not have to be over positions. A subtler prior asks, for each *piece*, how often it ends up serving a **scarce demand**: a cell whose north and west colours can be served by only one or two pieces in the whole set. The strong boards increasingly satisfy the same scarce demands as they climb, so a piece that is often the one spending itself on a rare colour earns a high weight. Ranking candidates by matched edges plus a small multiple of that weight nudges the search to commit rare pieces early, before they are stolen. This is the [LODESTONE](/research/lab/experiments/raphael-anjou/learning/lodestone) experiment, and it connects to the puzzle's [rare-colour geography](/research/why/rare-color-geography). ## Why it stays a tiebreak The load-bearing detail, for both the positional and the scarce-demand prior, is that the multiple on the prior must be *tiny*. A prior is a pure tiebreaker: it decides between equally-matching moves and nothing more. Turn the weight up even slightly and the search starts trading a real matched edge against a statistical hunch, and the score collapses, LODESTONE falls from 451 to 422 to 380 as the weight climbs. That collapse is not an implementation bug; it is the shape of the whole family, and it has its own page: [when learning collapses](/research/build/learning/when-learning-collapses). ## Related - [Learned move-ordering](https://eternity2.dev/research/build/learning/learned-value-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. - [When learning collapses](https://eternity2.dev/research/build/learning/when-learning-collapses) — The sobering half of learning from strong boards. A learned signal reaches the top of a search's own range reliably, and then stops. Over-trust it and the search collapses; even used perfectly it does not raise the ceiling, because the ceiling is not a thing the corpus knows. - [PRIOR](https://eternity2.dev/research/lab/experiments/raphael-anjou/learning/prior) — Build a board from nothing, breaking ties by where pieces tend to sit in the strong boards we already have. It reaches a high score with no starting board to copy. - [LODESTONE](https://eternity2.dev/research/lab/experiments/raphael-anjou/learning/lodestone) — A faint compass for a from-scratch search: nudge it to commit the rare pieces early, where they're needed. It doesn't raise the ceiling; it makes the search reliably reach the top of its own range. - [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. - [The rare colors live on the frame](https://eternity2.dev/research/why/rare-color-geography) — 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.