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.
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.
Take every board in the corpus that scores above a threshold, and tally a matrix : how many strong boards place piece in cell . Normalised per cell, 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.
The prior is only ever used to break ties. A 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 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:
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 experiment, and it connects to the puzzle's rare-colour geography.
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.