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.
Reproducestochastic — won't reproduce exactly; the board is verifiable·re-verifies a stored board·Budget: not logged (exploratory run, not the standardized single-core bench)
Pipeline
1
beam producernot yet written up
Build from empty; a scarce-demand weight breaks ties as a pure tiebreak, never the objective
carries: Rare-colour prior: a per-piece weight for how often a piece serves a scarce north-west demand
Complexity
Time
prior build O(B) over the corpus; beam construction as PRIOR/KEYRING, plus a tiebreak lookup
Space
one scarce-demand weight per piece (a 256-vector)
The prior is a per-piece scalar; it costs nothing at search time beyond a tiebreak, which is exactly why over-weighting it (making it part of the objective) is what breaks it.
Hardware & run
Native runCPU only
Cores
8
RAM
16 GiB
GPUs
0
CPU
Apple M1
Machine
MacBook (Apple M1, 8 cores)
Budget
not logged (exploratory run, not the standardized single-core bench)
KEYRING added signals;
LODESTONE, the third experiment in the
study, pares back to a single
deliberately faint one, and in doing so exposes the knife-edge every learned
signal here balances on. The strong boards quietly agree on something: as the
score rises, they increasingly satisfy a particular set of scarce demands, places
where only one or two pieces in the whole set can serve a cell's north-and-west
colours. LODESTONE asks whether telling a from-scratch search about those demands
helps it commit the right rare pieces before they're stolen.
From the corpus of strong boards, LODESTONE builds a prior: for each piece,
how often it ends up serving one of those scarce north-west demands in a good
board, weighted up sharply for the rarest, so a piece that is the only possible
server gets the biggest boost. The beam search then ranks its candidate
placements by the usual matched-edge score plus a small multiple of this
prior, so among otherwise-equal moves it prefers to place the pieces the good
boards learned to spend early.
The crucial detail is the size of that multiple. The prior has to be a pure
tiebreaker, not part of the objective: it breaks ties between equally-matching
moves and nothing more.
▶Interactive: rare-colour attraction mapExplore →
Watch the rare pieces drain away
A real solver fills this 6×6 board. After each piece, we count the still-empty cells whose colours can be met by only one or two of the pieces left in the box — the scarce demands. Watch them pile up as the board fills: the rare pieces a cell needs get spent elsewhere, and by the endgame almost every open cell is down to its last server or two. That scarcity, learned from strong boards, is the signal LODESTONE feeds the search.
At a tiny weight the prior gives a small, consistent gain: across five seeds
the median from-scratch score rises from 449 to 451, and, more usefully, the
spread tightens, from a 446–451 scatter to a reliable 450–451. It makes the
constructor land at the top of its range instead of sometimes stumbling.
Turn the weight up even slightly and it collapses (422, then 380) because
chasing the corpus's scarce demands then trades directly against matching the
edge in front of you. That failure is itself the finding: scarcity is a real
signal for which piece to prefer, but a weak one, and only safe as a
tiebreaker. LODESTONE is modest about its size: it improves construction
quality and consistency by a couple of edges, and does not touch the basin
ceiling that stops every method near the top.
The signal comes first, then the deliberately-tiny way it is used.
The measurement. A scarce demand is a cell whose north-and-west colours
can be served by only one or two pieces in the whole set. Over the corpus, the
count of scarce demands satisfied in at least half the boards rises
monotonically with score, roughly 1 → 2 → 3 → 10 as boards climb toward
458+. Strong boards don't just happen to place rare pieces well; they
increasingly satisfy the same scarce demands. That is a real, score-correlated
structure that no earlier constructor had wired in.
The prior. For each piece, weight how often it serves one of those scarce
north-west demands in a good board, boosted sharply for the rarest (a piece
that is the only possible server gets the biggest weight). The beam ranks
candidates by the usual matched-edge gain plus a small multiple of this weight.
The knob is the whole story. The multiple must be a pure tiebreaker:
it decides between equally-matching moves and nothing else. At a tiny weight:
median from-scratch score 449 → 451 across five seeds, and the spread tightens
from 446–451 to a reliable 450–451. Nudge the weight up and it collapses (422,
then 380), because chasing scarce demands then trades directly against matching
the edge in front of you. The collapse is the result: scarcity says which
piece to prefer, but weakly, safe only as a tiebreak.
Seeded; the reproducible artifact here is the tiebreak effect (a tighter
spread and a +2 median across five seeds), not a single board. The engine is the
shared beam producer;
the change this page describes is a faint scarce-piece-early tiebreak. That
tiebreak weight is derived from the piece set and a corpus, so the run is not
reproduced from scratch here.
Could the prior be made position-aware without becoming part of the objective,
strong only in the regions where scarce demands actually concentrate? Does
combining it with KEYRING's three signals add a fourth useful vote, or just
more noise? And is the consistency gain worth more than the median gain, given
that a tighter search is easier to build a ladder on?