Eternity II has two rules: edges must match, and each piece is used once. The first is generous. All the hardness lives in the second.
deterministic — reproduces byte-for-byte
just research-entropy-area-lawForget the use-each-piece-once rule for a moment and treat the 196 interior pieces as reusable tiles. Count the valid all-matched patches and they grow exponentially with size: there's no shortage of locally-valid ways to tile. The matching grammar is rich, not restrictive.
We can measure that richness exactly. For a strip of width , the growth rate per cell is an entropy density , and the sequence decreases toward the true two-dimensional value as the strip widens.
Now put the use-each-piece-once rule back, and ask how often a random color-valid patch actually uses distinct pieces. Call it . It collapses, and it collapses in the area, not the perimeter:
An area-law decay is brutal because area grows quadratically. The fraction of realizable patches drops below one in a thousand at around eighty cells.
The same idea on the real pieces, exactly counted. Step the block size and watch how many colour-valid blocks survive the use-each-piece-once rule.
Two exact counts on the real Eternity II pieces. Reusable: how many colour-valid n×n blocks exist if pieces may repeat — the richness of the matching rules. Distinct: how many survive the use-each-piece-once rule. Step n up and watch the gap open.
Even with reusable pieces, each added cell adds less freedom than the last — the grammar alone tightens as blocks grow.
The distinct-realizable fraction collapses with the AREA of the patch, not its perimeter — and area grows as n². By around eighty cells it has dropped below one in a thousand: the size of the smallest moves between the best known boards. The hardness isn't in matching colours (that grammar stays rich); it's in the quiet once-each rule, whose cost compounds over area.
Eighty cells is not arbitrary. It's the size of the smallest moves that separate the best known boards. The matching grammar stays rich up to about that scale, then the distinctness rule collapses it. So the wall isn't in the part that looks hard, matching colors; it's in the quiet rule that each piece is used once, whose cost grows with area, on a board just large enough for it to bite.
See why basin-hopping is impossible.
The per-width entropy has a well-defined limit. Joining an -wide and an -wide strip side by side only adds a seam constraint, so the eigenvalues satisfy
Taking logs makes subadditive, and Fekete's lemma gives the limit as an infimum, which is exactly why the curve above decreases:
The upper bound is the purely horizontal rate: ignoring vertical constraints only adds patches, so
with the spectral radius of the horizontal color-compatibility matrix. Positivity holds because the grammar supports exponentially many chains, so the density is strictly between zero and 1.6645, measured near 0.67.