Analysis
These do not try to solve the puzzle: they measure it, which is how the community knows where the walls are. Parity arguments give one-pass impossibility proofs; solution counting pins down how many full solutions exist to within a factor of two, without anyone having seen one.
Count anything on an edge-matching board twice, once from each side, and the totals must agree, giving impossibility proofs for the price of one pass. The 479 story shows both the power and the trap: a clean parity claim, true for every interior move, defeated through the sixty border edges nobody scores.
Nobody has ever seen a full Eternity II solution, yet the community knows, to within a factor of two, how many exist. This page is the story and the craft of that number: exact censuses on small boards, the expectation formula and its twenty-year convergence on 14,702, and the culled-search estimates that were trusted only when four independent runs agreed.