The Catalogue of Authentic Positions
Every catalogue raisonné exists to answer one question, and the question is never whether a thing is beautiful. It is whether a thing is real. A canvas arrives at the desk. The pigment is right, the ground is right, the hand looks right, and none of that settles anything. What settles it is provenance: an unbroken account of where the object has been, back to the moment it left the maker’s hands. A work with no such account is not a lesser work. It is not a work at all.
The fifteen-puzzle has a maker, and the maker is the solved board. Every legitimate position descends from it by a chain of single slides, and that chain is the position’s provenance. So the catalogue that follows is not a list of arrangements. It is a list of descendants — the positions that can trace themselves home.
The surprise, the whole reason this foreword is longer than a page, is how many arrangements cannot.
Lay out the tiles of the smaller eight-puzzle every way they will go, and there are three hundred sixty-two thousand eight hundred eighty arrangements. Exactly half of them descend from the solved board. The other half, one hundred eighty-one thousand four hundred forty of them, are forgeries. Not clumsy forgeries. Perfect ones. Set a genuine position beside a forged one and there is nothing to see: the same tiles, the same tidy grid, a configuration a child could reach for and a scholar could not fault by eye. Connoisseurship is helpless here. You may study a forged position for the rest of your life and never find the tell, because the tell is not in the position. It is in the history the position does not have.
What authenticates a board, then, if not the look of it? A single bit. Number the ways the tiles sit out of order, counting each pair that stands in the wrong sequence, and ask only whether that count comes out even or odd. Pair it with one more fact, the parity of how far the blank square has wandered from its home corner. Add the two. The sum is the seal. Every legal slide breaks the tile order by exactly one pair and moves the blank exactly one step, so every slide flips both halves of the sum at once, and the sum comes through unchanged. It cannot be otherwise. A quantity that survives every move is a quantity no sequence of moves can forge.
This is the mechanism the eye keeps missing. The forger’s difficulty is not that the target is hard to reach. It is that the target sits in the wrong world, and the two worlds do not touch. The authentic positions are sealed together, closed among themselves; the forgeries are sealed off in a coset of their own, and the wall between is one parity bit thick and infinitely tall.
I have verified the seal the only honest way, which is exhaustively. For the eight-puzzle every one of the 362,880 arrangements was checked twice: once by walking outward from the solved board along legal slides until no new board remained, and once by reading the parity bit off the arrangement directly. The two accounts agree on every board without exception: reachable exactly when authentic, and the wall never once crossed. The genuine corpus numbers 181,440, half of every board there is. Restrict to the positions that also bring the blank home, and 20,160 remain, which is the alternating group in another costume, the even permutations of eight tiles and nothing else. For the full fifteen-puzzle the authentic corpus is larger than any catalogue could print: six hundred fifty-three billion, eight hundred thirty-seven million, one hundred eighty-four thousand positions, and I have not walked all of them. I do not need to. The seal is proven to survive every move; that half the positions carry the maker’s bit and half do not is fixed the moment the tiles are laid, before a single slide is made.
Which brings us to the catalogue’s most requested entry, and its rejection.
No. 14–15. The solved board with the final two tiles exchanged. Rejected. This is the position for which Sam Loyd offered a thousand dollars to anyone who could supply its provenance, and the offer was safe, because the provenance does not exist and no amount of sliding will manufacture one. The exchange of two tiles breaks exactly one pair. That is a single transposition, odd where the maker is even, and the seal reads accordingly. The prize could never be paid. It had already been shown unpayable a decade before Loyd printed it, by two men named Johnson and Story, who counted the parities and closed the door.
There is a second forgery in the record, and it is Loyd himself. He claimed, late and loudly, to have invented the puzzle. He did not; a postmaster named Noyes Chapman set the tiles moving years before Loyd took an interest. So the same man offered a board with no provenance and a story with no provenance, and both fail the same test for the same reason. Authorship, like a position, is a claim about descent, and a claim about descent is either true or it is a forgery wearing the face of the thing it isn’t. The catalogue does not care how convincing the face is. It asks where the work has been, and it counts.
