The Puzzle With Six Worlds

Click any tile next to the hole. A legal shuffle is only a faster sequence of those same moves. Watch the current-world label before and after: no legal slide can change it. Deal another world may cross the boundary because it places a board directly. It is explicitly not a legal move.

The census is running.

The ordinary wall

On a bipartite graph, every edge goes from one vertex colour to the other. A slide does two things at once: it swaps the hole with one tile, flipping the sign of the full permutation, and it moves the hole to the other colour. Those two flips cancel inside one combined bit. Half the boards have one value of that bit, half have the other, and no legal move changes it.

The ordinary grid puzzle lives here. The familiar parity obstruction is not merely a trick for the numbered square box. It is the bipartite branch of a classification for sliding puzzles on graphs. Switch the instrument between Bipartite K2,3 and Odd cycle + chord: the same census sees two equal components in the first and one connected universe in the second.

The objection: parity already explains everything

It does not. Parity has only two values. It can split a universe in half, but it cannot produce six mutually unreachable classes. Wilson's exceptional graph does exactly that. It is a hexagon with a new middle vertex joining opposite corners, or equivalently two endpoints joined by three internally disjoint paths of lengths 3, 3, and 2.

There is a counting subtlety worth keeping, because dropping it makes a true result look false. The full state graph includes every possible position of the hole. Its connected components therefore contain more than 120 states. The classical puzzle group instead asks what permutations of the six tiles can be made while returning the hole to one chosen home vertex. The live atlas reports both counts. Each full world contains all seven possible hole positions above one 120-arrangement home slice.

Parity fails inside every exceptional world

Return attention to the small line at the bottom of each world card. The browser counts even and odd tile permutations separately, only among arrangements with the hole at home. In the bipartite example each world is pure: one contains one parity, the other contains the other. In theta_0, every world contains both parities in equal number. The exceptional obstruction is not a finer version of the ordinary even versus odd fence. It cuts across that fence.

The group hiding behind the hole

Move the hole around a closed route and return it home. What remains is a permutation of the six numbered tiles. The tile permutations induced by all such closed routes form the puzzle group. The exhaustive search measures its orbit from any home arrangement, so the order of that action is the computed size of one hole-at-home world. Its index is the number of such worlds.

Wilson identifies the exceptional puzzle group as a transitive subgroup of S6 isomorphic to S5. The exceptional six-point action is often written as PGL(2,5) acting on the six points of the projective line. That abstract identification is cited group theory. The browser establishes the safer finite claim directly: the orbit order and index shown in the atlas. It does not infer a group isomorphism from a matching order.

Where Wilson's theorem applies

The classification is powerful because its hypotheses are narrow and exact. The graph must be finite, simple, and 2-connected, also called nonseparable. Removing any one vertex must leave it connected. A single cycle is excluded. Subject to those conditions, Wilson's theorem gives the connected case, the bipartite two-world case, and the one exceptional six-world case.