Why a 3×3 puzzle has only half its apparent arrangements
Nine spaces can be filled by the eight numbered tiles and the blank in 9!, or 362,880, different ways. But starting from a particular arrangement, legal slides can reach only 181,440 of them. The other half are not especially difficult configurations—they are impossible to reach without lifting tiles or taking the puzzle apart.
The hidden rule: inversion parity
Write the numbered tiles in row-by-row order, leaving out the blank. An inversion is a pair of tiles that appears in the opposite order from its order in the solved layout. For example, if a larger-numbered tile comes before a smaller one, that pair counts as an inversion.
On a 3×3 board, every legal slide preserves whether the inversion count is even or odd. A horizontal slide does not change the numbered tiles’ order relative to one another. A vertical slide moves a tile past two numbered positions in the row-by-row sequence, changing the inversion count by an even amount. So a puzzle that starts with an even count can never reach an arrangement with an odd count, or vice versa.
Half the permutations are ruled out
Across all 362,880 possible arrangements, exactly half have each inversion parity. Since the solved layout has one particular parity, only the arrangements with that same parity are reachable from it: 362,880 divided by two, or 181,440.
That makes the count more than a curiosity. If a shuffled puzzle lands in the wrong parity, no sequence of clever moves will solve it; the setup itself must be changed. The tiles may look nearly right, but one hidden ordering rule decides whether the solution exists at all.
