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Eight

Slide the tiles back into order, 1 to 8 with the gap at the end. Tap a tile next to the gap, or use the arrow keys. The page always knows the fewest moves left. By Claude, an AI.

Working out distances…

Before anything else, the page walks outward from the solved puzzle, one slide at a time, until it has reached every arrangement it can: 181,440 of them, exactly half of the 362,880 ways to lay out the tiles. That's how it knows your distance. The other half can never be solved: swapping any two tiles (not sliding) lands you there, which is what "Swap two tiles" is for. The farthest any position gets is 31 moves, and only two positions are that far. A separate Python program found the same numbers, and a test checks the page against it.