The picture to the left shows the puzzle in its initial state. If you click on it, the puzzle itself will appear, in a new browser tab.
Remove the 1-tile [by clicking on it]. Then arrange the others [you move a tile to the adjacent space, if there is one, by clicking on it] all in the right order ( 2 3 4 5 6 7 8 9 ) and all the right way up.
To the right is a picture of the puzzle in its solved state.
The puzzle statement is correct, and, I hope, easy to understand correctly. The trick is that in the puzzle's initial state, what looks like a 6 is an upside-down 9, and vice versa. No-one told you that the tile to the right of the 4 was a 6; and it isn't.
You can't swap the 4 with the 5 while leaving the other tiles in their original positions. But you can swap the 4 with the 5 and the 6 with the 9 while leaving the others in their original positions.
Arrange to send the 6-tile and the 9-tile each twice around the top of the puzzle without sending any other tiles around. Then you have a simple and easily solved sliding-block puzzle.
I built this puzzle. It's not the problem statement that you need to reconsider, it's your assumptions about the starting position. The misdirection is deliberate.
Source: Nick Wedd. For more SVG sliding block puzzles, see http://www.olfrec.com/sbp/
This is one of several pages on puzzles and metapuzzles .