Solution:

Here are three simple move sequences that, applied to a row of seven numbered tiles, can be used to swap around three of the tiles. The solution that follows will use these sequences.

Outer cycle:	Inner cycle:	Side cycle:
7 2 3 1 5 6 4 Swap two left blocks 1 5 6 7 2 3 4 Swap two right blocks 1 2 3 4 5 6 7	1 2 4 5 3 6 7 Swap outer blocks 3 6 7 5 1 2 4 Swap two left blocks 5 1 2 3 6 7 4 Swap two right blocks 5 6 7 4 1 2 3 Swap outer blocks 1 2 3 4 5 6 7	2 6 3 4 5 1 7 Swap two left blocks 4 5 1 2 6 3 7 Swap outer blocks 6 3 7 2 4 5 1 Swap two left blocks 2 4 5 6 3 7 1 Swap two right blocks 2 3 7 1 4 5 6 Swap outer blocks 4 5 6 1 2 3 7 Swap two left blocks 1 2 3 4 5 6 7

Note that Inner Cycle is an Outer Cycle sandwiched by swaps of outer blocks, and that a Side Cycle is an Inner Cycle sandwiched by swaps of the left blocks. These move sequences use only 7 of the 9 tiles, so you can apply them to the left, right or middle seven tiles.

Phase 1: Solve tile 9

1. Make sure all the numbered pieces are visible, i.e. the rectangle tile is not in the row.
2. If tile 9 is in location 1, 2, 7, or 8 then a single swap can bring it closer to the centre, to one of the locations 3,4,5, or 6.
3. If tile 9 is in location 3, 4, 5, or 6 then a single swap puts it into position at location 9.

Phase 2: Solve tile 8

1. If tile 8 is in location 1, 2, 6, or 7 then a single swap can bring it closer to the centre to one of the locations 3,4,or 5.
2. If tile 8 is in location 3, 4, or 5 then a single swap puts it into position at location 8.

Phase 3: Solve tile 1

1. Place the rectangle piece at location 5,6,7.
2. By alternately pushing the plunger at locations 1,2,3 and then at 2,3,4, you can cycle around the remaining 7 tiles until tile 1 is in location 1.
3. Take out the rectangle piece again, by pushing the plunger at location 5,6,7.

Phase 4: Solve parity

1. If the plunger of the puzzle is in the up position, then swap the tiles at locations 2,3,4 with those at 5,6,7.

A note about parity: From now on we use the 3-cycle move sequences to solve the puzzle. These are even permutations, so they can only solve the puzzle if it is in a position that is an even permutation. On the physical puzzle there is an easy way to determine this: If the plunger is down it is an even permutation, and if it is up it is an odd permutation. Phase 4 above fixes the parity by making a swap. This assumes however that the plunger was down when it was originally in the solved position. While this is true for most physical versions of the puzzle, the online version on the manufacturer's home page does not model this properly. In that case you cannot determine from the plunger position what the parity of the permutation is. This may mean that after following the rest of the solution you end up with a position in which exactly two tiles need to be switched. In that case just apply the swap mentioned in phase 4, regardless of the plunger position, and try again to solve it.

Phase 5: Solve tile 2

1. If tile 2 is at location 5,6 then apply the inner cycle once or twice to locations 5,6,7 to bring it to location 7.
2. If tile 2 is at location 4 then apply the inner cycle once or twice to locations 3,4,5 to bring it to location 3.
3. If tile 2 is at location 3,7 then apply the side cycle once or twice to locations 2,3,7 to bring it to location 2.

Phase 6: Solve tile 3

1. If tile 3 is at location 7 then apply the inner cycle to locations 5,6,7 to bring it to location 5.
2. If tile 3 is at location 6 then apply the inner cycle to locations 4,5,6 to bring it to location 4.
3. If tile 3 is at location 4,5 then apply the inner cycle once or twice to locations 3,4,5 to bring it to location 3.

Phase 7: Solve tile 4

1. If tile 4 is at location 7 then apply the inner cycle to locations 5,6,7 to bring it to location 5.
2. If tile 4 is at location 5,6 then apply the inner cycle once or twice to locations 4,5,6 to bring it to location 4.

Phase 8: Solve tiles 5, 6, 7

1. If tile 5 is at location 6,7 then apply the inner cycle once or twice to locations 5,6,7 to bring it to location 5.
2. Tiles 6,7 should now be correct. If they are not, then plunger is in the wrong state. To fix that, swap 2,3,4 with 5,7,6 and go back to phase 5.