# Tri-Trick

Tri-Trick is a puzzle made by Hidden One Design. It contains nine square tiles numbered 1 to 9, and one blank 1 by 3 rectangular tile. The puzzle has a horizontal slider that is 1 by 9, and a vertical plunger that is 2 by 3. They have a 1 by 3 area in common. Initially the horizontal slider has the tiles 1 to 9 in order. By moving the slider any three adjacent tiles can be brought into the plunger. The plunger is initially down, and contains the 1 by 3 rectangular tile. By moving the plunger up, any three adjacent number tiles can be swapped out for the rectangular piece.

If your browser supports it, you can click on the link below to play with a Javascript version of Tri-Trick.

## The number of positions:

There are 8 places where the rectangular piece can be: hidden away on the plunger, or in one of 7 locations within the 1 by 9 row. The 9 numbered tiles can then be in any order. This gives 8·9! = 2,903,040 positions.

I ran a computer search to find the number of positions at each distance from the solved position. Only moves of the plunger are counted as moves. The results are shown in the tables below. It shows that it can always be solved in at most 17 moves (12.428 on average). The second table shows the results if tile 9 is never moved, i.e. a shorter version of the puzzle is played. Similarly the last table is for when only 1-7 are moved and 8 and 9 are kept fixed.
1-9 1-8 1-7
Moves   # Positions
01
17
220
352
4184
5638
61,983
75,991
818,292
955,011
10155,996
11393,194
12761,970
13893,572
14510,122
15103,051
162,952
174
Total2,903,040
Avg Depth12.428
Moves   # Positions
01
16
212
322
468
5184
6423
7970
82,330
95,533
1012,329
1125,843
1248,243
1370,155
1468,227
1537,674
169,468
17733
1819
Total282,240
Avg Depth12.975
Moves   # Positions
01
15
26
37
417
528
636
750
874
9105
10139
11193
12283
13390
14518
15704
16964
Moves   # Positions
171,286
181,710
192,226
202,788
213,306
223,620
233,605
243,190
252,327
261,383
27742
28339
29132
3047
3112
325
332
Total30,240
Avg Depth21.042

The four antipodes for the standard puzzle are:

The two antipodes for the 1-7 puzzle are:

## 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.