I've run an analysis of all of the positions in the Triple Cross puzzle. I count the number of positions with the horizontal slider centered and the left slider in the down position. There are four sequences that change the tiles and return to this position: LUCD, ULDC, RUCD and URDC. All puzzle positions can be solved within 24 moves. There are only 7 positions (out of 2.9 trillion total) that take 24 moves to solve.
The program took 2 hours 44 minutes to run on a 2.2GHz Core 2 Duo processor (using one core) and uses about 1.4GB of RAM while running.
Here's a list of how many positions are at each number of moves away from a solution. I've included an example of one position at each level.
total positions = 2940537600
0 1 ..---..12.....34ba
1 4 .1.--..32b.-...4a.
2 12 13b.-...2a.-...4.-
3 36 3.ab.1..2..-...4--
4 104 3..b.2..4.a.1-..--
5 303 3.b..a4.....21-.--
6 884 3....b.2....a41---
7 2579 ....3a2....b.41---
8 7521 ...42a......b31---
9 21937 4..2...3.b..a.1---
10 63923 ..a.3b4......21---
11 186196 .....a43.b...21---
12 542124 ..4.b......3a21---
13 1577700 ......4.3a.b.21---
14 4585298 b.......4a..321---
15 13282991 .ab.......4.321---
16 38149858 ...b.a...4..321---
17 107133350 .....a....b4321---
18 283555419 ........a.b4321---
19 644626804 .......a..b4321---
20 1014637237 .........ab4321---
21 719688345 .........ba4321---
22 111515380 .a....b...4.321---
23 959587 ..4...ab..32..1---
24 7 ...b4.--.3..1a-.2.
Update:
Here are the seven positions that are 24 moves from solved:
-b2..4-...1...-a.3
-..42.-...1a..-b3.
-.2.....-3.ba..-14
b-2....-a.1..-..43
.-24.13.-....b-a..
..3..a---.4..b..21
...b4.--.3..1a-.2.
