First, what do we know about this puzzle?

 - There are 5*7*8*11 = 3080 possible positions.
 - Moves do not affect each other, so if you have a good set of moves, you can do them in any order.

Given this, it is a relatively simple algorithm to find the shortest sequence of moves.

What I did was to start with a list containing only (0,0,0,0), then for each item in that list, I did a +5,-5,+7,-7,+8,-8,+11,-11. For each item produced by this, I check if it is in the list and if not, add it to the end.

I had it stop when it got to (3,3,3,3).

(0,0,0,0) -> +5<br>
(1,2,2,2) -> +7<br>
(4,3,4,4) -> -8<br>
(1,5,3,2) -> -8<br>
(3,0,2,0) -> -8<br>
(0,2,1,9) -> -8<br>
(2,4,0,7) -> -8<br>
(4,6,7,5) -> -11<br>
(1,1,1,4) -> -11<br>
(3,3,3,3)

So the minimum number of moves is 9. One example is(just reversing my list):<br>
+11,+11,+8,+8,+8,+8,+8,-7,-5

I am going to analyze this a little more and add further findings.