In the 12 May material you wrote:

> Here's a rolling slab maze I created for the US Puzzle
> Championship (but was too long).

It didn't seem too hard to me.  I was able to carry out a
breadth-first search on paper, and got the following 46-move
solution.  The branching was not very bad.

+++++++++   ++++4555+   +++++++++   +++66544+   ++++33444
++11000++   ++0+4555+   +++111+++   +++66544+   +++122+++
+@223++++   +@077666+   +@+111+++   +@+66544+   +@+122+++
++223+++@   ++988+++@   +++22233@   +++77+++@   888122++@
++223@+++   ++988@+++   +++++@+++   +++++@+++   9990+@+++
+++++++++   ++988++++   +++++++++   +++++++++   9990+++++

   0-3        4-10        11-13      14-17        18-24

+++++++++   ++88899++   +++766+++   ++4555+++   +++++++++
++7776555   ++++100++   +++766+++   ++4555+++   +++++++++
+@7776555   +@++100++   +@+766+++   +@322++++   +@+666+++
++++++++@   +444100+@   999855++@   ++322+++@   ++++++++@
+++++@+++   +3332@+++   9998+@+++   ++322@+++   +++++@+++
+++++++++   +3332++++   000++++++   +++11++++   +++++++++

 25-27        28-34      35-40        41-45        46

However, I don't know of any good heuristics that would make 
it more of a reasoning puzzle and less of a clerical exercise.  
I did notice, though, that the shapes tend to follow the cycle

    1x3 -> 1x2 -> 3x2 -> 3x1 -> 2x1 -> 2x3 -> 1x3

or its reverse, with only two U-turns (at moves 15 and 26).  
That is to say, those are the only times where the slab goes 
in the same direction for two successive rolls.

Dan Hoey