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