Hey Bob, welcome to the den. It's been a while since I've checked this site.
This puzzle, at first glance, seemed similar to "Groundhog in a Hole." -- http://brainden.com/forum/topic/11943--/
I have posted a couple variations, too. -- http://brainden.com/forum/topic/12010--/ and http://brainden.com/forum/topic/18524-groundhog-in-a-hole-yet-again/
Anyway, I thought I'd ask for some clarification. If the rat is in hole 10 on day 1, does that mean on day 2 it has an equal chance (20% each) of having gone to holes 8, 9, 10 (stayed there), 11, or 12? If the rat is in hole 2 on day 1, on day 2 is there an equal chance (25% each) of being in holes 1, 2, 3, or 4? And a third chance each for holes 1, 2, and 3 if starting right on the edge in hole 1? And does the rat have an equal chance of being in any given hole on day 1 (1% each)? Just thought I'd nail down the way the probabilities move.
Here's a quick guess before checking... anything really...