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Message: <blockquote data-quote="The Sigil" data-source="post: 1207477" data-attributes="member: 2013">The original question seems to be, "at a given time T, what are the odds of the dragon being in room X in map matrix M, after starting from point Y, with a restlessness period of Z?"Where T is very small (a few multiples of Z), it is quite important to analyze the structure of matrix M, especially as it refers to Y and X.  However, this analysis is relatively easy and straightforward and does not need a general solution.As T gets very large (many multiples of Z), the starting point Y becomes less important except that the number of steps between Y and X is important (though not Y and X themselves; merely their separation) to determine whether this is a bistatial problem.  Once we've determined that, the solution is only trivially different than the (possibly bistatial) M/N solution I posited earlier.  You are welcome to run the numbers yourself for *any* matrix M that you like and see that it is indeed the correct solution.Another way to think about the importance of the starting point is to handle the problem "in reverse" - if the dragon is at X now what are the odds he was at Y a few Z's ago?  Obviously, the fewer Z's, the fewer possible solutions.  As the number of "steps backwards" gets larger, the possible number of permutations gets larger and larger until you get (quickly, I might add) to a point where you have no real certainty of where the starting point was.  Thus, for a time elapsed that is large since the start, the starting point is of little import.Simply put, this is a problem that DOES have a simple general case.  The *only* thing that matters in terms of the structure of M is whether that structure demands a bistatial solution.--The Sigil</blockquote>

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