Some weeks ago I asked a question about a problem I found in the book "Thinking Mathematically" by Mason, Burton and Stacey. It's called "Coin Slide" and can be found on page 149. The problem deals with $n$ large ("B")and $n$ small coins ("S") ordered as a $BSBSBS$ block (here for $n=3$) and the reader's task is to convert this block to the shape $BBBSSS$ under manipulations by certain allowed moves. The allowed (legal) moves are explaned precisely in the linked question.
Firstly, let me make some important remarks:
in order to keep the text shorter (it is long enough) I decided to outsource a couple of important informations which are relevant for this problem (like rules for allowed moves, my efforts on the orginal problem, etc) which can be found agian in original problem, since the rules are the same und the differences are explaned explicitely below
at the question's end (the grey text) I summarized findings and important results gained on dealiing with original problem. I also contains the essential motivation behind this question.
That is I think we can start how to tackle the actual Problem I would like to discuss in this question. We are going to simplify the original problem from linked question in natural way in order to try to avoid the weird story from linked problem with gaps of different lenghts.
Assume we don't have two kinds of coins with different size, but two coins of same(!) size, but different colors, say black (B) and white (W). The rest of the problem stays the same as in original "Coin Slide" problem.
This problem looks simpler since we have now more options and flexibility when we dealing with gaps: a gap of the size of $B$ has the same size as a gab of the size of $W$. Let denote a gab of the size of a coin (now no matter black or white) by "$-$".
So for example we can now move into the gap $W--W$ every pair $WW, BB, WB$, and indeed recall, in the original exercise this was a big problem.
Question: If we now consider this simplified problem as I have explaned, does there exist a solving strategy to some arbitrary $n-n$ block $BWBWBWB...WBW$ in a similar manner like in the Leapfrog exercise?
That is a strategy which works as: assume I have already performed a couple of legal moves and before next move, I can "look up in my strategy instruction" which tells me something like "move A is compatible with winning strategy, ie brings me closer to solution, because ..." and "move B is not efficient because ..." (see the elaboration of the solution in Leapfrogs exercise in the 4. image of the linked question & the 'AHA' part; essentially this part unfolds exactly a clear strategy, how to do right moves: the colors have to alternate).
Of course it cannot be expected for the actual problem that there exist also a such simple strategy like in Leapfrogs based on an elementary observation. But I'm would like to know if there exist a strategy with 'same spirit', a kind of valuative character which allows me to valuate which next move is better and which not; that's exactly what happens in 'Leapfrogs' from abstract viewpoint.
Here are some additional informations & summary about results and obstacles of the original question:
In the linked question I suggested an accessable formalism which uses the notions $B$ for big (=lange) coin, $S$ for small coin, "$-$" for gap with lenght of diameter of big coin, and "$.$" gap with lenght of diameter of small coin. For example a legal move is to take the pair $SB$ (positions 2 and 3) for $BSBSBS$ to $B.-SBS--SB$.
Now a closer look into discussions on the linked question provedes general solutions for $n=3,4,5$ and $n=4k+i$ with $i=0,1,3$ but a couple of questions stayed unsolved, especially it's relation to another exercise, "leapfrogs", as remarked in a hint. Drawing parallels to the "leapfrogs" exercise (see hint) I hoped to find a general strategy in the sense that has I similar spirit as the strategy for "leapfrogs" (see again in the linked question what I mean precisely by this strategy).
Shortly I'm looking for a strategy which literally works like that for every actual position of coins the strategy contains criteria which allowed me to decide before every next move which move I have to do next, and which not.
(recall: in leapfrogs it was literally: move in that way, that after the move the colors of pegs alternate). And hoped to find something similar for this problem. But I think that such strategies may fail here since the problem is too complicated to be attacked in that way and indeed antkam's approach via recursion seems to be most general.
As one of the weirdest obstacles in this problem pops up the fact that we have to handle gaps of different lenghts, ie "$..$" or "$--$" or "$-.$" were all considered as different, for example in the first one I can move the pair $SS$, but not the pair $BB$ or $BS$ since the gap "$..$" is to small for the last two.