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Thanks for the new puzzle!
The example doesn't make complete sense, however. Are you sure A and B start where indicated:
Suppose, 1st player (Alice) starts at the second cell of the first (topmost) row. The 2nd (Bob) starts at the lower-right corner.
...as opposed to the other way around (Alice at bottom right, Bob in the top row's island)?
Vladimir, thanks for speedy proof-reading! Verily so, it seems I have specific talent in messing up any "A" and "B", equals and not-equals etc. Fixed now :)
This was a fun one! I do recall playing similar games with friends as a child myself, often followed by arguments about whether a given 'maze' was actually solvable in the first place :)
Well, here I confess that the current version of maze generator does not care about possible existence of cells without way to enter or leave them... Congratulations and thanks for proof-solving to both Kevin and Vladimir!
BTW if you can recall in more details the games you mention you played in childhood, it would be interesting to compare! as you see the question of game origin is an open one, I still suspect it travelled somewhere from abroad.