Reachability and route counts

A specially moving pawn can reach only certain squares, each along a number of paths. Enumerate its reachable squares, and for each count the routes by the same additive rule: a square's path count is the sum of the counts of the squares that can step to it.

Predicting a count before computing sharpens intuition: bet on a square, then verify by adding its predecessors. Reachability and counting are separate questions. First decide which squares can be reached at all, then, among those, how many ways lead there. The additive (Pascal) structure returns whenever moves combine independently.