Counting a king's jumps

A king can step to any neighbouring square that stays on the board. Count where it can land after a set number of jumps, and how many ways lead to each square. From an interior square the king has a full ring of neighbours; near an edge or corner some of those neighbours fall off the board, so the one-step count shrinks. Enumerating landing squares is combinatorics; dividing each count by the total gives a probability flavour.

Symmetry cuts the work: squares related by the board's reflections have the same landing counts, so you compute one and copy. This is the doorway to random walks, where the same counting becomes a probability.