The most non-attacking knights

How many knights fit on the board so that none attacks another? Start from one fact: a knight always jumps from a light square to a dark square and back. Knights standing on squares of a single colour can therefore never attack each other, because every square they threaten is the other colour.

That turns the question into counting. The board splits into two colour classes of the same size, so one entire class is already a safe arrangement, and its size is the count you are after. Any knight added past that must stand on the other colour, where the class you just filled already reaches it. The colouring builds an arrangement and shows in the same breath why nothing larger exists, and the argument survives on any board size.