The concept of a "safe check", i.e. a check delivered by a piece that cannot result in a trivial capture by an enemy king, was first used by Taylor in 1876. Taylor reasoned that the probability associated with delivering a safe check on an empty board featuring just the piece in question and the enemy king should be proportional to its strength.

Let's take a Rook for an example. Ok, place a Rook on a1,a2,a3...a6. It can safely check a king on a8. It cannot safely check a king on a8 when it is on a7, since Kxa7 violates "safe check".

You essentially "sum" these safe checks over the entire board, placing the king on each square, and computing the number of squares on which a rook resides. The ratio of safe checks to total arrangements on an 8x8 board is 1:6 for the Rook.

Bishops get a little messy in the computation (explanation is not too intuitive) but basically varying diagonal lengths as a function of bishop location and king placement make it very recalcitrant to derive. Knight computations are easy, so are the other pieces.

So, I set out to do the same for a board of dimensions Y by Z, not just a square board like Taylor did.

More on the algebra of the Rook computation in the next post...