In how many ways can you arrange the black and white kings on an empty chess board to get an acceptable position (according to standard chess rules)

Number of ways = 3612 ways

Step-by-step explanation:

I have attached a diagram representing two scenarios on the chess board. The scenarios are; when the first king takes away 4 squares (label A) while the second scenario is when it takes 6 squares (label B).

Now, If the white king is in a corner, it has 4 possibilities while the black king can be at any of (64 − 4 =) 60 positions.

If the white king is at the boundary, but not in a corner which is 24 possibilities, the black king can be at any of (64 − 6 =) 58 positions

Finally, in all other cases, for the white king which has 36 possibilities, the black king can be at any of (64 − 9 =) 55 positions.

So, in total, the number of ways would be;

(60 × 4) + (58 × 24) + (55 × 36) = 3612 ways