I'm really sorry about my late reply but due to some rather unfortunate events I wasn't able to post one earlier.
"P(64,2) = 4,032 where we assume two kings can be next to each other."
Yes, I got so far.
"if we follow the rule that no two kings are next to each other. That is P(64,2)-420"
And I do get why that is correct.
What I don't get, however, is how you reached the 420.
Or to put it differently, how do you know that cardinality? (That I didn't know the answer to that question was the very reason I calculated the valid positions for a two king game in that intricate a way.)
"It shouldn't matter yet which diagonal the bishop travels on because combined the bishops travel on all the squares."
Right. Why do such easy things elude me? -.-
What he said."Meaning you're allowing the possibility for one of the kings to be in check?"
Yes, 1 king in check is possible and we are also counting the checkmates for example, a white king at A1 with a black queen on B2 and the black king on C3 would be checkmate.
Where we could have a problem, however, would be in preventing both kings' being checked/mated at the same time (later).
"Given the constraints of the problem are 20 rooks on the chess board possible?"
No. For 20 rooks, all pawns would have to be promoted to rooks, which is impossible as they block each other (a black pawn must leave to enable a white promotion and v.v.).
"We start with 2 rooks how many rooks on a chess board are possible?"
"How many pawns can make it to their last row?"