Two mathematics puzzles for you.

I will be checking these puzzles to make sure I got them correctly.

One:  How many white and black queens can you fit on a chess board without them attacking each other.  I think the answer is 9 and 10.  Can you find the solution with that number?

Two:  You have 13 coins if I remember correctly, and one is counterfeit and weighs less than or more than the others.  The other 12 weigh the same.  You get to make three weightings on a scale; can you identify the coin no matter what?

I can do 9 white queens and 8 queens.

If we know whether the coin is lighter or heavier, I can do 15 coins.

For the queens, a white queen can attack a white queen, and a black queen can attack a black queen, but they can't attack the opposite colour? If no queen can attack any other queen, the answer is 8 :p

You can do 9 white queens and 8 black queens if the white queens are allowed to attack each other, the black queens are allowed to attack each other, and the white queens can't attack the black queens.

I also did eight queens so that none of them attack each other.