Logic Puzzles

 I just finished reading Alice in Puzzle-Land by Raymond Smullyan, and was inspired to come up with a couple logic puzzles of my own.

When she was a girl, the White Queen practiced believing impossible things for half an hour every day. Once, when she was practicing, she overheard two bishops having the following conversation:

B1: At least one of us is red.

B2: I never step on squares of my own colour.

B1: I never step on red squares.

B2: We are both the same colour.

The White Queen believed everything they said. Assuming the court contains only those pieces normally found in a single chess set, and follows the normal rules of chess, how many impossible things did she believe?

Solution 1A

The next day, the White Queen overheard another conversation between two knights:

K1: Knights always tell the truth.

K2: Bishops always lie.

The White Queen believed these statements, but no longer necessarily believed what the bishops said on the previous day. Did she believe any impossible things?

Solution 1B