Two different coins are placed on squares of a standard 8 8 chess board; they may both be placed on the same square. Let us call two arrangements of these coins on the chess board equivalent if we can...

Two different coins are placed on squares of a standard 8 8 chess board; they may both be placed on the same square. Let us call two arrangements of these coins on the chess board equivalent if we can move the coins diagonally to get from one arrangement to another. For example, the two positions shown on the two boards in the figure are equivalent.

How many different (inequivalent) ways can the coins be placed on the chess board?

1. Please redo the previous problem, this time assuming the coins are identical.

2. Let A be a set and let P be a partition of A. Is it possible to have A = P?