Two different coins are placed on squares of a standard 8x8 chessboard; 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.
How many different (inequivalent) ways can the coins be placed on the chess board? Redo the problem, assuming the coins are identical.
For example, these two positions shown on the two boards in the figure are equivalent.
Partition the board into 'odd squares' and 'even squares', like black and white on a chessboard. A coin being on an odd square is equivalent to being on any other odd square, and the the same with even squares.
So there are are 4 inequivalent ways the coins can be arranged: OO, EE, EO, OE, where OE means blue on an odd square, black on an even square.
For two coins of the same colour there are 3 possibilities, OO, EE, OE, since OE is indistinguishable from EO.