PAWN (excluding capture, en passant, and post-promotion moves)

Out of the 6×8=48 squares the pawn can move (it can’t touch the first rank, and by the last rank it will have promoted), we have 8 squares (the second rank) with two possible moves, and the other 48-8=40 have one possible move. Thus, our average is .

KNIGHT

This one is more complicated, as we obtain the following possible number of moves for different positions:

Thus, we end up with:

.

BISHOP

This is another tricky one. We can use systematic counting to obtain another table:

Thus, we end up with:

.

ROOK

This one is surprisingly easy. No matter where the rook stands, it dominates the seven squares in it’s row, and the seven squares in it’s column, giving an average of .

QUEEN

The queen “combines” the bishop and the rook. Since the rook’s movement range is ALWAYS 14, we just add an extra 14 to the bishop’s average, ending up with .

KING

In the four corner squares, the king can move to 3 places, 5 places for the 6×4=24 edge squares, and 8 places for the 6×6=36 middle squares. Thus, our average is .

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