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The moves described are only valid when no other piece is interfering. If any piece is in the way, then the moving piece cannot skip over it. The moving piece can only move to any square before it.
If it is an enemy piece, it can also be captured. To capture the opponent's piece, one moves his own to the enemy piece's square, then removes the enemy piece from the board. (It is good to capture enemy pieces because opponent's strength is reduced as a result).
For example, suppose a white Queen is positioned on a1 (Qa1) and a black Bishop is placed on d4 (Bd4). If it is WHITE'S move, he can capture the Bd4, provided the squares b2 and c3 are empty; to do this, he removes the enemy bishop and places his queen on d4. We denote this move by "Qa1 x d4" (or simply "Qxd4").
This means "The Queen at a1 captures (something) at d4". If there was no capture we would write "Qa1 - d4" (or simply "Qd4").
There are some exceptions to the basic rule regarding captures. A pawn can not capture pieces positioned on the square(s) the pawn can advance to; instead it may capture an enemy piece that is placed one square diagonally and in front of it. For example, suppose WHITE has a pawn located at e2.
The pawn can move both to e3 and e4 if no piece interferes, but it cannot capture on these two squares. Instead it can capture on d3 or f3.

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