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Chess axioms are helpful, but never 100% correct. But remember-able expressions like "a knight on the rim is grim" and "passed pawns must be pushed" do lend us some true helpful hints generally.
Well, here are some more:
- Same colored bishops for both sides: Only the bishops should focus on their colored squares. All other pieces should play opposite colors. For example; if both sides have dark colored bishops, we should put all the other pieces on white squares.
- Two bishops versus bishop and knight: The side with the two bishops should play on the colors where the opponent does not have a bishop. The side with the bishop and knight should play on the color of the bishop.
- Opposite colored bishops: Both sides should play on the colors of their bishops.
- One bishop versus one knight: The side with the knight should play on the opposite color of the opponent's bishop. The side with the bishop should utilize his other pieces on the opposite color of the bishop.
- Both sides have both bishops: When the central pawns are fixed on a particular color, we should play on the opposite color of our opponent's centralized pawns, and try to exchange the opponent's bishop of that same color. For example; if the opponent's center pawns were fixed on the light squares, then we should exchange the dark colored bishops and fight for the dark colors with other pieces.