Bishops and knights

Though these 2 minor pieces have been long acknowledged to be equal in power, it has been found that bishops are marginally stronger as they can control large parts of the board at a time. Despite this, both the pieces function well in certain conditions.
Bishops
Bishops operate well when the board is uncluttered and when the center is usually free of blocked pawns. As many players have advocated, bishops also work very well in tandem, or in pairs. The 'bishop pair' is a formidable advantage in the game.
Bishops should generally be placed on a safe, distant squares(there are exceptions) where they can exert diagonal control. The hypermodern approach of chess talks about 'fianchettoeing' bishops, which involves placing them on the long diagonals (on the g2, b2, g7, and b7 squares).  From these squares they remotely control the center and are tough to dislodge.  Bishops should also be placed on 'holes' in the opponent's position, where they cannot be attacked by pawns and are equally difficult to dislodge.
Care should be taken when exchanging a bishop for a knight or even when sacrificing the exchange. This would mean that your control over those respective squares is weakened. Exchanging a light square bishop for a knight(as seen many times in the Ruy Lopez), means your light square control is weakened. The player should then try to compensate by using pawns to control those squares, or using the queen as a pseudo bishop.
Despite their many merits, bishops tend to be useless in closed, blocked positions, typically in the endgame. Many games end in a good knight versus bad bishop situation, where the bishop gets blocked in by its own pawns and the knight is free to attack them.  A bishop getting blocked in by its own pawns is the biggest problem. This is seen in openings such as the Slav defense and the Queen's Gambit declined. Systems such as the Sicilian defense lead to a more or less open and flexible game.
Knights
Knights are the magical pieces that can weave through others as if they weren't there. Because of this unique property, they can operate the best in closed positions which are dominated by blocked pawns. Knights are associated with the concept known as 'outposts'. These are useful square either in the center of the board or deep in the opponent's territory from which the knight cannot be ejected. Knights are very annoying to deal with because they can 'fork' pieces, especially involving the king and rooks, owing to their configuration on the board.
Knights are strong on squares such as d4, c4, d5, e5, etc from where their movements become unpredictable and strong.
Knights also have an unexpected benefit over bishops. Their movements cannot easily be tracked.  A knight can maneuver from a square like b1 to e5 and it may seem mysterious how it got there. When players cannot track knight maneuvers, they have a condition known as 'knight blindness'. Backward knight movements are crafty and require long term plans.
Knights, however, have obvious weaknesses in open positions, when they can't get around to both sides of the board quickly. For these reason, they are quite bad at controlling passed pawns. Their awkward maneuverability at the corner of the board can lead to bizarre cases of the pawn passing the knight as if it weren't there. Smart players will make the position conducive to the movements of knights if it is facing a bishop. This would involve blocking the pawns up and putting your pawns on squares where the bishop cannot attack them. The king must be used in close conjunction with the knight.

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