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The Opposition

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In the endgame, one king is stronger than the other by “having the opposition.”
Offensively, this means the ability to push the opposing king aside.
Defensively, this means the ability to prevent the opposing king from reaching advantageous squares.

2k5/8/8/8/1PK5/8/8/8 b - - 0 57

The black king can prevent the white king from entering key squares (a6, b6, c6) required to promote the white pawn.

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King on the 6th, Pawn on 5th Wins

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In a king and pawn endgame…

King on the 6th rank, pawn on the 5th rank (not on the a or h file) wins regardless of whose turn it is to move. The king on the 6th rank uses the opposition to push the opposing king aside.

4k3/8/4K3/4P3/8/8/8/8 w - - 0 1

The white pawn will promote.

The black king can stop the white pawn if the pawn is on the a or h file.

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Castling on Same Side

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King Safety

Castling on same side creates a quieter game than when kings are castled on opposite sides.
Pushing pawns weakens king safety.
Initiate an attack with pieces rather than a pawn storm.

r2q1rk1/pp3ppp/2pp1n2/4p3/4P1P1/1P1P1P2/P1P3BP/R2Q1RK1 w - - 0 1

White’s move g3-g4 does more harm to his position than to his opponent’s. It weakens white’s king safety.

Closed Center

If you control the center, or the center is closed, you may initiate an attack with a pawn storm.

r1b2rk1/2qnbppp/p2p1n2/1ppPp3/4P3/2P2N1P/PPB2PP1/R1BQRNK1 w - - 0 1

Because the center is closed, g4 is a good move.