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Can a rook and a king force a checkmate against a bare king that starts in the center if the opponent was allowed to pass?

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Of course not. This is Basic Endgames 101. The main principle lies in forcing the lone king back by getting the opposition. If the lone king can pass then the stronger side can never get the opposition.

Here's how the standard checkmate works. Notice how white can use a rook move as a pass to force the opposition.

[fen "8/8/8/3k4/7R/4K3/8/8 w - - 0 1"]

1. Rg4 Kc5 2. Kd3 Kd5 3. Rg5+ Ke6 4. Ke4 Kf6 5. Ra5 Kg6 6. Rf5 Kg7 7. Ke5 Kg6 8. Rf1 Kg7 9. Ke6 Kg6 10. Rg1+ Kh5 11. Kf5 Kh4 12. Rg8 Kh3 13. Kf4 Kh2 14. Kf3 Kh1 15. Kf2 Kh2 16. Rh8#

Now let's look at what happens if black is allowed to pass:

[fen "8/8/8/3k4/7R/4K3/8/8 w - - 0 1"]

1. Rg4 null 2. Kd3 Ke5 3. Ra4 null 4. Ke3 Kd5 5. Rg4

and black has no problem maintaining the king in the middle of the board. There is absolutely nothing white can do to make progress.

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