Can you checkmate with a king?

Two kings, one rook, and everything depends on where the white king is standing.

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The rook gives the check. The king on h6 took g7 and h7 away before the rook ever moved. Now park that same white king on d1 instead and try the identical idea:

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Same rook, same black king, and no mate anywhere in sight. So a king cannot deliver mate, and two bare kings is a draw before either player has thought about it, but that is the least interesting thing about it. The rook covers a rank or a file; the three squares beside the king on the next rank belong to somebody else, and the only piece that can take them without being captured is your king, because it cannot be taken by his king.

The king is a piece, and in the endgame a strong one

Beginners spend the opening hiding the king and then carry on hiding it once the queens are off, which costs them more half points than any opening mistake. In every forced mate on the list, from king and queen down to bishop and knight, your king walks up to within one square of his and does a third of the work.

The rule of thumb I give students: once the queens are exchanged, the king moves towards the action, not away from it. In a king and pawn ending it is the strongest piece you own.

The one time a king move gives check

There is a loophole, and it is worth knowing. Move your king off a line and the rook or bishop behind it is suddenly attacking the enemy king. That is a discovered check, it can be mate, and it looks for all the world as though the king delivered it. The checking piece is the one that was standing behind, which is exactly the distinction drawn on whether a king can check a king.

If you want to feel all this in your own hands, set up a rook and two kings, mate from the middle of the board, then try the same thing with your king glued to h1. The second attempt never ends, and that is the whole lesson. The method for the first one is on king and rook against king, and a round of mate in one puzzles will show you how often the winning move is a quiet king step that takes the last square away.