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# Two knights vs king for checkmate?

• #1

Is it possible, in terms of a helpmate, for two knights to checkmate a lone king?  I'd like to know what people will say.

• #2

It's possible if Black errs at the very end, but otherwise White must stalemate before he can mate.  Mate can only be forced if Black has a pawn, and even then only if it is blockaded by a White Knight and only within certain ranges of squares.  The basic idea is White's King and one Knight restrict the Black King to near a corner, then the other Knight rushes in.  White can then give the "stalemate" because Black's pawn can move - but he must do it before the pawn promotes, or to deliver mate on the very next move, even then.

The great endgame composer A.A. Troitzky wrote a treatise on K+2N v K+P a century ago, and it is one of the few endgame analyses which the computer "Table bases" could not improve upon.

• #3

Sure, help-mate is possible. It doesn't even have to be in a corner:

They do need assistance of their own King for this, though.

• #4
[COMMENT DELETED]
• #5
Estragon wrote:

It's possible if Black errs at the very end, but otherwise White must stalemate before he can mate.  Mate can only be forced if Black has a pawn, and even then only if it is blockaded by a White Knight and only within certain ranges of squares.  The basic idea is White's King and one Knight restrict the Black King to near a corner, then the other Knight rushes in.  White can then give the "stalemate" because Black's pawn can move - but he must do it before the pawn promotes, or to deliver mate on the very next move, even then.

The great endgame composer A.A. Troitzky wrote a treatise on K+2N v K+P a century ago, and it is one of the few endgame analyses which the computer "Table bases" could not improve upon.

Wow, I didn't know this.

• #6

this happened to me and the result was funny http://www.chess.com/livechess/game?id=765978737

• #7

It would be interesting to compose a study where three knights have to try and force mate (can it be done?) and can't be helped by the king... something like this with white to move and subpromote with check -this isn't it by the way , so don't try and solve

• #8

Ziggy - it can't be done without a helpmate!

The centre of the board won't work; the knights obviously have to be near the K, so two of them are busy protecting each other, but this always leaves at least 4 flight squares for the K.

The edge of the board also doesn't work for the simple reason that the king's square and the 5 squares around it are 3 light and 3 dark squares. 2 knights can take care of 2 light and 2 dark squares, but the 3rd knight can't cover the remaining 1 light and 1 dark square simultaneously, so the K will always have 1 flight square.

So it has to be the corner, say h8. h7 and g8 must be covered by one knight, a second must cover g7, and the 3rd can deliver mate. The question is, how to force the king to h8?

This is the position we want:

The K must have been on g7 before, which means that one of the other knights must have been protecting the one on g7:
However you set up the protecting knight, the K has at least 4 flight squares (not including the corner square), which obviously cannot all be covered by the 3rd remaining knight. So the K will always be able to escape away from the corner.
• #9

"Helpmate" is a term I've never heard before. Can someone provide it's definition?

• #10

Helpmate-the opponent helps you to checkmate them

• #11
bouncing_check wrote:

Ziggy - it can't be done without a helpmate!

The centre of the board won't work; the knights obviously have to be near the K, so two of them are busy protecting each other, but this always leaves at least 4 flight squares for the K.

The edge of the board also doesn't work for the simple reason that the king's square and the 5 squares around it are 3 light and 3 dark squares. 2 knights can take care of 2 light and 2 dark squares, but the 3rd knight can't cover the remaining 1 light and 1 dark square simultaneously, so the K will always have 1 flight square.

So it has to be the corner, say h8. h7 and g8 must be covered by one knight, a second must cover g7, and the 3rd can deliver mate. The question is, how to force the king to h8?

This is the position we want:

The K must have been on g7 before, which means that one of the other knights must have been protecting the one on g7:

However you set up the protecting knight, the K has at least 4 flight squares (not including the corner square), which obviously cannot all be covered by the 3rd remaining knight. So the K will always be able to escape away from the corner.

Good work

• #12
• #13

That is some tactic Ziggy!

• #14

I've just put half a dozen K v K+3N positions into stockfish and it found forced wins every time, average about 16 moves.

• #15

oh arse, old thread.

• #16
nigelnorris wrote:

oh arse, old thread.

Yes, but this is good information: I never knew that two knights could mate on the edge, which means the basic endgame table I made has an error in it. It also means the Wikipedia page on this topic (https://en.wikipedia.org/wiki/Two_knights_endgame) is slightly misleading, since all of its diagrams show only corner positions in that basic checkmate.

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