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Is it really possible to checkmate with two knights and king against a king?
There is a mating position with this but no extra tempo to do it in real game.
What are the explanations, why knights cannot create an extra tempo around 64 squares on the board?
Is it really possible to checkmate with two knights and king against a king?
Theoretically, the checkmate is possible, but you can not do it in practice unless the weaker side allows you to. This is related to a drawback in the way knight moves.
There is a mating position with this but no extra tempo to do it in real game.
What are the explanations, why knights cannot create an extra tempo around 64 squares on the board?
There are two reasons:
They can not do it because they can not move to another square while keeping the previous one in control ( like bishop and many other pieces can ) and because they need too much time to position themselves properly.
DEMONSTRATING KNIGHT'S INABILITY TO "LOSE TEMPO" :
Take the following position as an example:
[Title "This position is won with a bishop"]
[fen "8/8/8/8/4B3/8/P7/K1k5 w - - 0 1"]
This position is won for White because he can play with bishop and retain control over c2 square ( 1.Bh7for example ), which will create zugzwang. As you say in your question, White will "lose tempo" and force Black to move his king away.
Now analyze this position with a knight instead of a bishop:
[Title "If it is White to move this is a draw"]
[fen "8/8/8/8/3N4/8/P7/K1k5 w - - 0 1"]
Why is this endgame only a draw? Because White can not move and keep control over c2 square. This means that he can not create zugzwang, or as you say he can not "lose tempo" to force Black king away.
Without the ability to create zugzwang you are just not able to corner the opposing king.
DEMONSTRATING KNIGHT'S INABILITY TO POSITION ITSELF PROPERLY :
As for the positioning problem, we shall take a look at the below diagrams:
[Title "Agile and swift bishop"]
[fen "8/8/8/8/8/1K1N4/3B4/1k6 w - - 0 1"]
1.Nb4 Ka1 2.Nc2+ Kb1 3.Na3+ Ka1 4.Bc3#
Here, bishop needed 1 move to reach the mating square. Notice that the endgame would be drawn, just as the one with two knights, if bishop was not able to reach c3 in one move.
In the following diagram you will see that knight needs more moves to reach the mating square, which enables the weaker side to draw:
[Title "Clumsy knight"]
[fen "8/8/8/8/8/1K1N4/4N3/1k6 w - - 0 1"]
1.Nb4 Ka1 2.Nc2+ Kb1 3.Na3+ Ka1
and here Ne2 needs 2 moves to reach c2 and deliver checkmate.
Remember this: Knights experience great difficulties when they need to reposition themselves between short distant squares in a short notice. This is often impossible task for them. The following example demonstrates clearly what I mean:
[Title "Slow knight"]
[fen "8/8/8/8/8/1K5p/6N1/1k6 w - - 0 1"]
In this position we can see how hard it is for the knight to position himself fast to a square nearby like f3 or g4. Again he needs 2 moves to do this and thus he fails to save the game.
Knight's way of moving gives him ability to "do the impossible". He can "jump over" obstacles, his attacks can not be stopped, his line of fire can not be covered or intercepted, but all these beautiful features come with a price. He is slow and clumsy when he needs to reposition himself to a square in its proximity, and if player can not "buy" him enough time to do that then knight fails to accomplish this task.
Hopefully this post answers your question. If you have further questions leave a comment.
Best regards.
Is it really possible to checkmate with two knights and king against a king?
It is possible, but it is not forced unless the position or a mistake from the opponent allows it.
There is a mating position with this but no extra tempo to do it in real game. What are the explanations, why knights cannot create an extra tempo around 64 squares on the board?
Very Simple Reason -
A knight always moves to a square of the opposite color. If the knight is on a white square it moves to a black square and if it is on a black square it moves to a white square.
Thus, knight on white square moves to -
1. Black square
2. White square
3. Black square
4. White square
...forever
In order to create an extra tempo, you would need the capability for a knight to move to a square of the same color twice in a row, but that is not possible. Thus, you cannot create an extra tempo with a knight.
Wikipedia has a long article on this topic.
Unlike some other theoretically drawn endgames, such as a rook and bishop versus rook, the defender has an easy task in all endings with two knights versus a lone king. The player simply has to avoid moving into a position in which he or she can be checkmated on the next move, and always has another move available in such situations
Not only it is showing a lot of explanatory examples, but also speaks about really strange paradox Troitzky line which explains how the material advantage for the weaker size can cost him a game. Having one pawn against 2 knights can give white a chance to win.
Also you can read another helpful resource here.
