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Martin Gardner mentioned the following "chess" problem in his column, Mathematical Games, for Scientific American (he got it from Smullyan, who learned about it from a high school math teacher):

... Assume that a game is played with the new rule that a knight can on a single turn move twice. It is not clear if this more powerful knight move is an advantage to the first or second player. The first player, White, offers to play the game without his queen. How can he be sure to win?

After thinking a bit, I could not make progress, so I look at the solution:

White jumps queen's knight forward twice. Checkmate!

That's all it says. How is this a checkmate? I must be missing something.

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    The knight should be on b5. Note that the knight is attacking the king (it can jump to e8 via d6). But the king cannot move, and the knight cannot be taken. So it's checkmate.
    – Mathmo123
    Nov 25 '20 at 22:15
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    There's a small confusion here. On reading the question, I understood it that a knight can move twice on any one single move (that is, once it has used its double move, it can't use it again). It might need to be clarified that this "double move" can be performed as often as you like. Nov 26 '20 at 7:22
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    I believe Brian's answer, but then what is the narrative purpose of the statements "It is not clear..." and/or "White offers to play without his queen"? White's missing queen doesn't factor into the solution, right? Is it just that it's not clear to Black what's going on, and meanwhile White is displaying confidence by offering to play at a handicap? Would the puzzle be any different if White had offered to play with nothing but his king and his queen's knight? Nov 26 '20 at 16:28
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    @Quuxplusone Indeed, I assumed there'd be some strange reason why 1. Nb1-c3-d1 was winning for White :) Nov 27 '20 at 16:48
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According to the FIDE Laws of Chess:

1.4 The objective of each player is to place the opponent’s king ‘under attack’ in such a way that the opponent has no legal move.

1.4.1 The player who achieves this goal is said to have ‘checkmated’ the opponent’s king and to have won the game

That means that if white has a knight which is 2 normal knight moves away from the opponent's king then the opponent's king is in check, under attack.

Then consider white starting the game like this:

[fen ""]

1. Nc3 null 2. Nb5

Then you can see the black king is in check. On the knight's next move it can go Nb5-d6-e8. But there is no way for black to get out of check. Hence it is checkmate.

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    Oh my god, I completly forgot that the knight can "move" twice in the check scenario. Thank you!
    – Favst
    Nov 25 '20 at 22:49
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    Note that Nc3-Nd5 won't work because then it can be countered by Nf6-Nxd5 which of course ruins your plans. Nov 26 '20 at 7:25
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Ok, I think I got it. Your first move is Nc3-Nb5, threatening Nxc7 and Nxe8 in its next move, and since Black can't move their King they just can't prevent check mate.

EDIT: just to make it clearer, actually the first move is Nc3-Nb5+, since the Knight is attacking the king because of the special way it moves.

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I guess that can White play 1. Nb1-a3-b5+ or Nb1-c3-b5+ or Nb1-a3-c4+, in order to be in range of the bK but not in range of the bKN.

I suppose we clarify that if the first knight move is a capture, the second move can still take place. Otherwise 1. ... d6! defends.

But what happens if the first knight move is a check? Does that mean that White doesn't get to make a second move? And does that affect the definition of check. Because wNb5/wNc4 cannot threaten to capture bK, as the first move wN-d6+ checks bK and that would terminate the turn!

If this is correct it means that a knight only gives check when one move away from bK. Can the original poster make clear how this works, please?

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