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We have this Existing Puzzle which has got the valid and invalid cases listed ; the Accepted Answer is along the lines of what is invalid and what is valid.

But there was a flaw or fault in the Puzzle (not in the Answer, which satisfies the given Conditions) which has been corrected now, by adding a single word.

If you wish to check and see the flaw yourself, you can check the latest change there.

A king can not move itself out of attack, but it can help some other king come out of attack.

Existing Examples in the earlier Puzzle are all invalid according to this new correction :

There are N White Kings on the Chess Board.
There are M Black Knights.
There is no Black King and no other Piece. Only M+N Squares are occupied.

Each White King is attacked by at least 1 Black Knight, but it cannot move itself or some other king out of attack.

Maximize N. In the Possible Solutions, Minimize M.

Needless to mention that the valid Answers there become invalid here:

Invalid Examples:

In the Accepted Answer,

The kings around c7 can take c7 ; then a8 and e8 are not in attack.

In the Other Answer,

The kings around b6 can take b6 ; then a8 & c8 are not in attack.

Valid Examples:

N=3 & M=4

N=9 & M=12

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  • $\begingroup$ Essentially, all kings must be attacked by at least two knights? $\endgroup$ Jul 4, 2022 at 11:44
  • $\begingroup$ Not necessarily ; There might be no King near a Knight (which can not be taken) ; This Knight can be the only Knight attacking some King(s) , @DanielMathias $\endgroup$
    – Prem
    Jul 4, 2022 at 11:48
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    $\begingroup$ @DanielMathias ... or attacked by a single knight that cannot be captured by any white king. $\endgroup$ Jul 4, 2022 at 11:49
  • $\begingroup$ Exactly !!! @JaapScherphuis $\endgroup$
    – Prem
    Jul 4, 2022 at 11:51
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    $\begingroup$ In your example N=9 M=11, c2 can take c3 leaving b1 unattacked. Isn't that a problem? $\endgroup$
    – Florian F
    Jul 4, 2022 at 16:06

3 Answers 3

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I'll stand by my initial comment: All kings should be attacked by at least two knights. If any king is attacked by only one knight, the attacking knight must not be threatened by another king. This greatly reduces the number of kings that can be placed.

Doubling up the knights in loopy walt's answer to the earlier question is a good start, but there is room for improvement.

This solution has 24 knights attacking 36 kings. The four vacant squares are guarded, providing no safe haven for nearby kings. All knights are guarded, so no king can save himself by capturing a knight. All kings are attacked by at least two knight, so none can be saved by another with the capture of a single knight.

enter image description here

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  • $\begingroup$ +1 , Currently the Best Answer ! I will wait a while before checking whether somebody else can get a Better Solution ! $\endgroup$
    – Prem
    Jul 5, 2022 at 9:58
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UPDATE Saved two knights

38 Kings and 24 Knights enter image description here

If I'm not missing something these are

38 Kings and 26 Knights enter image description here

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  • $\begingroup$ +1 , It looks valid to me !! Currently the Best Answer !! $\endgroup$
    – Prem
    Jul 6, 2022 at 4:44
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The answer I got is :

24 knights and 36 kings

The position(s):

Position number 1:

enter image description here Or if you want to see the lichess version click here https://lichess.org/editor/K1KKKKKK/KKnKnnK1/KnnnnnnK/KnnKKnKK/KKnKKnnK/KnnnnnnK/1KnnKnKK/KKKKKK1K_w_-_-_0_1

Position number 2:

enter image description here Or if you want to see the lichess version click here https://lichess.org/editor/KKKKKKKK/K1nnnn1K/KnnnKnnK/KnKKKnnK/KnnKKKnK/KnnKnnnK/K1nnnn1K/KKKKKKKK_w_-_-_0_1

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    $\begingroup$ +1 , It is matching the Other Solution , with a Slight Change ! $\endgroup$
    – Prem
    Jul 5, 2022 at 17:17

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