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Pandolfini's Puzzler #11 - All the King's Knights

  • NM brucepandolfini
  • | Oct 11, 2013
  • | 3511 views
  • | 14 comments

“I’ve been thinking a lot about knights recently,” said Professor Pando to an attentive Zephyr. 
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“The other day I came upon an interesting problem in a math book.  It asked the reader to calculate the number of knights that could be positioned on a clear chessboard in a definite way. Funny thing is, it gave the wrong answer.”
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Question: How many knights can you place on an empty chessboard so that no knight is in position to capture any other knight?
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knight dancing black.png
Zephyr wasn’t sure she understood the problem. But as she started to ponder a bit more, she got the idea. 
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“May I try a few possibilities on an actual board?” Zephyr asked.
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The professor preferred it when students tried to do things in their heads. After all, in real chess games, opponents don’t let you move pieces around to see what works and what doesn’t work before playing your next move. But it was time for an exception.
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“Sure,” the professor relented. “Go right ahead, if you think it will help you.”
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So Zephyr cleared a chessboard and began using all the pieces, as if each one was a black knight, even white pawns and pieces, because there weren’t enough black knights to use as markers.
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After some time, shifting markers this way and that, she thought she had arrived at an answer. Zephyr had arranged three rows of knights. She had filled up the a-file, the d-file, and the h-file with substitute knights. 
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“The answer is 24,” she said with some confidence.
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Here’s what it would have looked like if she had a bucket of knights to use.
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Knight24.png
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The Professor wasted no time in responding: 
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“No, that’s wrong. But it is the same wrong answer given by the math book.”
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Zephyr seemed dumbfounded. Then, as people often do, she tried to defend her answer. 
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“I know it’s possible to place the knights on different files and ranks, other than the lines I’ve used here,” Zephyr shot back. 
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“Is that what you mean by saying that it’s wrong?”
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“No, that is not what I mean,” retorted the Professor. 
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“Let me give you a clue. Whatever color square a knight starts on, it must move to a destination square of the other color. So, if it starts on a light square, it must move to a dark square. If it starts on a dark square, it must move to a light square.”
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For an instant, Zephyr continued to look perplexed. But then her face began to beam.
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knight giving a thumbs up black.png
“I understand completely now. I love this problem. It’s so elegant.” Zephyr liked using big words. 
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“May I be allowed to elucidate?” Zephyr asked.
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What was Zephyr’s answer and why did she think it was so elegant?
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Answer Below - Try to solve ProfessorPando's puzzle first!
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ANSWER #11
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Zephyr’s answer was 32, and she actually had two different correct answers. The knights could be placed entirely on light squares, or entirely on dark squares. 
Knight32d.png
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Knight32L.png
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Why did Zephyr call the problem elegant? Because the correct answer was based on a concept. It didn’t need a practical illustration.
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Take note
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Many problems are solved by first trying out specific solutions, to see if they work. But there’s an entire class of problems that can be better solved by first getting a good idea that takes into account all the possible solutions. Scientists, mathematicians, and chess players, at different times, use both methods. When calculating variations, chess players tend to get specific. When making plans, chess players tend to get general.
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Of course, good chess play requires that players do both. And the best plans are always supported by concrete and specific analysis.
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RELATED STUDY MATERIAL

  • Watch PlayfulSquirrel's video on using knights in the endgame;
  • Check out our entire library of videos on how to use knights properly.

Comments


  • 6 months ago

    greenbossu

    never specified colour and the prof said in POSITION to capture

  • 6 months ago

    Big_J24680

    ah, this is a good article

  • 6 months ago

    Rebro

    @dennyhan - the knights in your central two files are attacking each other, so it's not a solution.

  • 6 months ago

    dennyhan

    The firs

    t diagram can easily be changed so that it gives the correct answer, with two rows of knights instead of one. Zephyr is not really smart, is she? So in the end, there is 4 different answers- Zephyr's two answers, 4 rows of knights and 4 columns of knights. Hope it is taken into account by Dr. Pandolfini!

    4 rows of 8 knights: 4x8=32 (the answer)

    4 columns of 8 knights: 4x8=32 (the answer)

  • 6 months ago

    Rebro

    @dennyhan - the problem posed in the article is to arrange as many knights on the board as possible so that none is standing on an attacking square of any of the others.

  • 6 months ago

    dennyhan

    The only problem about this puzlle is: I really don't get what this question is asking me to do :C, and it hasn't explained it.

  • 6 months ago

    Rebro

    @kcsmith169 - your dual question is interesting, but as you say, it seems pretty clear that it can't be done. The question then becomes, how many squares can be covered by knights without any overlap? I used 4 knights to cover 24 squares (in your sense of the word cover, where a square is covered if a knight stands on it) and I can't put down any more knights or overlap will happen. Can anyone do better?

     

    edit: goodness me, I just found a much better solution to this problem...9 knights can be placed so 46 squares are covered without overlap!

    edit2: okay, wow. The solution has got to be 12 knights covering 48 squares. The knights are distributed in 4 formations of 3 to the corners of the border (one at a corner, one at an adjacent square to the corner, and one at the diagonal square touching the corner).  The resulting arrangement has fourfold symmetry. It's very beautiful!

  • 6 months ago

    GSHAPIROY

    GOOD PUZZLE!!

  • 6 months ago

    Genghis_McCann

    This is a variation on the "dominoes on a chessboard" puzzle. A domino covers exactly two chessboard squares. Is it possible to arrange the dominoes so that all the squares on the chessboard are covered except two squares on opposite corners?

    The answer, of course, is "no". Every domino has to cover both a white and an adjoining black square, so it is impossible to leave two squares of the same colour at opposite corners of the board.

    Quod erat demonstrandum!

  • 6 months ago

    JuanoC

    a

  • 6 months ago

    upen2002

    nice

  • 6 months ago

    kcsmith169

    An intereseting dual question is: what is the fewest number of Knights that can be placed on the board to cover every square only once? Or can this even be done? A necessary simplification is the assumption that the Square a Knight resides on is considered covered.

    My initial take is that it cannot be done. The Knights keep stepping on each other's Squares

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