How many ways can two knights threaten each other on a chessboard?

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Discussion Overview

The discussion revolves around the problem of determining the number of ways two knights can threaten each other on a chessboard. Participants explore various methods of counting configurations, addressing the implications of knight placement and the effects of board position on the number of threats.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant describes an initial approach involving counting the number of squares a knight can threaten from each position, but expresses uncertainty about the correctness of their method.
  • Another participant questions the rationale behind multiplying by 2, suggesting that each configuration is counted twice and should be divided by 2 instead.
  • A different perspective is introduced, emphasizing that the number of threatened squares varies depending on the knight's position on the board, leading to a need for separate calculations for different squares.
  • One participant asserts that color does not matter when counting threats, proposing that counting threats from one color to another suffices without needing to adjust for color differences.
  • There is a suggestion that the initial confusion about color and counting may have led to misunderstandings in the calculations.

Areas of Agreement / Disagreement

Participants express differing views on whether the multiplication by 2 is necessary, with some agreeing that it may not be needed while others defend their original reasoning. The discussion remains unresolved regarding the correct counting method.

Contextual Notes

Participants highlight that the number of threatened squares changes based on the knight's position, indicating that assumptions about uniformity in threat counts may not hold across the board.

Who May Find This Useful

This discussion may be of interest to those studying combinatorial problems in chess, mathematical reasoning related to game theory, or anyone exploring the intricacies of knight movements on a chessboard.

ShayanJ
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Find the number of ways of placing two knights in a chessboard that they can threaten each other.

I tried to solve it like this but it was wrong because the answer was not among the four options.
I wrote the number of squares that that the knight can threaten on every square that we place it.Then multiplied the number of squares with the same number by the number in it.then calculated the sum of the results and multiplied it by two.But as I said it was wrong although I'm still sure its right.

thanks
 
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Why are you multiplying by 2?
 
Well, for each square, find the number of ways a knight can threated the square. After you have counted the sum of ways for all squares, you realize you have counted each way twice. Divide the total number by 2, and there you go.

Tell us your way of solving the problem.
 
I thought like below:
I should at first find the number of ways that I can place a knight on the chess board then multiply it by the number of squares it threatens when placed in a square then multiply it by two because I can do that with picking the white knight first or the black knight first.But as you know the number of squares a knight can threaten differs in differente areas of board.so I found the ways for each square seperatly.And the rest is obvious.And I don't think any thing is counted twice hear!Could you explain more?
thanks
 
Shyan said:
And I don't think any thing is counted twice hear!Could you explain more?
thanks

Consider a chess board with only two squares such that a knight on one threatens the other. The count is two: both squares threaten one other square. But there's only one configuration where the knights attack each other: the configuration where both knights are on the only two squares.
 
CRGreathouse said:
Consider a chess board with only two squares such that a knight on one threatens the other. The count is two: both squares threaten one other square. But there's only one configuration where the knights attack each other: the configuration where both knights are on the only two squares.
No, the knights have to be of opposite colours or they don't attack each other.
 
There was a slight confusion here. I assumed that color was irrelevant and any two knights were opponents. If you start out with a black knight, and for each square count the ways it can be threatened by a white knight you have counted all possibilities. There is no need in this case, as far as I can see, to multiply or divide by 2.
 
Agreed.
 
So you people mean my solution is all right except the part I multiply the number by 2?
 

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