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OPEN PROBLEM #1

This is an open question. There is currently no known answer. Source.

On a chessboard of any size, is there a configuration of kings and knights such that each king attacks exactly 2 kings and 2 knights, and each knight attacks exactly 2 kings and 2 knights?

(Sub-problem -- if it is possible, what's the smallest board needed?)

Details: Kings can attack any adjacent square, including diagonally. Knights move in an L shape as shown below.

Note by Jason Dyer
1 month, 2 weeks ago

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I'm inclined to think so too with about 80-20 confidence. The problem is to prove it!

Jason Dyer Staff - 3 weeks, 1 day ago

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Yes, I know. I tried in this way,

  • Each king attacks 2 kings and 2 knights.

  • Each knight attacks exactly 2 kings and 2 knights.

Considering a \(4 \times 4\) chessboard, where it doesn't satisfies the condition.

In other words, it could be possible for a certain number of knights and kings. But not possible for each knights and kings.

Munem Sahariar - 3 weeks, 1 day ago

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This looks like a good start, but it doesn't exhaustively cover every single configuration of kings and knights. To begin with, you might want to think of what configurations with the kings will meet "every king must attack two other kings" condition before adding on the knights.

Jason Dyer Staff - 3 weeks, 1 day ago

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@Jason Dyer What is maximum size of configuration? \((\text{e.g} ~ 2 \times 4, 3 \times 2)\)

Munem Sahariar - 3 weeks, 1 day ago

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@Munem Sahariar From the very start it mentions: On a chessboard of any size

Admittedly, one approach that might get some traction is to start with specific board sizes (that is, if you can eliminate 4x4 very thoroughly, then 5x5, that might be useful and suggest a general method).

Jason Dyer Staff - 3 weeks, 1 day ago

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@Jason Dyer I had checked it with configurations like \(8 \times 8\) and \(6 \times 6\) etc. I didn't see any hopes which will make it possible. Currently I have no ideas to prove that.

Munem Sahariar - 3 weeks, 1 day ago

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@Jason Dyer Well, let me find which configuration is suitable to prove. :)

Munem Sahariar - 3 weeks, 1 day ago

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I think the answer is no.

Munem Sahariar - 3 weeks, 1 day ago

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