**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.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

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

Log in to reply

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.

Log in to reply

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.

Log in to reply

Log in to reply

of any sizeAdmittedly, 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).

Log in to reply

Log in to reply

Log in to reply

I think the answer is

no.Log in to reply