I'm going to start doing updates of our project weekly. This will allow me to summarize what we've learned so far and suggest which avenues of research are most useful. Also, please put new general discussion as responses to this post, rather than the first one; that way the discussion thread doesn't get too long.
To reiterate, here is our open question. There is currently no known answer in the finite case.
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? Also, what's the smallest board needed?
Details: Kings can attack any adjacent square, including diagonally. Knights move in an L shape as shown below.
We've had a great start to the group! Some highlights include:
Novak Radivojević pointing out right away there was a solution on an infinite board. He wrote a problem to go with this insight that we've featured as a Problem of the Week.
Lots of great discussion from Mike Harding, João Areias, Marcus Luebke, Stefan Van der Waal, and others.
A long post by Stefan Van der Waal springing from the discussion which explores the problem at the edge of the board.
Please note the new wiki page dedicated to this problem. Feel free to add results there!
Based on the discussions, these seem the most promising areas of study:
A complete description of all possible king configurations on a finite board. The current data is on the wiki here.
We could use a more systematic description of the infinite cases. Is there a specific way to describe all infinite board solutions? Does anything from an infinite board solution suggest why the problem might not be solvable on a finite board?
This may not be the complete list though - feel free to go in an entirely different direction!
One question people might be wondering is: how long will this open question last for before we go on to another one? I'm not sure on the timing yet (this is the first time we've done this!) but I expect we will likely switch to a new question every month. You are very welcome to post in the thread suggesting open problems.
Also note: I have turned off moderation for new questions / posts - I haven't seen any issues, so any new posts you make to the feed will now show up immediately.