Final push! I'm going to declare next week the final full week of Open Problem #1, and post Open Problem #2 on December 11th.
Please compile any information at the wiki here. After December 11th I'm going to forward what we've found to the mathematician (Erich Friedman) who posed the problem in the first place.
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.
Stefan Van der Wall came up with a solution on a cube and Sizheng Chen posted a solution on a torus.
Ahmed Amrani wrote a computer program checking all cases from 5x5 to 17x17, with no solution.
Still possible routes of study:
A complete description of all possible king configurations on a finite board. The current data is on the wiki here.
Ahmed Amrani's program can be used to check cases larger than 17x17.
Thank you everyone for your efforts so far!