# Knights on a Chessboard II

It is well-known (and has been asked here earlier) how many knights can fit on a chessboard without any knights attacking any others. But what if each knight can attack at most one other?

Idea from "The Mathematical Knight" by Noam Elkies and Richard Stanley.

• This has the same answer as the previous question... Jun 28, 2015 at 23:28

The maximum possible is number of knights is:

32

This can be achieved by:

Putting a knight on every white square (thanks to Timtech for the graphic).

There's also other nontrivial configurations, like:

A knight on every edge or corner square (28), plus the four very center squares.

This is optimal because of existence of a construction with the following properties:

Partition the chessboard into 16 sets of 4 squares, with each set forming a a loop of four connected by knight moves.

One instance of this construction is:

Splitting the 4x4 square into four sets as follows

1342
4213
3124
2431

Then, split the chessboard into four 4*4 quadrants each with a copy of this pattern.

This construction proves optimality because:

Each set of 4 can only have 2 knights, since if there are 3, one knight must be attacking two knights. So, this gives a maximum of 2*16=32.

• Nice graphic (spoiler): mathworld.wolfram.com/images/eps-gif/KnightsMax_700.gif Nov 18, 2014 at 12:11