pigeonhole principle proof for 10 kings in a chessboard

Question

There are 10 kings in a chessboard. prove that there is one square in the board that creed with 2 kings.

Answer 1

Divide the board into 3x3 regions as much as possible, like this:

a a a b b b c c
a a a b b b c c
a a a b b b c c
d d d e e e f f
d d d e e e f f
d d d e e e f f
g g g h h h i i
g g g h h h i i

Each region obviously can contain at most one king (else they would be attacking each other, or both cover the central square of the region). There are 9 regions, so at most 9 kings are possible.

Answer 2

Hint: When you "place a king" on the board, instead of tracking exactly where you put the king, trace how many squares that king touches. Might need to handle some special cases when the king is placed in a corner or on an edge, but I believe that this approach might give you a solution.