In how many ways can two kings be placed on an $8 \times 8$ chessboard so that they do not attack each other, that is, the squares occupied by the two kings do not share a common corner?
The task text does not specify whether the kings are of different colors or not - although kings of the same color do not typically occur on the chessboard. Let's assume, then, that we need to place a dark and a light king in the desired manner.

Fixing the position of the dark king, the light king cannot be placed on the same square, nor on the squares adjacent to this one at the corner or edge. However, it can be placed anywhere else.

For the 4 corner squares, this means excluding 4 squares each; for the further $4 \cdot 6$ edge squares, 6 squares each; and for the 36 inner squares, 9 squares each from the possible 64. The number of valid placements is thus

$$
4 \cdot(64-4)+24 \cdot(64-6)+36 \cdot(64-9)=3612
$$

If the kings are indistinguishable, then the number of possible placements is half of this, i.e., 1806.