How many fields are there on an infinitely large chessboard that can be reached from a selected field with the king in $k$ steps, but not in fewer? (In chess, the king can move to the 8 fields that share either a side or a corner with the field the king is on.)
We place a coordinate system on the chessboard such that its origin is on the square from which the king starts, and its axes are parallel to the appropriate sides of the square grid. The king changes at least one of its coordinates by one with each move. We will show that the point with coordinates $(i ; j)$ can be reached by the king in a maximum of $(|i|,|j|)$ moves, but not in fewer. (Here, max $(|i|,|j|)$ denotes the larger of the numbers $|i|$ and $|j|$.) Indeed, if $|i|>|j|$, then the king should move $|j|$ times such that with each move, the first coordinate increases by one if $i>0$, or decreases by one if $i<0$. The second coordinate should change similarly depending on the sign of $j$. Then, it should make $|i|-|j|$ more moves such that the first coordinate changes as before, while the second remains unchanged. Thus, it reaches the point $(i ; j)$ in a total of $|i|$ moves. It cannot reach this point in fewer moves because with each move, the first coordinate can change by at most one. The situation is similar if $|i|<|j|$.

Therefore, the squares that can be reached in $k$ moves but not in fewer are those whose at least one coordinate has an absolute value of $k$. These are located on the sides of a $(2 k+1)$-sided square, and thus their number is exactly $8 k$.