On an $n \times k$ chessboard ( $n, k \geq 2$ ), some figures are placed. (A figure can only stand on a square, and at most one figure can stand on each square.) We know that on the 4 squares where any 2 rows and 2 columns intersect, there is always exactly 1 or 3 figures. What are the limits within which the number of figures can vary?
We will show that the condition of the problem can only be satisfied when $k=n=2$. It is obvious that in this case, either 1 or 3 dolls can be placed.

For example, if $n \geq 3$, select three rows and two columns on the chessboard.

On the resulting $3 \times 2$ section, count how many figures are in each row. Among the three numbers obtained, there are two with the same parity. If we consider these two rows and the two columns, the number of dolls in the $2 \times 2$ section is even, so it cannot be 1 or 3.

Similarly, we reach a contradiction when $k \geq 3$.

Therefore, the sought bounds are 1 and 3, and the only chessboard that meets the conditions is the $2 \times 2$ one.

Based on the work of Gergely Horváth (Fonyód, Bálint Magyar Secondary School, 8th grade)