Problem 2169

Answer: 160.
In our torus each cell is determined by coordinates $(i, j)$, $1 \le i, j \le 20$, the two cells being neighbours if one of their coordinates is the same, and the others differ by $\pm 1 \bmod 20$.

Example. Select the following 160 rectangles
$$(a-1,b); (a-2,b); (a-3,b); (a-4,b) \pmod{20}, \quad \text{where } 5 \mid (a+b), \quad \text{and} \\ (a,b-1); (a,b-2); (a,b-3); (a,b-4) \pmod{20}, \quad \text{where } 5 \mid (a+b-1).$$
If the sapper says that the treasure belongs to only one of the rectangles, then the cell is uniquely determined, because each rectangle contains the unique cell not covered by the other rectangles. If the sapper says that the treasure belongs to two of rectangles then the treasure is in their intersection cell.

Estimation. Suppose that we select 159 rectangles only. It is clear that the torus is fully covered by the rectangles except at most one cell. Therefore at least $399 \cdot 2 - 159 \cdot 4 = 162$ is covered by only one rectangle, and hence two of these cells belong to the same rectangle. If the rectangle contains the treasure we can not distinguish on which of these cells it is hidden.