Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le M$ and $1\le c\le d\le N$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready.

In terms of $M$ and $N$, what is the minimum $Q$ required to gaurantee to determine the location of the treasure?
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   - We need to ensure that each cell in the grid is uniquely identified by the set of detectors.
   - We can achieve this by setting up detectors to scan specific rows and columns.

4. **Vertical Detectors:**
   - We set up detectors to scan vertical strips of the grid.
   - For \( 1 \leq k \leq \lceil \frac{m}{2} \rceil \), we query the subgrid \([k, k + \lfloor \frac{m}{2} \rfloor - 1] \times [1, n]\).
   - This ensures that each column is uniquely identified by the set of vertical detectors.

5. **Horizontal Detectors:**
   - Similarly, we set up detectors to scan horizontal strips of the grid.
   - For \( 1 \leq k \leq \lceil \frac{n}{2} \rceil \), we query the subgrid \([1, m] \times [k, k + \lfloor \frac{n}{2} \rfloor - 1]\).
   - This ensures that each row is uniquely identified by the set of horizontal detectors.

6. **Combining the Detectors:**
   - By combining the vertical and horizontal detectors, we can uniquely determine the cell containing the treasure.
   - The total number of detectors required is \( \lceil \frac{m}{2} \rceil + \lceil \frac{n}{2} \rceil \).

7. **Bounding the Minimum Number of Detectors:**
   - We need to show that this number of detectors is both necessary and sufficient.
   - Consider the edges of the grid and the requirement that each cell must be distinguishable from its neighbors.
   - Each rectangle (detector) can overlap at most 2 edges, and we need to cover all edges to ensure unique identification.
   - This leads to the conclusion that at least \( \lceil \frac{m+n}{2} \rceil \) detectors are required.

8. **Handling Special Cases:**
   - If both \( m \) and \( n \) are odd, we need to ensure that every corner cell is covered by some rectangle.
   - This requires additional consideration, but the argument still holds that we need at least \( \lceil \frac{m}{2} \rceil + \lceil \frac{n}{2} \rceil \) detectors.

9. **Conclusion:**
   - The minimum number of detectors required to guarantee finding the treasure is \( \lceil \frac{m}{2} \rceil + \lceil \frac{n}{2} \rceil \).

The final answer is \( \boxed{ \lceil \frac{m}{2} \rceil + \lceil \frac{n}{2} \rceil } \).