Let $m,n > 2$ be even integers. Consider a board of size $m \times n$ whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to find the parity of the number of adjacent cell-pairs whose colors are different. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)
1. **Define the Problem and Setup:**
   We are given an \(m \times n\) board where \(m\) and \(n\) are even integers greater than 2. Each cell on the board is colored either black or white. The Guesser can ask the Oracle about the color relationship (same or different) between two adjacent cells. The goal is to determine the parity (even or odd) of the number of adjacent cell-pairs with different colors.

2. **Score Definition:**
   Assign a score to each cell: 1 if the cell is black and 0 if the cell is white. For any two adjacent cells, the Oracle provides the parity of the sum of their scores, i.e., \(u + v \pmod{2}\), where \(u\) and \(v\) are the scores of the two adjacent cells.

3. **Objective:**
   The Guesser needs to compute the parity of the sum of the scores of all adjacent cell-pairs. This is equivalent to finding the parity of the sum of the scores of all cells with an odd number of non-boundary (non-red) edges.

4. **Partitioning the Board:**
   The cells with an odd number of non-boundary edges can be partitioned into disjoint non-adjacent rectangles:
   - Two \(1 \times (n-2)\) rectangles along the top and bottom rows.
   - Two \(1 \times (m-2)\) rectangles along the left and right columns.

5. **Domino Partitioning:**
   Since \(n-2\) and \(m-2\) are even, each rectangle can be partitioned into dominoes (each domino covers two cells). The Guesser can ask the Oracle about the sum of scores of cells in each domino.

6. **Number of Inquiries:**
   Each \(1 \times (n-2)\) rectangle has \((n-2)/2\) dominoes, and each \(1 \times (m-2)\) rectangle has \((m-2)/2\) dominoes. Therefore, the total number of dominoes (and thus the number of inquiries) is:
   \[
   \frac{(m-2) + (n-2) + (m-2) + (n-2)}{2} = m + n - 4
   \]

7. **Optimality:**
   If fewer than \(m + n - 4\) inquiries are made, there will be at least one cell in the four rectangles for which the Guesser has no information. Consequently, the score of that cell could be anything, making it impossible to determine the parity of the sum of the scores.

8. **Conclusion:**
   The minimum number of inquiries required to guarantee finding the parity of the number of adjacent cell-pairs with different colors is \(m + n - 4\).

The final answer is \(\boxed{m + n - 4}\).