Example 5 A $98 \times 98$ chessboard is displayed on a computer screen, with its squares colored like a chessboard. You are allowed to select any rectangle with the mouse (the sides of the rectangle must lie on the grid lines), and then click the mouse button, which will change the color of each square in the rectangle (black to white, white to black). Find the minimum number of operations required to make all the squares on the chessboard the same color.
(1998, United States of America Mathematical Olympiad)
Analysis: If we consider "all cells of the same color," it would be very difficult (because too many objects can obscure the essential relationships), leading to the idea of adopting a local thinking strategy.

First local thinking: Transform "all cells of the same color" into "any two adjacent cells of the same color." Thus, we only need to focus on some adjacent cells. For convenience, we call two adjacent cells a "pair."

Under what circumstances will a pair be of the same color? This is clearly equivalent to the parity of the number of flips of the two cells being different. Therefore, we define the sum of the number of flips of two cells as the "characteristic value" of the pair. Then, we only need all pairs to have an odd characteristic value. This way, the study of the problem's nature becomes a numerical study: we only need the characteristic values of all pairs to be odd after several operations.
Second local thinking: Note that
the number of operations
$\geqslant$ the sum of the number of times the characteristic values of all pairs change $\div$
the number of times they are counted repeatedly.
If we consider all pairs, some pairs are counted too many times, making the estimation difficult, so we should appropriately select some pairs to study: only consider the pairs around the edges of the board (called "edge pairs").

Third local thinking: Considering all edge pairs also encounters difficulties; we further divide them into two parts: separately examine all vertical and horizontal edge pairs. This solves the problem.

Solution: Call two adjacent cells a pair, and the sum of the number of flips of the two cells in the pair is called the characteristic value of the pair. Since the characteristic value of all pairs is initially 0 (even), after several operations, all cells on the board will be of the same color. Therefore, the parity of the characteristic values of all pairs must change an odd number of times (at least once) during the operations.
A $98 \times 98$ board has $2 \times 97$ vertical edge pairs, and the parity of their characteristic values must change at least $2 \times 97$ times.

Consider each operation; to change the parity of the characteristic value of a pair, the pair must have exactly one cell within the selected rectangle. However, for any rectangle, at most 4 cells outside the rectangle are adjacent to cells inside the rectangle, thus changing the parity of at most 4 edge pairs. Therefore, the number of operations to change the characteristic values of vertical edge pairs (called vertical operations) is at least $\left[\frac{2 \times 97}{4}\right]+1=49$ times.

By symmetry, the number of operations to change the characteristic values of horizontal edge pairs (called horizontal operations) is also at least 49 times.

Thus, to make all cells on the board the same color, at least 98 operations are required.

Next, we need to construct 98 operations to make all cells on the board the same color. Consider the condition for equality. If a total of 98 operations are performed, then from the above discussion, it must be that there are 49 horizontal operations and 49 vertical operations, and each horizontal operation (or vertical operation) changes the parity of 4 horizontal pairs (or vertical pairs). This leads to a specific method of operation:

First, change the color of the odd-numbered rows (the first operation changes 2 vertical pairs, and the rest change 4 vertical pairs), then change the color of the even-numbered columns (the last operation changes 2 horizontal pairs, and the rest change 4 horizontal pairs), making all cells the same color, with a total of 98 operations.
In summary, the minimum number of operations is 98.