Example 6 Let $n$ be a given positive integer, $6 \mid n$. In an $n \times n$ chessboard, each square is filled with a positive integer, and the numbers in the squares of the $i$-th row from left to right are $(i-1) n+1,(i-1) n+2, \cdots,(i-1) n+n$. Now, take any 2 adjacent (sharing a common edge) squares, add 1 to one of the numbers, and add 2 to the other, which is called one operation. Question: What is the minimum number of operations required to make all the numbers on the chessboard equal? (Original problem)
Let the sum of the numbers on the chessboard be denoted as \( S \).
Obviously, when all the numbers on the chessboard are equal, each number is at least \( n^2 \), so the sum of the numbers on the chessboard \( S' \geq n^2 \cdot n^2 = n^4 \). Each operation increases \( S \) by 3, and the initial sum of the numbers on the chessboard \( S_0 = 1 + 2 + 3 + \cdots + n^2 = \frac{n^2(n^2 + 1)}{2} \). Therefore, when the numbers on the chessboard are all equal, \( S \) must increase by at least \( n^4 - \frac{n^2(n^2 + 1)}{2} = \frac{n^2(n^2 - 1)}{2} \). Thus, the number of operations required is at least \( \frac{1}{3} \cdot \frac{n^2(n^2 - 1)}{2} = \frac{n^2(n^2 - 1)}{6} \).

Next, we prove that it is possible to perform \( \frac{n^2(n^2 - 1)}{6} \) operations to make all the numbers on the chessboard equal. This is equivalent to proving that it is possible to perform a certain number of operations to make all the numbers on the chessboard equal to \( n^2 \).

Divide each row of the chessboard from left to right into groups of 3 consecutive cells, resulting in \( \frac{n}{3} \) groups, which we label as the first group, the second group, ..., the \( \frac{n}{3} \)th group.

First, note that for any 3 consecutive cells, it is possible to perform 2 operations to increase each number by 2. We call these two operations a large operation \( A: (a, b, c) \rightarrow (a+2, b+1, c) \rightarrow (a+2, b+2, c+2) \). Now, perform \( \frac{n}{2} \) large operations \( A \) on each group in a row, so that the numbers in that row all increase by \( n \). Therefore, for each row, it is possible to perform a certain number of operations to make it identical to the \( n \)th row.

Next, note that for any 2 consecutive cells, it is possible to perform 2 operations to increase each number by 3. We call these two operations a large operation \( B: (a, b) \rightarrow (a+2, b+1) \rightarrow (a+3, b+3) \). Now, assume that each row of the chessboard is identical to the \( n \)th row. Consider the \( i \)th and \( (i+1) \)th rows, and for the \( j \)th group in these 2 rows, perform a large operation \( B \) on the 2 cells in the same column, so that the numbers in the \( j \)th group of these 2 rows all increase by 3. Since the numbers in the same row but different groups differ by multiples of 3, it is possible to perform similar operations on each group of these 2 rows a certain number of times to make all the groups in these 2 rows identical to the \( \frac{n}{3} \)th group, which is \( (n^2 - 2, n^2 - 1, n^2) \). This group can be transformed into \( (n^2, n^2, n^2) \) with one operation, so these 2 rows can all become \( n^2 \). Perform such operations on every 2 consecutive rows, and all the rows will become \( n^2 \).

In conclusion, the minimum number of operations required is \( \frac{n^2(n^2 - 1)}{6} \).