At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner?
1. **Understanding the Problem:**
   - A knight on a chessboard moves in an "L" shape: two squares in one direction and one square perpendicular, or one square in one direction and two squares perpendicular.
   - We need to determine the minimum number of moves for a knight to travel from one corner of the chessboard (e.g., $(1,1)$) to the opposite corner (e.g., $(8,8)$).

2. **Analyzing the Knight's Move:**
   - Each move of the knight changes its position by $(\pm 2, \pm 1)$ or $(\pm 1, \pm 2)$.
   - This means each move changes the sum of the coordinates $(x + y)$ by $\pm 3$ or $\pm 1$.

3. **Initial and Final Positions:**
   - Starting position: $(1,1)$, so $x + y = 2$.
   - Ending position: $(8,8)$, so $x + y = 16$.
   - We need to increase the sum from $2$ to $16$, which is an increase of $14$.

4. **Calculating the Minimum Number of Moves:**
   - To achieve a total increase of $14$, we need to consider the possible changes in the sum $(x + y)$ per move.
   - The smallest number of moves to get a sum at least as big as $14$ is calculated by considering the maximum change per move, which is $3$.
   - If each move added $3$, then the minimum number of moves would be $\lceil \frac{14}{3} \rceil = 5$ moves.
   - However, since $14$ is not a multiple of $3$, we need at least one additional move to adjust the sum to exactly $14$.

5. **Adjusting for Modulo 3:**
   - The sum $14$ modulo $3$ is $2$.
   - We need to ensure that the total number of moves results in a sum that is congruent to $2$ modulo $3$.
   - Since $5$ moves would result in a sum that is a multiple of $3$, we need one more move to adjust the sum to $14$.

6. **Conclusion:**
   - Therefore, the minimum number of moves required is $6$.

The final answer is $\boxed{6}$.