9.6. In a day, 1000 dwarfs wearing red or blue hats meet each other in pairs. Dwarfs wearing red hats tell lies, while those wearing blue hats tell the truth. Each dwarf may change the color of their hat several times (i.e., red to blue, blue to red). It is known that when any two dwarfs meet, they both say that the other is wearing a red hat. Find the minimum total number of hat changes in a day.
9.6. Clearly, when two dwarfs meet, they say that the other is wearing a red hat if and only if their hat colors are different.

Thus, in a day, if three dwarfs never change the color of their hats, then two of them must have the same hat color, and they cannot both say that the other is wearing a red hat when they meet. Therefore, at least 998 dwarfs must have changed the color of their hats, and the total number of hat color changes is greater than or equal to 998.
Below is an explanation that 998 times is achievable.
Let the dwarfs be $1, 2, \cdots, 1000$. Initially, 1 wears a blue hat, and $2, 3, \cdots, 1000$ wear red hats. After 1 meets $2, 3, \cdots, 1000$, 2 changes the color of his hat to blue and then meets 3, $4, \cdots, 1000$, $\cdots \cdots$ 999 changes the color of his hat to blue and then meets 1000, for a total of 998 hat color changes.