10. (10 points) In the Land of Dwarfs, there are 2011 dwarfs, each of whom wears either a red hat or a blue hat. When a dwarf wears a red hat, they tell the truth; when they wear a blue hat, they can lie; and they can change the color of their hat at any time. One day, every two of them met exactly once, and each said that the other was wearing a blue hat. How many times did they change the color of their hats at least? $\qquad$
10. (10 points) In the Land of Little People, there are 2011 dwarfs, each of whom wears either a red hat or a blue hat. When a dwarf wears a red hat, they tell the truth; when they wear a blue hat, they can lie; and they can change the color of their hat at any time. One day, each pair of them met exactly once, and they all said that the other was wearing a blue hat. Therefore, they changed the color of their hats at least 2009 times in total that day.
【Solution】Solution: $2011-2=2009$ (times)
Answer: They changed the color of their hats at least 2009 times in total that day.
Therefore, the answer is: 2009.