1. Each of the 10 dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?
Solution. The dwarves who always tell the truth raised their hand once, and the dwarves who always lie raised their hand twice. A total of $16=10+5+1$ hands were raised. If all the dwarves told the truth, 10 hands would be raised. If one truthful gnome is replaced by one liar, then the number of hands raised will increase by 1 . Since six "extra" hands were raised, six gnomes lied, and four told the truth.
Answer: 4 .