4. 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: butter, chocolate, or fruit. First, Snow White asked those who love butter ice cream to raise their hands, and everyone raised their hands, then those who love chocolate ice cream - and half of the dwarfs raised their hands, then those who love fruit ice cream - and only one dwarf raised their hand. How many of the dwarfs are truthful?
# Answer. 4.

Solution. The gnomes who always tell the truth raised their hands once, while the gnomes who always lie raised their hands twice. In total, 16 hands were raised (10+5+1). If all the gnomes had told the truth, 10 hands would have been raised. If one truthful gnome is replaced by one liar, the number of raised hands increases by 1. Since 6 "extra" hands were raised, 6 gnomes lied, and 4 told the truth.