One day in Rohan, King Théoden discovered a chest that had belonged to his father, Thengel. Inside, there were five helmets. Three of them were plated with silver, and two were plated with gold. Théoden called in Éowyn, Éomer, and Theodred, and showed them the helmets. He asked them to arrange themselves in a line. Theodred, at the back of the line, was able to see both of his cousins, Éomer and Éowyn. But Éomer, in the middle, could see only his sister Éowyn. Éowyn was in front, and could not see either of the men.
Théoden then proceeded to place a helmet, at random, on each of the youths' heads. Afterward, he asked Theodred whether he could tell what color his helmet was: gold or silver. Theodred could not tell. Théoden then asked Éomer whether he could tell what color helmet he was wearing. He admitted that he could not tell either. Finally, Théoden asked Éowyn whether she could tell which helmet she was wearing. She said yes, and named the color. She was correct.
What color helmet was Éowyn wearing, and how did she know?
A Dwarven Dilemma
Once, a party of thirteen Dwarves set out on a quest to kill a dragon. It was agreed between them that Thorin, their leader, would claim half of the riches, and the rest would be divided evenly among the remaining party members. Unfortunately, the plan did not entirely pan out. The dragon was destroyed, but soon neighboring communities of Elves and Men began demanding gold for themselves; and to make matters more confusing, Thorin and his two nephews were killed in battle. The result was a terrible disorder in the matter of distributing wealth.
Ultimately, each of the ten surviving Dwarves received a certain number of pounds of the dragon's gold; but now that they are headed home, there is no way for any of them to know how much gold was distributed all together, or whether the portions they received were equal. None of them is willing to tell the others (or anyone else) how much gold he received, for fear of being robbed. But each Dwarf wants very badly to know whether he has been cheated. Using only a bit of parchment and a quill, how can the ten Dwarves calculate the average number of pounds of gold they received, without divulging the amounts of their individual portions?