logical deduction - Who has mud on their forehead?

This is probably my favorite puzzle. It uses logic, but it's not one of those "logic table" puzzles which i dislike immensely. What is this type of puzzle called and are there more like it that are not just variants?

Three children are playing on the beach and all three have mud on their foreheads. An old man walks by and asks them, “After looking at both of your friends foreheads, raise your hand if one or both of them have mud on their foreheads.“ All three children look at both their friends and raise their hands.

He then asks, “For a dollar, without touching your own forehead or looking in a mirror or anything similar, tell me if you have mud on your own forehead, and how you know that.” All three children are bewildered. Finally, one raises his hand and says, “i have mud on my forehead.” He explains how he knew this and receives the dollar.

What was his reasoning?

Note: The old man and the children are honest, able to employ logic, and want the dollar.

Answer

As with klm123, I don't know of a specific name for this type of problem (oracle riddle?). However, I've usually seen this question in the context of an island with people of two eye colors and an oracle.

In this case, the old man is the oracle, whether or not the children have mud on their face corresponds to whether the person has brown or blue eyes, and raising the hand is equivalent to leaving the island.

This is a vaguer retelling in a more natural setting, however. Only one child raises his hand, even though all of them should be able to reach the conclusion at the same time. Likewise, the oracle problem has a defined amount of time that has to pass, rather than waiting for some arbitrary "Oh, he didn't react fast enough, which gives me information" point.

Gilles' explanation of the oracle problem is a great answer, and I'll quote a bit of it here:

Suppose that only Alice has blue eyes. Before day 0 [when the oracle speaks], she never knew anyone with blue eyes. On day 0, she learns that someone has blue eyes; since nobody else does, it has to be her and only her, so she takes the ferry that night.

Now suppose that only Alice and Bill have blue eyes. Before day 0, Bill already knew that there was someone with blue eyes, but he did not know that Alice knew. If Bill had had green eyes, Alice would have been the only blue-eyed person and would not have known. On the first night after the guru, Alice doesn't leave; this tells Bill that Alice did not know the color of her eyes, so learn that she was the only blue-eyed person. Since Bill knows that either Alice is the only blue-eyed person or Bill and Alice are the only two, Bill now knows that both he and Alice have blue eyes.