QOD #10 — Solution

Solution

Since all three raised their hands, either one has a yellow sticker, or none have a yellow sticker. If two were to have yellow stickers, then one wouldn't have their hand raised. So Adama reasons this way. "I can see two read stickers, so maybe mine is yellow. But if it were yellow, then Peter or Jeff would know that theirs were red because there can be only one yellow sticker. Since they haven't figured it out yet, and Peter is pretty smart, mine must be red."

Adama could have also figured it out if there were four people playing. In that case, there can be, at most, 2 with yellow stickers for everyone's hand to be up. She would then reason, "Suppose I have a yellow sticker. The next smartest student would see my yellow sticker and wonder whether she has one. But she would kno that, if she did, the other 2 would know instantly that theirs were red as there can only be 2 yellow stickers and they were seeing them both. Since the next smartest student hasn't made that conclusion yet, she must not see a yellow sticker on me, so mine must be red."

In fact, this reasoning can continue indefinitely. If there are 5 people playing, the smartest can reason that the remaining four are in the same situation as the previous scenario, and thus can figure it out. Through induction, you can prove this is the case for any number of players.

Of course, the real answer is that Adama saw her card reflected in Jeff's glasses, but this one is ok, too.

Comments

This QOD as an absolute gem. I tell this one all the time
The induction in this case breaks down because no one can reasonably assume that four other people have thought to a certain level before he/she can say, "I have a red sticker!"
When people understand this QOD (or even if they don't), I like pairing it with the oracle problem (which I've modified slightly):
Imagine a mythical island inhabited currently by 70 green-eyed and 30 blue-eyed creatures. The blue-eyed creatures are hunted by the green-eyed creatures. However, one day, an oracle descends from heaven and gives these creatures the following commands:
1. Thou shalt not commit murder.
2. If you have blue eyes, and you find out by any way, you must kill yourself at midnight on the day you find out. (Apparently, heaven doesn't really like blue-eyed creatures either.)
3. No one may speak of eye color or in any way hint that someone should kill himself/herself. Punishment consists of death by stoning. (No one likes dying, so this is to prevent false accusations. And stoning is not murder; it's law enforcement.)
4. All reflective surfaces are hereby banished. (Don't ask how that happens.)
Now, like the average God-fearing creature, they follow these laws fanatically. So fanatically in fact, that when the oracle repeated the laws, the creatures stoned him death on the spot as per law #3. (Oh well.) Anyway, these creatures are really smart and logical to an extreme, and the question is, "What happens next?"
- What is the fate of all the blue-eyed creatures that are doomed to die?
- If any of the blue-eyed creatures die, how many and when?
"Adama saw her card"? What about her sticker? =) Weird slip

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