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# Dots, not hats...

## Question

So bonanova's most recent puzzle, Another hat problem, reminds me of one of my favorite riddles that my dad always told me. It's very similar, but slightly different, and I don't think I've posted it before, so here it goes (he was a much better story teller than I, so pardon my narrative):

A king put out a decree saying he will give his entire kingdom to the person who could solve all of his riddles, thereby proving to be the smartest person in the land. Obviously many showed up, but after all of the trials, only 3 people remained. The king told them to get some rest for tomorrow they would face their final challenge. Only one person would remain after it, the other two would be killed. That threat didn't dissuade them, and they all went to bed.

When they woke up, they found themselves all in a room together. Any and all reflective surfaces were covered, and there were armed guards surrounding them. Before any of them could speak, the king told them their challenge:

"While you were sleeping, we painted either a red dot or a blue dot on each of your foreheads. You are not allowed to speak or communicate with each other in any way shape or form. Anyone who does so will be killed immediately. Now, raise your hand if you see a blue dot on someone else's head."

The three people looked at each other and all three raised their hands.

"Good. Keep your hands raised. You may only lower your hand when you can tell me what color dot is on your own forehead. If you lower your hand and do not answer, you will be killed. if you answer incorrectly, you will be killed. If you answer correctly, you will gain all of my wealth, and the other two will be killed. Good luck."

The three looked at each other...what seemed like hours went by with no one answering. You could see they were all straining themselves, trying to figure out how they could know what color dot they had. Their arms started getting tired...

Finally, one of the men got a big smirk on his face. He did some final double checking of his logic and slowly lowered his hand.

"The spot on my head is ... "

Given that the man successfully answered the question, and it clearly wasn't a "guess", what color dot was on his head, and how did he know?

## Recommended Posts

• 1

blue.

Rationale for observer

* Everybody can see at least 1 blue dot (as they all raised their hands when asked)

* At least 2 of the contestants are blue or else 1 of them would see 2 reds and not raise his hand

Rationale for the contestant who can see 2 blue dots (all contestants but I have not proven that yet)

* If I had a red dot on my head the other two contestants would have seen 1 blue (their opposite) and 1 red (me) and known that their opposite would not have raised their hand if they saw two reds. Thus they should have lowered their hand by now and I should be dead

* As their hands are tired and show no sign of lowering, I must not have a red dot on my forehead

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