<p>Line of Geniuses</p>

13.

Line of Geniuses

This question is from my geometry class, it only involves logic not probability. Please help me solve this! It is only middle school, so I don't think it would be TOO hard. Anyways, here it is. Reply as soon as possible, please.
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Three geniuses stand in a line (one behind the other). Each can see only to the front, so the rear person can see the middle and the front, the middle person can see the front, and the genius in the front cannot see anyone.

You have five hats. Two are white, and three are red. You blindfold the three geniuses, who are utterly truthful, and put a hat--at random--on the head of each. Then you hide the other two hats and remove the blindfolds.

You then ask each genius to name the color of his hat (which he cannot see).

The rear one says, "I don't know." The middle one says, "I don't know." Then the front one says, "I know."

What color hat is the front genius wearing?????? How does he know with 100% certainty what color hat it is?????

Submitted by Akira970 · Added 11 January 2008 · Updated 5 July 2026

Show Solution

Solution:

The front genius is wearing a red hat. The rear genius, seeing the hats of the two in front, would only say 'I don't know' if he saw one red and one white hat, as he could not determine his own hat color. The middle genius, knowing the rear genius saw two hats and still said 'I don't know', concludes that he must be wearing a red hat. Therefore, the front genius deduces that he must also be wearing a red hat.

Comments (2)

toothslooth ★ Solved 8 February 2008

The front genius is wearing a red hat. Here's how he knows...
The front 2 were NOT both wearing white or the back 1 would know he was red. So the front 2 must either be wearing both red or 1 red 1 white. If the front was wearing white the middle 1 MUST be wearing red. As he does not know, the front 1 MUST THEREFORE be wearing RED.

Enjoy!

mdbbarilla 11 August 2009

list the possibilities.

then cancel out.

man 1: red / white
man 2: red / white
man 3: red / white

if man 1 : red
then man 2 : red
man 3 will not be able to know his hat because there is more than one possibility
then man 2: white
man 3 will still not know
but if man 1: white
man 2 : white
man 3 will know that his hat is red.

if man 2 was red man3 will not know his hat

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