Interesting Hat Problem

This is a logic puzzle that I really enjoy, but it doesn't translate well to the "input your answer" format. That aside, I thought people on Brilliant might enjoy it too.

The logic mastermind has taken 100 mathematicians hostage. The mathematicians are told that they will each be given a hat out of 100 possible colors (the mathematicians are told the possible colors in advance). The mathematicians must then simultaneously guess their own hat colors. (Note that they all must guess; no one may abstain.)

The mathematicians are able to see the other 99 hats but not their own. Moreover, they are not allowed to communicate any information to the other mathematicians (e.g. by physical movement or change in tone) on pain of death - they must only guess.

The mathematicians will be set free if EXACTLY ONE mathematician guesses correctly. Unfortunately, they are told that their hat colors may not necessarily be distinct - for example, while there are 100 possible hat colors, they might all be given yellow hats.

The mathematicians are allowed to devise a strategy before they don their hats and guess. Can they succeed and win their freedom? If so, what is their strategy?

This problem is not original.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
list

Note: you must add a full line of space before and after lists for them to show up correctly

paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

Math

Appears as

Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

And there are several other hat problems like this and this and this. That's amazing! Thanks Maggie!

Calvin Lin Staff - 5 years, 11 months ago

I have a solution for this problem. But I don't see how the mathematicians can do it without communicating. The solution involves assigning numbers, which involves communication. I suppose that means I don't have a solution.

Jihoon Kang - 5 years, 10 months ago

The mathematicians are allowed to devise a strategy before putting on hats, so they may communicate at that time. They can't communicate once the hats are on.

Maggie Miller - 5 years, 10 months ago

Okay! Thanks for posting this problem, it was interesting

There are 100 caps, that might be distinctively colored or similarly. Since there is no observed pattern of slaying the riddle, nor is there any mention of the colours, its practically impossible to find the right colour of the hat unless guesses are allowed.

Oppositions to my answer are most welcome! :)

Vanya . - 5 years, 3 months ago

Well they are each guessing - no one certainly knows their own hat color, but they do know that exactly one person will guess correctly. There is a solution :)

Maggie Miller - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$