Tricky twist on hat problem!

Logic Level 3

There are $100$ people standing in a circle. Each person is given one hat to wear. A hat is either colored $\color{royalblue}{\text{blue}}$ or $\color{#D61F06}{\text{red}}$ . A person cannot see his own hat color, but he can see the colors of the hats of all other people standing in the circle.

At the same time, each person is asked to guess their own hat color and all of them have to answer at the same time . They only win if all $100$ people guess their own hat color correctly.

The $100$ people are allowed to devise a strategy before they are given their hats. They agree on a strategy that maximizes their chance of winning, $P$ . If $P$ can be written as $P=\frac{m}{k}$ with coprime positive integers $m$ and $k$ enter your answer as the last $5$ digits of the sum $m+k$ .

Note: If you need a calculator to write your result in the given expression, feel free to use one ;)

The answer is 3.

1 solution

Simon Kaib
Feb 28, 2019

One strategy that maximizes the chance of winning is the following:

If you see an even amount of $\color{royalblue}{\text{blue}}$ hats, guess $\color{royalblue}{\text{blue}}$ . If not, guess $\color{#D61F06}{\text{red}}$ .

Note that every person that wears a $\color{royalblue}{\text{blue}}$ hat gives the same answer, as they see the same amount of $\color{royalblue}{\text{blue}}$ hats. Someone who wears a $\color{#D61F06}{\text{red}}$ hat sees $1$ $\color{royalblue}{\text{blue}}$ hat more than a person who wears a $\color{royalblue}{\text{blue}}$ hat and therefore guesses the other answer. Thus there are only two equally likely possible outcomes left: all people with $\color{royalblue}{\text{blue}}$ hats guess $\color{royalblue}{\text{blue}}$ and all people with $\color{#D61F06}{\text{red}}$ hats guess $\color{#D61F06}{\text{red}}$ , or all people with $\color{royalblue}{\text{blue}}$ hats guess $\color{#D61F06}{\text{red}}$ and all people with $\color{#D61F06}{\text{red}}$ hats guess $\color{royalblue}{\text{blue}}$ .

This yields a probability of $\frac{1}{2}$ and therefore the answer $1+2=\boxed{3}$ .

I have actually asked myself a similiar question before. Wonderful solution !

Robin Wal - 2 years, 3 months ago

very nice solution, tho how do we know this strategy maximizes the chance of winning?

Mehdi K. - 2 years, 3 months ago

If there was a strategy with a higher chance of winning, the chance would be greater than $\frac{1}{2}$ . However, this would mean that the average probability that someone guesses their own hat color correctly is greater than $\frac{1}{2}$ , which is impossible as every single guess is a coin flip. Whenever someone guesses their color, it is equally likely to be the other color. You cannot gain any information about your own hat color from the other hats.

Simon Kaib - 2 years, 3 months ago

does this mean if there are 3 colors instead of 2, there is a strategy that has a $\frac{1}{3}$ chance of winning?

Mehdi K. - 2 years, 3 months ago

I think there is a typo 'red hat sees 1 blue hat less' it should be more. Excellent problem and solution.

Mr. India - 2 years, 3 months ago

I corrected it. Thanks!

Simon Kaib - 2 years, 3 months ago

Tricky twist on hat problem!

The answer is 3.

1 solution

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