The Impossible Bet

A riddle.

Imagine 100 people at a house, with 3 rooms. They are all betting $100 each. Each box has a unique piece of paper with a number ranging from 1-100.

Every one is assigned a random number. If all 100 people find their corresponding number (e.g Person 52 finds Box 52), all get their dollars back.

You can search 50 boxes only and you cannot report your findings to your teammates, by shouting, or marking them.

If at least one person fails, everyone will fail and the dealer keeps all the dollars.

I know this is supposed to be a problem but the answers to this question needs to come in a set of paragraphs.

#Logic

Easy Math Editor

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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

the answer to WHICH question?

What's the purpose of there being three rooms at the house?

Also, about the people being assigned a "random" number... Can the assigned numbers overlap? -.-

C . - 2 years, 9 months ago

I think this has been asked before in Brilliant. Here's a reference to the best strategy. You can win about 31% of the time.

Patrick Corn - 2 years, 9 months ago

Nice problem... too bad the OP left out so many crucial details. :-|

C . - 2 years, 9 months ago

I think it is related to This TED Video

Mohammad Farhat - 2 years, 8 months ago

There is a great video on this by Minute Physics.

Explanation - Randomly picking 50 boxes will reduce your chances to $0.5^{100}$ . This as you can see, is even lesser than my IQ(and that's saying a lot). There is no technique yet, that guarantees a $100\%$ win, but the best one at least gives you a $31\%$ chance of winning.

Steps -

Start with the box number with your bill number on it (For example - If your bill is 52, then got to box 52)
If your bill is there, Awesome! If not, take that bill's number, and go to that box.
Continue step 2. This will create a chain until you reach your bill. A 50+ box chain is not very common so you will find your bill easy.
If all 100 use this strategy then there is a $31\%$ that all of you will win!
Take 10,000$ home, or waste a dollar, its a 31 : 69 chance :)

A Former Brilliant Member - 10 months, 3 weeks ago

I'm 2 years late to this :)

A Former Brilliant Member - 10 months, 3 weeks ago

Okay, Before we start we need to do something I learned elementary school. Let’s Break down the question: Imagine 100 people at a house - There are 100 people participating in the bet

with 3 rooms. - This is useless information

They are all betting $100 each. - So, we now how much they are betting and how much the host can gain

Each box has a unique piece of paper with a number ranging from 1-100. - 100 participants...

Every one is assigned a random number. If all 100 people find their corresponding number (e.g Person 52 finds Box 52), all get their dollars back. - So, if everyone matches their box everyone gets their money back.

You can search 50 boxes only and you cannot report your findings to your teammates, by shouting, or marking them. - You can search half the boxes - No communication (i.e. not telling your friend that your box is in the corner)

If at least one person fails, everyone will fail and the dealer keeps all the dollars. - 1 person fails, host wins.

Let’s break it down. This is just probability. What is the chance everyone gets their money back? Let’s say I am person 38. I can see 50 of the 100 boxes in order to attempt to get my box. So, I have a 1/2 chance of finding my box. There are 100 people that have a 1/2 chance of winning. So, the probaility goes to 1/200. An equivalent to 0.5%.

So, long story short the host is probably going to earn $10,000 that night. But, if you are a player, good luck, you will need it.

Trevor Greenlee - 2 years, 9 months ago

I think you want $(1/2)^{100},$ a much smaller probability.

Patrick Corn - 2 years, 9 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}`	$\boxed{123}$