Riddles

Here are some riddles, I didn't come up with them but I don't know the original authors. Let me know if you find any copyright infringing content on this page and I will remove it. I won't post the solutions here to avoid spoilers, but I will be happy to share them via. email or otherwise. I did post some hints so don't skim the page. If you have more riddles I would also love to hear them in the comments or via. email etc. Thank you.

100 Prisoners

With hats

100 prisoners are told that the next day they will be lined up in the prison yard. Each prisoner will wear a white or black hat, and all will be facing east so that they can see the hats of the prisoners in front of them. Starting from behind an executioner will ask them one at a time to guess their hat color. If a prisoner guesses correctly he is spared. A wrong guess will lead to immidiate execution of the prisoner. Any other communication or moving around will result in the execution of all 100 prisoners.

The prisoners are free to coordinate an optimal strategy beforehand. What should they do?

Hints and clarification

The optimal strategy saves at least 99 prisoners.

In numbered cells

100 prisoners are staying in cells numbered 1-100. They are told that the next day they will in turn be called to the wardens office. In the office will be 100 numbered boxes, in each box one of the cell numbers with no duplicates. Each prisoner will have a chance to look in 50 of the boxes. If he can find his own cell number the boxes will be put back as they where and the next prisoner will be allowed his turn. If all prisoners succede the prison closes and they are released. If a single prisoner fails they all remain imprisoned.

The prisoners cannot communicate during the next day, but can coordinate an optimal strategy beforehand. What should they do?

Hints and clarification

The best strategy yields much better than $2^{-100}$ probability of release.
There is a strategy that gives 50% chance of the prisoners being released.

2 shoelaces

You have two shoelaces that have been in dipped in wax, a scissor and a lighter. Each shoelace takes one hour to burn, but the wax is not uniformly distributed and some parts of the shoelaces burn faster than other.

Measure 15 minutes, i.e. You have to be able to say "start" and "stop" with 15 minutes between the two.

Hints and clarifications

You can light a shoelace more than one place.

The doctor

A father and son are driving along the highway. There is an accident and the son is hurt. He is rushed to the hospital, but when he gets to the operating room the doctor exclaims "I can't do it, that's my son!". How is it possible?

Rush to the gate

You are at the airport, late for the plane and rushing to get to the gate as quickly as possible. You need to tie your shoelace before getting there. Every now and then there is a rolling pavement. Should you tie it on the rolling pavement, off the rolling pavement or is it irrelevant when you do it. You run at a constant speed whenever you are not tying the lace, you will not be faster after tying it.

Hints and clarifications

You can ignore the time for stopping and starting, getting on and off pavements etc.
Imagine there where two of you doing different things.