I want to thank my fellow PhD student Arvinder Palaha for helping to distract me at ALICE Physics week in Paris last week, when I was horrendously ill. He also gave me a fresh reminder of what makes scientists so strong.
It was a very busy week of deadlines and talks and fancy meals, and I was hit with a wave of nausea at the most incredible restaurant during our group dinner. As each course was placed in front of me, all I could think about was the smell. I needed a distraction or I might be sick. Arvinder knew that there are only two surefire ways to distract me. One was chocolate, and that would probably not have the desired effect at this point. The other was a good juicy problem – when I hear one I can’t think about anything else until I have solved it. This one was excellent because despite being deceptively simple, it epitomizes what I love about physics – mathematics and logic help you through when instinct fails completely.
I warn you – if you read on, and if you are anything like me, you won’t be very sociable for a while. Good luck. I will post the answer in a few days. If you think you have the answer, comment. If you get it right or come close then I will temporarily block your comment because of spoilers, but I will credit you later. 🙂
There are 100 prisoners. They are told by a prison guard that in 15 mins time they will be blindfolded, and then each given a cap. Some will be black, some will be white. The amount of each will be arbitrary (in fact there may not be any white caps at all, for example). They will not be allowed to talk or make any contact with each other. They will be lined up, each one behind the next, so that all but the one in front are facing the back of another person’s head. Following so far?
Their blindfolds will then be taken off. They will still not be allowed to communicate in any way. They will all be able to see every head in front of them (a line of black and white caps in some random order) but they will not be allowed to look back/around. The prison guard will then, starting at the back (with the prisoner who can see the most heads), ask each prisoner in turn what their hat colour is. If they get it wrong, they get shot. They will not be able to say anything other than the words “black” or “white”, and they can’t risk making a code with intonations in their voice or something similar because he might pick up on it, and if anything fishy goes on AT ALL he will shoot them all.
After telling them all this, the prison guard leaves and the timer starts. So, obeying the prison guard’s rules, how can they (in the 15 minutes they have to discuss their predicament before being blindfolded) come up with a plan to save as many of them as possible?
Just to show you why I love this problem so much, I can tell you that they do come up with a plan that guarantees the safety of 99 of the prisoners. The other one has a 50:50 chance of survival.
PLEASE NOTE: There are now spoilers in the comments 🙂
