Good morning. I am back on ALICE night shifts for the last time (*sniff*) and after a mad dash to make it here by midnight it has been a very slow and quiet night. Hopefully it will stay like this all week so I can write some thesis! 🙂

The response to last week’s problem was great, and many QD readers seem to have made light work of it. Well done. It is a sign of a logical mind. You are named and congratulated at the end. I’d be interested to know how many of you feel your interest in/knowledge of science/mathematics played a part in you getting it right. I am willing to bet that all of you do, despite the answer being very simple to compute once you know how (explanation at the bottom for those who want it). When I first started my undergraduate degree I was given problems like the “three children” story and whilst the actual maths involved was basic, I found myself infuriated by the seeming impossibility of it. This is the trick that scientists learn – be determined, defy your instinct and attack the problem logically (you can’t really give up on finding a way to fix your contaminated data sample because it seems like there is no way to do it). By the time I came round to the ALICE PhD “initiation problem” (again, no predetermined sciency/mathematical knowledge required) I was a dab hand – if only for my confidence that I would find a satisfactory answer. I’ll save that problem for another time because the solution – which I got after stopping in the middle of dinner and running to the nearest whiteboard for 4 hours – might be a little tricky to convey!

Now, if you don’t have the answer to the prisoner’s plight yet, I strongly recommend you don’t read on – you will kick yourself. Give your brain a little more thinking time, it will be worth it. If you think you have the answer, or you are ready to give up, here’s the solution:

Arvinder, the friend that gave me the problem, said that there was one word that could give the whole game away. The word he was referring to was “parity”. When I got the solution, I thought his word was “binary”. Some could say simply “operators”. But you don’t even need to know what any of those are to get this right.

Consider first of all, the man at the back, looking at the row of hats. He has to convey to the person in front what their hat colour is, and he can do that by either saying “black” or “white”. But “black” does not need to mean “your hat is black”. They can predefine a new definition for these words.

Now consider the information the second prisoner along receives, and what he does with it. Clearly he will determine his hat colour and be saved, but what then? That needs to be enough to allow prisoner three to save himself. How can this be possible?

So now, finally, consider the most information there is available to anyone – the prisoner at the back can see all hats except his own. Call this the whole system – a string of blacks and whites. The person in front of him sees the same system minus one hat. From there on, every prisoner will know, by the time it is his turn, the colour of all hats except his own. Here comes the binary/parity/charge/whatever you want to use (even symmetry of a wavefunction if you like!) All you need to consider is that what ever black does to the system, white does the opposite. The initial statement from the first prisoner needs to convey what state this whole system is in, in such a way that if only one of these hats was unknown, it would be clear from the state of the system what that one hat must be. If they were 1s and 0s and being multiplied, whether the result was 1 or 0 would determine what the one unknown hat was doing to the system. Similarly if they were -1 and +1, or if you were adding the 1s and 0s for an “odd” or “even” system. Yes, it’s as simple as adding up 1s and 0s.

Thanks so much for the responses. Well done to: Emlyn, Jacques Distler, Cavendish McKay, odd man out, M Kneebone, Sourabh and Nic. Your comments are all approved and you win the “Yes, I am a scientist at heart” prize! (It is not a real prize but it makes you feel nice).