Learning to say ‘I don’t know’

An evil wizard has captured 15 dwarves of rare mathematical genius. He informs them that, the following day, he will make them stand in a circle and then from behind will place a hat, randomly either black or white, on each of their heads. He will then go to the dwarves in turn and ask each to state the colour of his hat. While each dwarf can see the colour of the others’ hats, he can see nothing of his own. If a dwarf answers correctly he can go free, otherwise he will be incarcerated for life. No signalling is allowed between the dwarves — they cannot meaningfully pause before answering, for instance — but they do have a night in a communal cell in which to agree the optimum strategy. How many dwarves can be certain of going free if the best strategy is ^–adopted?

This question was set to my daughter as homework. I quickly saw that it would be easy to secure the release of seven (seven is the Dunbar-number for dwarves, I suspect) by instructing each dwarf to answer with the colour of the hat worn by the dwarf eight places in front of him. However the correct answer is 14*.

I hope this wonderful maths teacher does not set more questions like this. Not because I disapprove — it’s a beautiful problem — but because the damned thing kept me up until two o’clock in the morning and made me start smoking again. But the exercise did lead me to propose an idea for improving maths and science teaching in Britain. My suggestion is that 40 per cent of science and mathematics problems posed to schoolchildren and students should be not merely fiendishly hard, like the one above, but actually impossible to solve.