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. And hence the correct answer would be ‘I don’t know’.
Why? Well, when you think about it, most science exam questions are a slightly cheeky contrivance. They invariably provide the examinee with just enough perfect information from which to derive a single ‘right’ answer. But this situation is rare in the real world. You almost never have enough reliable information, or else the variables that matter are uncertain, non-linear or not numerically expressible. Artificially neat exam questions encourage in all but the shrewdest students an undeserved confidence in the power of numbers to answer -anything.
This overdependence on maths in modern decision-making leads to a kind of stupidity where reality is sacrificed on the altar of mathematical neatness. Economics is especially guilty of this sin, massively oversimplifying human behaviour the better to create cute theoretical models.
But the other problem with trying to run the world through numbers is that people can ‘game’ the metrics. If you have ever wondered why you can’t book a doctor’s appointment 48 hours in advance, it’s because GPs are assessed by the speed at which they see patients. By allowing only last-minute appointments, their metrics look great, even though it’s a pain for the patient.
I suspect this is what happened in the recent West Coast rail franchise bidding. Since no one in government has the balls to make a decision, responsibility was offloaded to a mathematical bid-model to lend an illusion of objectivity. But no mathematician on earth can predict the traffic between London and Manchester in 2024. So bidders ended up gaming the model.
A similar pseudoscientific approach will be adopted in ‘deciding’ (i.e. delaying until after the next election) about new runway capacity for London. It will take ages, and add nothing to the quality of the decision except a spurious plausibility. These models are used ‘as a drunk uses a lamppost. For support rather than illumination.’
*Explanation to be supplied in a fortnight
Comments