Solving fiendish problems

William Byers, in How Mathematicians Think (Princeton University Press, 2007), recalls the interview with Andrew Wiles on Nova. Wiles is the mathematician who proved Fermat’s last theorem after seven years of dedicated work, focus, determination, and, yes, a little help from his friends. (By the way, for Malcolm Gladwell fans, there’s a video from the 2007 New Yorker conference in which he talks about the importance of stubbornness and collaboration in Wiles’s triumph.)

Wiles described the process of solving what I have dubbed fiendish problems: “Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of—and couldn’t exist without—the many months of stumbling around in the dark that precede them.” (p. 1)

Problems in the financial markets aren’t nearly so fiendish as proving Fermat’s theorem. But haven’t we all been in that dark unexplored mansion? Well, that question is incorrect. Aren’t we all still in that mansion? Perhaps we’re in room two or three, perhaps in room six or seven, but I’d wager to say that most of us still spend more time bumping into furniture than seeing the light.

And this reminds me of the puzzle of four switches and a light bulb from Paul Wilmott’s Frequently Asked Questions in Quantitative Finance (Wiley, 2007; a second edition is now available). “Outside a room there are four switches, and in the room there is a light bulb. One of the switches controls the light. Your task is to find out which one. You cannot see the bulb or whether it is on or off from outside the room. You may turn any number of switches on or off, any number of times you want. But you may only enter the room once.” (p. 383) Tomorrow I’ll share the “trick” to the solution, the day after I’ll outline the steps necessary to identifying the correct light switch.