## Tuesday, April 6, 2010

### 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.

1. Can we make assumptions about the initial conditions? Is the light bulb on or off to start with? I take it we know the on and off positions of the switches. Are we talking incandescent or compact fluorescent?

Can we make some assumption about the rate at which a lightbulb heats up? For instance, if it takes a little while to get too hot to touch I see a solution. What if it is a compact fluorescent? I think these get warm and take a while to get up to full light output?

I like your blog by the way - I will stop by regularly!

2. Assume that the light bulb is off to start with and that all light switches are initially in the off position. Assume further that we're talking about a good old fashioned incandescent light bulb.

Brenda

3. I know ho to do this with 3 switches and a bulb that heats up but four makes it harder...

4. Can we assume that it doesn't take longer than a few seconds to get back into the room?

5. ...and do you want the solution in the comments or do you prefer to reveal it tomorrow?

6. I have outlined all the relevant assumptions.

I'm trying to stretch this poor puzzle out. Tomorrow is "clue" day, although it's already been more or less revealed in the comments. Thursday is "solution" day.

If you'd like to post the solution as a comment I'll just hold the comment until I publish the solution. That way you can take credit for being "fiendishly" clever and still let others keep working away.