LOCKERS, MOTHS AND FLICKERING CANDLES

My friend Emmie set a puzzle the other day, in the full knowledge that I can't leave puzzles alone until I've solved them. It's one of the things I find most annoying about myself - like a moth and a flame, I dance around it while it taunts me, flickering away in its hypnotic mystery.

So last night, I grabbed a piece of paper and a pencil and I resolved myself to extinguishing the flame, solving the puzzle, and then forgetting about it so I could get a good night's sleep.

I solved it. In a moment I'm going to write out the problem, and then tomorrow I'll explain the answer. If you're also a fluttering moth with a penchant for puzzles, don't read the next post until you're squarely confident that you know. If on the other hand, you're not fussed by this kind of thing and you couldn't care less about this abstracted candle, then maybe you'll follow my logic and be grateful that I spared you the time thinking about it.

Here is the problem, as Emmie presented it:

There are 1000 lockers in a high school with 1000 students. The problem begins with the first student opening all 1000 lockers; next the second student closes lockers 2,4,6,8,10 and so on to locker 1000; the third student changes the state (opens lockers closed, closes lockers open) on lockers 3,6,9,12,15 and so on; the fourth student changes the state of lockers 4,8,12,16 and so on. This goes on until every student has had a turn. How many lockers will be open at the end?

This is the point where we lose some of you. Quite understandably you wonder who on Earth really cares about this weird school ritual and why it matters to you. You wonder what kind of establishment this is where students pass the wall of lockers one by one in such regular fashion, routinely opening and closing locker doors. You wonder why you're being forced to endure a headache by trying to imagine it, and it reminds you once again that you never cared about that silly gameshow with the goats, and you certainly didn't give a button about the quantum state of Schrödinger's hypothetical cat.

If you're a moth, like me, you'll already have begun to think about how to work out the answer. And that's the key with these things, it's all about the how. In one sense, the answer doesn't matter at all - it's all about the process of figuring it out, and maybe finding a rule that extends way beyond just 1000 lockers, way beyond a puzzle about odd students at Hypothetical High and way beyond what you thought was a tricky maths problem. It's a kind of Schrödinger's cat all of its own.