### Are we trendsetters, Adam?

I'm beginning to think this math blogging thing may catch on. They're even doing "West Wing" math blogging over at The Volokh Conspiracy. Check out their puzzler; it's kind of intriguing.

It's similar to a puzzle involving a mountain climber that I used to enjoy torturing people with. It goes like this:

On Saturday, a man climbs a mountain. He begins at the base of the mountain at 9 AM and arrives at the summit at 6 PM. Other than that, we can make no assumptions about his route or his rate of progress. He might have gone at a steady, even pace, or in bursts, with frequent rests. He might have even lingered for nearly nine hours near the bottom and then sprinted up to the top during the last 10 minutes. Who knows.

The next day, Sunday, he comes back down. Again, he starts out at 9 AM and reaches the bottom at 6 PM. We know nothing of what happens in between. My assertion is this: There is some time on Sunday during which the climber is at the *exact same height* as he was at the same moment Saturday. It could be noon, it could be 9:37 AM, it could be 4:10 PM, could be anything. But there exists some moment at which the dude is at the same elevation as he was exactly 24 hours earlier. Again, this is independent of his route or his pacing.

Do you buy it or not? It seems non-intuitive at first, but it's true. Think about it, and if you give up you can click below for the answer.

This doesn't constitute a formal proof, but it's a hand-waving kind of argument that you'll hopefully find convincing nonetheless. Instead of one hiker going both ways, imagine there are two *different* hikers, one going up and one going down. Also assume that they're climbing/descending on the same day rather than one day apart. If they each start out at 9 and each finish at 6, they will certainly pass each other at some point. They may not see each other, because they may be on opposite sides of the mountain, but there will be some instant during which they are both at the same elevation.

**UPDATE:** Paul made me realize I did a crappy job of relating the two-climber case to the original problem. Let me try to bridge the gap like this.

You've got two movie screens, side by side. On one screen, you're watching Climber A climb the mountain in real time. On the other, you're watching Climber B simultaneously descend the same mountain, also in real time. At some point, their paths will "cross" in the sense that they will both be at the same elevation.

Now imagine that instead of watching the two climbers, you're watching two different videos of the *same* climber taken exactly 24 hours apart. That gets us back to the original problem, but the stuff you see on the movie screens is essentially no different from the two-climber case. There exists some point (some moment in time) during which the descending dude is at the same elevation he was the day before.

And no, there's no way to predict when that moment will occur, only that it will. That's obvious in the two-hiker case, but it seems less so when you state the problem in terms of one hiker over two days.

Oh well, I'm doing a sucky job of explaining this. Nevermind.

## Comments

I guess this is why I never studied higher math. You said one climber, not two, and two days, not one. Even if there were two, there would be no way to predict where they would meet. If you ran the test several times you would get different results, as the climbers would learn the best routes and become physically conditioned. I suppose now a dozen math whizzes will show up to explain to me why I'm wrong, and I still won't get it.

Posted by: Paul Moore | March 16, 2006 06:49 AM

The point isn't the the result is the same in every test, just that the result must, in fact, occur.

I never would have guessed that math blogging would be soothing to my brain, but straight logic with undisputable rules is a lot less work than politics, when you get right down to it.

Posted by: OTTami | March 16, 2006 09:33 AM

>

I never would have guessed that math blogging would be soothing to my brain, but straight logic with undisputable rules is a lot less work than politics....'Zackly. I think what I've been doing here is my own version of "taking a break." :)

Posted by: BNJ | March 16, 2006 09:36 AM

Can we be like those indie-music people, who always whine about how they liked a band before it was big?

I was mathblogging

beforeit was instalanched!Posted by: Adam | March 16, 2006 07:58 PM

Oh, I think so. I'm going to start practicing my "bored and disaffected" look.

Posted by: BNJ | March 16, 2006 08:06 PM