Tuesday, June 7, 2011

Never enough time

I'm about halfway through a couple of busy weeks...posting has been and will continue to be light.  Bear with me!  In the meantime here is something from the mental archives to tide you over.

You have an infinite number of ping pong balls, numbered 1, 2, 3, ... and so on. You also have a bottomless urn which is, for now, empty.

Here's the game. You start a stopwatch, put the balls numbered 1-10 in the urn, and take number 1 back out. After 30 seconds, you put in #11-20, and remove #2. After 15 more seconds, you put in #21-30 and remove #3. You continue this process, each time waiting half as long between each iteration.

When a total time of 30+15+7.5+... seconds = 1 minute has elapsed, you're done. And the question is, what's in the urn now?

Well on the one hand, at each stage you're putting in 10 balls and taking out 1 -- which seems equivalent to repeatedly putting in 9 balls -- and you're doing it an infinite number of times. So, shouldn't there be an infinite number of balls in the urn?

This is what basic human intuition tells us. On some level it feels obvious. But intuition can be misleading  So to convince me that "infinitely many balls end up in the urn," you'll have to tell me which balls are in there.

Is ball #3 in there? No, it was removed after 3 iterations, clearly 3 is too small. How about ball #47? No, you took that one out after iteration 47. OK but 47 is still pretty small, what about a really "big" number, like ball #232525609e86434574385? Again though, we know exactly when you took it out, so it's definitely not in the urn.

Indeed, any ball you can name is not in the urn. So it seems that nothing is in the urn after a minute, even though you're continually putting 10 in for every 1 you take out!  Sometimes infinity is strange.

Here's why this came to mind. Though this blog is pretty new, I'm already having a serious problem with overflow. For each post I write, I accumulate several posts I want to write. This would be no problem if I had an infinite amount of time (and patience)! But alas, time is our most desperately scarce resource.

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