I was at work for about 12 hours today.
Got moving earlier than usual to take Rob to the San Francisco airport; traffic was light in that direction, and it only took about half an hour to get there, quicker than expected. But it took about another half hour to get back down 101 to work, which is about a 10-15 minute drive if there's no traffic.
I started musing about an old math puzzle. (If you couldn't care less about math puzzles, you can skim the math stuff in the next few paragraphs.)
Point A and point B are 60 miles apart. You drive from point A to the halfway mark, 30 miles away, in an hour. How fast do you have to drive the rest of the way to average 60 miles an hour?
(For those unfamiliar with the puzzle: most people who don't think carefully about it say something like 90 mph. They figure 30 mph + 90 mph divided by 2 is 60 mph. But the actual answer is that it's impossible; to average 60 mph over a 60-mile distance, you have to travel it in an hour, and you've already used up that hour traveling the first half of the distance. You'd have to cover the last 30 miles in zero time. Why doesn't the "obvious" approach work? When I was a kid, I came to the sound-bite conclusion that it's because "you can't average averages." (A parent or teacher may've told me that.))
It's true that you can't generally average averages. But this morning it occurred to me that if the denominator is the same, you can. If you average 40% on the first five quizzes of the semester (out of ten quizzes total), and 60% on the last five, then your average across all ten is indeed (40+60)/2, or 50%; that only works because the denominator of both averages is the same: 5. So I was trying to draw an analogy to the math puzzle, thinking that the denominators in both halves of the trip were the same (30 miles), and I couldn't figure out why you can't average those particular averages.
Just as I was on the verge of figuring out the flaw in my reasoning, I passed a car with an interesting license plate. It said: DVD BY O. I thought, Must be someone who works for some company that makes DVDs. All those high-tech companies around here. Maybe O is the initial of the car's owner, or of the company.
And then I realized that the problem I was having with the math puzzle is that it's the numerators that are the same; the denominator of the first half of the trip is 1 hour, while the denominator of the second half of the trip would have to be 0.
In other words, to get the speed for the second half of the trip, you would have to divide by zero.
And then I figured out what the license plate really said, and I started laughing.