Monday, March 06, 2006

How to Plant a Money Tree

In my wallet there is a gift certificate from Border’s Bookstore given to me last December from my grandfather, and despite frequent use, I can never seem to spend the last remaining amount.

The reason is interesting.

At $100 it was a sizable gift. I couldn’t spend it all at once, and so every weekend I head over to the store and buy one or two items. When you buy a product at Border’s, your receipt becomes a coupon for the following weekend. The idea, obviously, is to lure customers back.

However, because I have a large gift card, I have to return anyway. By splitting up my purchases over several weekends, my buying power increases. Instead of $100, it has ended up being closer to $125. Border’s is banking on customers both spending and saving more. I am only saving more.

In addition, through various promotions and sales, the cost of my purchases have been reduced. So instead of $125, it’s more like $200.

On the road to California in January, I had a thought about deceleration. If I was driving 75 mph and I was exactly 75 miles from my destination, then it would take one hour to get home. But what if at every mile marker I instantly decelerated exactly one mph, so that at 74 miles away I dropped to 74 mph, and so on? How long would it take me to get home?

A long time.

Calculating in reverse, the last mile alone — where I would be traveling one mph — would take one hour to finish. Getting from mile marker two to mile marker one would take 30 minutes, because I would be drive one mile at two mph. Getting from mile marker three to mile marker two would take twenty minutes, or one-third of a hour. Each preceding mile back to the starting point would be a fraction of a hour. The first mile — going from mile marker 75 to mile marker 74 — would take 1/75 of an hour or 48 seconds.

Fine. I think this is what the “N” does on a fancy calculator.

Then I had another thought. Instead of decelerating only at each mile maker, what would happen if I decelerated constantly, so that I decelerated one mph over the course of each mile?

I would never arrive. At one inch away, I’m moving at one inch per hour. At 1/10,000th of an inch away, I would be moving at 1/10,000th of an inch per hour.

This is the same as the story about a frog crossing a pond. With each hop he covers half the distance remaining. Because space can be infinitely divided, that poor frog never arrives. He just hops less and less until he dies with his long tongue stretched toward the sandy shoreline.

This is similar to Zeno’s Paradox of the Tortoise and Achilles.

The tortoise — who is obviously getting big headed after his race with the hare — challenges Achilles to a race. The tortoise gets a 100 foot head start as a handicap. Whenever Achilles has moved 100 feet, the tortoise has also moved forward a little bit. When Achilles covers that distance, the tortoise has moved again. If space is infinite, then we always have to cover half the remaining distance. The paradox, therefore, states “You can’t catch up.”

Aristotle apparently solved this paradox, which sucks. Otherwise I could spend on my gift card forever.


Abby said...

oh man, that's just great :-)

Anonymous said...
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Ross said...

I just added you to google reader, and for some reason, this post was sitting at the top of the heap.

Regardless, a joke:

An infinite number of mathematicians walk into a bar. The first says, "I would like one beer." The second: "I would like half of a beer." The third says, "I would like one-fourth of a beer." The fourth one asks for one-eighth of a beer. Before the fifth can place his order, the bartender says, "You're all idiots!" and pours two beers.