Von Neumann, even cleverer than that
Mar. 23rd, 2010 02:30 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackThere is an old puzzle which goes "Two cyclists start out 50 miles apart, and head towards each other, each one going 10 mph. At the same instant, a fly leaves the first bike and flies at 30 mph towards the second. When it gets there, it immediately turns around and heads back to the first. Then it repeats, going back and forth between the two bikers. By the time they reach each other, how far will the fly have travelled?"
There's a systematic solution and a "quick" solution, although different people find different solutions more obvious. The systematic solution is to say that fly goes three times as fast as the first cyclist, so the first cyclist will be one third of the way between the start point and the fly, so when the fly turns at the second cyclist, the cyclists will be at 1/4 and 3/4 of the way along the course, and the fly has flown 3/4 of the route. By the same argument, on the fly's second leg, it will fly 3/4 of the remaining distance, ie. of half the course, and the remaining distance will halve again.
Assuming the fly can turn instantaneously, it will fly 3/4 * [1 + 1/2 + 1/4 + ... ] = 1.5 times the length of the course, or 75 miles.
This may or may not hint at the quicker way of solving it, which is to say "the cyclists approach each other at a net speed of 20mph, so meet in 2.5 hours. The fly flies at full speed for all that time, so covers 30 * 2.5 = 75 miles".
People are particularly prone to asking this question of professed mathematicians, since mathematicians like clever answers to interesting chestnuts like this one, even though they know the most efficient way in the long term is to keep a ready supply of sledgehammers handy and apply them liberally. After all, inventing clever theorems is all very well, but part of the point of a clever theorem is that you don't have to worry about it any more, and can go around saying "oh, that's trivial, by Whoever's theorem" whenever anyone asks you anything.
The point
If you've heard this puzzle before, and know people are likely to try to catch you out with it, you've no opportunity to look clever by guessing the answer, and may resort to jumping in:
A: OK, suppose there's two locomotives facing each other:
B: Wait, I know this, it's the one with the infinite series, isn't it? It's easier to calculate the total time and use the speed to calculate the distance travelled.
Unscrupulous people may *pretend* to figure out the quick answer on the spot.
However, it only just now occurs to me a more devious solution is to pretend to solve the infinite-sum on the spot.
A: ... how far will the fly have travelled?
B: Oh, that's easy. You just have to work out the series and sum it... [thinks] 3/2, multiply by the length, 75 miles, right?
A: Eek! You're supposed to... oh, never mind.
However, a famous anecdote is told about Von Neumann (or another famous mathematician)
"Von Neumann thought for a brief moment and gave the answer. The hostess was disappointed and said, oh, you saw the trick, most people try to sum the infinite series. Von Neumann looked surprised and said, but that's how I did it."
I assumed the joke was supposed to be, Neumann summed the series in the length of time it would take a normal person to divide the time by the speed. But maybe the joke is, he was thinking three moves ahead, and PRETENDED to sum the series.
The first is more mathsy, the second more devious. I'm not sure which I prefer.
(Some text from www.math.bme.hu/~petz/vnsumming.html)
There's a systematic solution and a "quick" solution, although different people find different solutions more obvious. The systematic solution is to say that fly goes three times as fast as the first cyclist, so the first cyclist will be one third of the way between the start point and the fly, so when the fly turns at the second cyclist, the cyclists will be at 1/4 and 3/4 of the way along the course, and the fly has flown 3/4 of the route. By the same argument, on the fly's second leg, it will fly 3/4 of the remaining distance, ie. of half the course, and the remaining distance will halve again.
Assuming the fly can turn instantaneously, it will fly 3/4 * [1 + 1/2 + 1/4 + ... ] = 1.5 times the length of the course, or 75 miles.
This may or may not hint at the quicker way of solving it, which is to say "the cyclists approach each other at a net speed of 20mph, so meet in 2.5 hours. The fly flies at full speed for all that time, so covers 30 * 2.5 = 75 miles".
People are particularly prone to asking this question of professed mathematicians, since mathematicians like clever answers to interesting chestnuts like this one, even though they know the most efficient way in the long term is to keep a ready supply of sledgehammers handy and apply them liberally. After all, inventing clever theorems is all very well, but part of the point of a clever theorem is that you don't have to worry about it any more, and can go around saying "oh, that's trivial, by Whoever's theorem" whenever anyone asks you anything.
The point
If you've heard this puzzle before, and know people are likely to try to catch you out with it, you've no opportunity to look clever by guessing the answer, and may resort to jumping in:
A: OK, suppose there's two locomotives facing each other:
B: Wait, I know this, it's the one with the infinite series, isn't it? It's easier to calculate the total time and use the speed to calculate the distance travelled.
Unscrupulous people may *pretend* to figure out the quick answer on the spot.
However, it only just now occurs to me a more devious solution is to pretend to solve the infinite-sum on the spot.
A: ... how far will the fly have travelled?
B: Oh, that's easy. You just have to work out the series and sum it... [thinks] 3/2, multiply by the length, 75 miles, right?
A: Eek! You're supposed to... oh, never mind.
However, a famous anecdote is told about Von Neumann (or another famous mathematician)
"Von Neumann thought for a brief moment and gave the answer. The hostess was disappointed and said, oh, you saw the trick, most people try to sum the infinite series. Von Neumann looked surprised and said, but that's how I did it."
I assumed the joke was supposed to be, Neumann summed the series in the length of time it would take a normal person to divide the time by the speed. But maybe the joke is, he was thinking three moves ahead, and PRETENDED to sum the series.
The first is more mathsy, the second more devious. I'm not sure which I prefer.
(Some text from www.math.bme.hu/~petz/vnsumming.html)