Von Neumann, even cleverer than that

[jack]

There 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)

Comments

[sunflowerinrain]

This demonstrates to me, yet again, why I have trouble with Maths.

Y'see, I visualise the situation. And the fly is caught by the wind and blown sideways and never gets anywhere near the other bicycle...

[simont]

... and since when can flies do 30mph in the first place, never mind make instantaneous 60mph changes of velocity while remaining functional? Generally, in my experience, when a fly makes even a 10mph change of velocity on meeting a solid object it accompanies it with a *splat* noise and does not thereafter move under its own power.

Not all maths is quite this bad in terms of setting up hypothetical situations massively at odds with reality, though. It seems to be mostly a problem with the physicsish end of the subject. (And, one must suppose, mostly confined to the bits accessible to laypeople – school exercises and puzzles for amusement, which need to keep the actual maths simple or nobody will be able to answer any of the questions at all. I'd certainly hope that serious mathematical physicists don't spend all their time working in models so obviously wrong that their answers are hopelessly irrelevant :-)

[sunflowerinrain]

Somehow this reassures me. :) x

I still can't do Serious Big Maths, though. I shall just gaze and admire, mkay?
In maths exams I had no trouble with equations and such things, but as soon as it got to the section of problems reflecting real life, I was lost and confused. Curiously, m'sister the statistician, who did maths at A level and also English, said the linguistically-inclined mathmos had arguments with the Physics-and-Applied ones over the several possible interpretations of some problems, where the physicists could only see one. I was told that the problems/models would make more sense further on in learning the subject, but I gave up and studied Latin instead.

[simont]

Correction, I think: after waving my hands in simulated fly-swatting motions last night and trying to estimate their speed, I might increase that 10mph impulse to about 25mph. Still no chance of a fly surviving 60, though. :-)

[jack]

Wow, actual experiments :)

the physicsish end of the subject

[jack]

Logic also has a problem: confined to theory its ok, but try to relate it into real life, and however much you state the assumptions, people just can't believe that a pirate would actually jump through all those game-theory hoops for $1, or that the liar lies all the time :)

[simont]

From what I've heard of von Neumann, he probably really did sum the series in his head.

The devious approach of doing it by a shortcut and claiming to have summed the series sounds more like something Feynman would have done. In fact, come to think of it, it sounds very like something Feynman did do.

[jack]

LOL. That sounds right. Feynman always sounds like someone who illustrates the advantages of optimism and seizing chances, but could be really annoying to know :)

Von Neumann: Could be. I assumed the story was entirely apocryphal, although come to think of it, it's attributed quite specifically for that.

[simont]

Of course the thing about that Feynman story is that it doesn't show that he wasn't incredibly quick-thinking and good at mental arithmetic. Just because he did clever-thing-B and claimed to have done clever-thing-A doesn't invalidate that basic point. (Which, I suppose, is what makes that anecdote an amusing prank rather than a reprehensible fraud. That and the fact that there was nothing at stake.)

It's one thing to know that e^2.3 is a bit under 10, e¹ = 2.718 and e^0.7 is just over 2, and to deduce from those facts that e^3.3 is a bit less than 27.18, e³ is somewhere around 20 and e^1.4 is a bit more than 4; it's quite another to be able to do the Taylor's theorem corrections in your head in real time and know that (e.g.) the real answer to the first one was 27.11 rather than 27.12 or 27.10 or 27.17.

[simont]

Speaking of getting lucky with the choice of numbers, I had an amusing incident once at primary school when somebody who I knew well but who wasn't in my class – and hence didn't already know what I was and wasn't good at – found out for the first time that I had a reputation as a maths whizz. "So," he said, "does that mean if I asked you what 64 times 64 was, you could do it straight off in your head?"

I couldn't believe my luck. If he'd picked any other number, I'd have had to either figure in my head for minutes or explain that mental arithmetic wasn't the bit I was really good at. But he'd randomly chosen a power of two, so I fired straight back "Sure, it's 4096" and enjoyed the noise his jaw made as it hit the floor.

('Course, he didn't check it either, so I suppose bluffing would have been an option too :-)