ISTR reading about a math problem that has been used since time immemorial to stump university mathematics students. I’m not sure of the exact wording, but the general idea goes like this: Two bicycles, one mile apart, begin moving toward each other, both at the same, constant speed (say 10mph). A bee, sitting at the foremost point on one of the bicycles, begins moving between the two bicycles at higher, yet constant, speed (say 20mph). Once he gets to the foremost point of the other bicycle, returns to the first bicycle. Repeat until the two bicycles meet.
Students, when given this problem, would set upon calculating how much distance the bee would cover in his first trip, then in his second trip (which would be shorter, relative to the amount of distance the two bicycles have moved, yada yada yada), etc., until they had pages of calculations.
The mathematician in question (I think it was John Nash), figured out a simple, yet ingenious, solution that apparently no one else had ever thought of.
Since the maximum amount of time the bee could be in flight was x minutes, where x represents the amount of time it would take for the two bicycles to meet in the middle; and he moved at a constant speed of y mph; then multiplying x by y gives us the distance he moved. So for example, if the bikes met after ten minutes (and I don’t feel like doing the math to figure out how long it would take the bikes in the problem to meet; math isn’t my strong suit), then moving for ten minutes at 20mph = 3.33 miles.
Do I have my facts straight on this problem, and was it indeed John Nash who figured it out?
Any competent mathematician could sum the infinite series in his or her head, but not instantly like von Neumann. There’s a similar story about Gauss summing the numbers from 1 to 100 when he was a child in school.
I’ve heard the question ending in the request for the distance traveled by the bee. The mathematician could sum the series, but should instead calculate the time for the two bicyclists to meet, then multiply the bee’s speed by that time - a much simpler way of getting the result.
I think I read that Ramanujan answered the question almost immediately, and his questioner said, “Very good - most people try to sum the series”, to which Ramanujan replied, “There’s a faster way than summing the series?”
ETA: Nevermind. Figuring out the length of a leg of the fly’s journey is easier than I thought, once I actually thought about it. FSR my first impression was that I’d have to solve a system of equations!
If I remember right, the story is that Gauss’ class was assigned an exercise to sum the numbers from 1 to 100 on paper — the slow way, adding in each number individually. Gauss however saw in his head that the numbers from 1 to N could be represented as a triangle of stacked rows of squares, and that this triangle would form exactly half of a rectangle measuring N by N+1.
So the sum of 1 to 100 is just (100 x 101)/2, or 5050, which was much, much easier and quicker to calculate. And so this is what he did.
But, as the story also goes, young Gauss then got into trouble with his teacher, who didn’t approve of his being such a smart ass.
I doubt it. It may well be a story told about John Nash (or other mathematicians) to show how they were especially smart or insightful, but the fact is that a reasonably smart teenager or university student would quite likely arrive at the same technique.
Anecdotes about mathematicians summing series in their heads are also fairly common, but not necessarily to be taken seriously - it’s just part of the legend of that person being particularly smart. Feynmann wrote amusingly about impressing people by using a series of tricks to rapidly answer a number of questions involving transcendental functions, and then compounding their amazement by saying “I just summed the series”. This tells us a number of things about Feynmann - he was genuinely smart, and playful, and enjoyed contributing to his own legend. But it also tells us that even a genius like him isn’t genuinely summing the series in his head.
I’d say more that it reflects the person thinking in a particular way, not that that way is necessarily smart. One might argue that there’s more real intelligence involved in being able to realize that there’s an easy way to solve a problem, than there is in being able to solve it the hard way.
What I find interesting is that everyone here attributes the story to Gauss… I have heard the exact same story many times, but only in reference to Einstein. (With some details usually added about being punished for a solution that used multiplication rather than addition, since the assignment was to add the numbers).
There may be people who tell the story about Einstein, but it was told about Gauss before Einstein was even born, as you can see in the link in my second post. The story gets attached to whoever has the current reputation of being a great mathematician. Let me note that I personally heard the story told about Gauss long before Nash became famous (outside the worlds of mathematics and economics).
That point wasn’t in reference to the OP’s flight time problem. It relates to more straightforward cases such as giving the natural log of 5.3 to 4 decimal places*, as described in Feynmann’s autobiography.
I don’t recall the real examples but they were problems of this kind.
Excuse me. The OP never claimed that the adding of numbers from 1 to 100 was true of Nash. He was claiming that the story about finding the length of the flight of the bee was true of Nash. But, once more, I also heard the story of finding out the length of the bee flight long before Nash became famous (outside the worlds of mathematics and economics).
The way I heard the bee story was that von Neumann took 5 seconds to answer. On being told that X did it in 4, he was said to have exclaimed, “No one can sum a series that fast!”
As for the Gauss story, I have heard it all my life. I had assumed that the teacher had once done it (it’s not that hard, you can sum the first ten digits and get 45 and realize that this would be repeated nine more times and then sum the tens digits and gotten 10*450 and add 100, which is the final total. Once the teacher had done it once he could use it in every year.
My understanding of the “sum the numbers up to 100” problem was not that the teacher already knew the answer and was honestly testing the students, but just that e was tired and needed a break, and so gave the students some meaningless work to keep them busy for a while.
Maybe, I’m too old, and when I first heard the story I admit I assumed the guy actually sums the numbers the hard way, but now I suspect he did it the easy way. The extra cleverness points are awarded for his mental agility in realizing it would be funny if he sustained the illusion he did it the hard way. I have never known a clever person who wouldn’t do that, just for the fun of it. (Plus, possibly he knew the guy he was dealing with was a suggestible tool). Honestly, now I think about it, I doubt he was really even aware of what method he used. If I asked someone what 4 + 7 was would they be able to tell me exactly how they knew the answer was 11? Or would they just know the answer?