I recall a math teacher in high school explaining that often, in the course of doing a proof, one simply gets stuck and doesn’t know where to go next, and a good thing to do at that point is to switch to working backwards from the conclusion in the general direction of the premise; sometimes the two paths can be made to meet in the middle. Usually this results in a step the two paths join involving doing something completely mystifying, like dividing both sides of an equation by the square root of .78pi.
“Of course, someone is bound to ask why you did that,” he continued. “So you look at them completely deadpan and reply ‘Isn’t it obvious?’”
I have forgotten everything I learned in that class. I remember that anecdote, though.
The standard proof of the Product Rule in calculus has this form. You add and subtract the same quantity, and then this allows you to regroup some things. But who would have thought to do that?
One of the characteristics of successful scientists is having courage. Once you get your courage up and believe that you can do important problems, then you can. If you think you can’t, almost surely you are not going to. Courage is one of the things that Shannon had supremely. You have only to think of his major theorem. He wants to create a method of coding, but he doesn’t know what to do so he makes a random code. Then he is stuck. And then he asks the impossible question, ``What would the average random code do?″ He then proves that the average code is arbitrarily good, and that therefore there must be at least one good code. Who but a man of infinite courage could have dared to think those thoughts? That is the characteristic of great scientists; they have courage. They will go forward under incredible circumstances; they think and continue to think.
Note that xkcd 759 is about something subtly different: you work from both ends and then, when they don’t meet in the middle, try to write the “solution” in such a way that whoever’s marking it won’t notice the jump.
I know someone who did that in an International Mathematical Olympiad. (He used an advanced variant of the technique, where you arrange for the jump to occur between two pages of your solution.) He got 6⁄7 for that solution, and the mark he lost was for something else. (Which was in fact correct, but you will appreciate that no one was inclined to complain about it.)
I recall a math teacher in high school explaining that often, in the course of doing a proof, one simply gets stuck and doesn’t know where to go next, and a good thing to do at that point is to switch to working backwards from the conclusion in the general direction of the premise; sometimes the two paths can be made to meet in the middle. Usually this results in a step the two paths join involving doing something completely mystifying, like dividing both sides of an equation by the square root of .78pi.
“Of course, someone is bound to ask why you did that,” he continued. “So you look at them completely deadpan and reply ‘Isn’t it obvious?’”
I have forgotten everything I learned in that class. I remember that anecdote, though.
The standard proof of the Product Rule in calculus has this form. You add and subtract the same quantity, and then this allows you to regroup some things. But who would have thought to do that?
--Richard Hamming
IIRC there was an xkcd about that, but I don’t remember enough of it to search for it.
EDIT: It was the alt test of 759.
Note that xkcd 759 is about something subtly different: you work from both ends and then, when they don’t meet in the middle, try to write the “solution” in such a way that whoever’s marking it won’t notice the jump.
I know someone who did that in an International Mathematical Olympiad. (He used an advanced variant of the technique, where you arrange for the jump to occur between two pages of your solution.) He got 6⁄7 for that solution, and the mark he lost was for something else. (Which was in fact correct, but you will appreciate that no one was inclined to complain about it.)
Is 759 the one you are thinking of? The alt-text seems to be relevant.
Yes.