Sylvester's theorem
There are a finite number of points on a plane. They are situated such that any line through any 2 of the points have at least one other of these points lying on it. Prove that all points are collinear.
From prior experience of hearing proof attempts, if you are giving a proof by induction, please consider your proof carefully before taking the effort to type it up.
General background: Sylvester was the first person to prove this, but his proof was a 7 page long proof! However, the current best known proof is very elegant.
~Grenade
From prior experience of hearing proof attempts, if you are giving a proof by induction, please consider your proof carefully before taking the effort to type it up.
General background: Sylvester was the first person to prove this, but his proof was a 7 page long proof! However, the current best known proof is very elegant.
~Grenade
posted by littlecow at 4/07/2006 01:00:00 PM
9 Comments:
Let there be n points in the chosen plane,such that the given condition(atleast one point lying on a line through any two points) be satisfied.
But now,let us also assume that not all the points are collinear, and one of the points,say X, doesn't lie on the line through all the other points.Thus, a line through X and any one of the other points,say Y, will not contain any of the remaining (n-2) points on it. But this is a contradiction to the given condition(see above).Thus all the points in the plane must be collinear.
QED,I hope???
So that means I basically have to extend the contradiction to 'k' points(k < n),right?
How about using induction? We see that the collinearity condition in the problem statement is valid for n = 1 and n = 2 points. Assume it is is true for n = k points so that we have a LINE connecting all these points. Then add a (k+1)th point somewhere on the plane and now we can use Sai's argument that basically proves that the (k+1)th point has got to lie on the original LINE connecting the first 'k' points. I am sure grenade has a counter argument to prove that this method is not sufficient or wrong :)
Oops, I didnt read the "be careful with induction proofs" note in the problem. But I dont yet see a mistake in my argument so please do post the catch :)
OK, I see it clearly now, thanks to grenade.
Looks like spammers are starting to show their skills here, and probably littlelow needs to add a word verification like lenscrafter to post comments at xyfactor.
Yup,I agree with Vand.
As for the puzzle,I've discussed it with my physics prof. and we had worked a solution out together,so I'll abstain from posting it till everyone has had a go.
BTW,what's the solution for the CH(n+1) problem?
I suggest that solutions be not provided on this blog unless the problem has been solved by one of the readers. It takes the fun out of solving the problems. More importantly, many of us come to xyfactor when we are jobless - which means that a problem could be attacked months after it has been posted. So, posting a solution within a time-frame assuming people have given up is based on an incorrect assumption! Clues and hints are ok until they are not too easy.
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