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

Sudoku

You must definitely know about sudoku,which has become a runaway hit all over the world.Exactly how many sudoku puzzles can be constructed?

~Sai

http://en.wikipedia.org/wiki/Sudoku

Angles without the angles...

I saw this in a Martin Gardner book a while ago. Shown in the figure are 3 squares, and 3 angles in the bottom left. Prove, without any trigonometry (or equivalent), that the sum of the two smaller angles equals the largest.

~ Grenade

Seven Eleven

Given a seven-digit number,whose digits add up to 59.Find the probability that the number is divisible by 11.

PS: Would this qualify as a puzzle?? And sorry for any bitter exam-related memories I may have inadvertently invoked in some of you...:)

~sai