Today is math riddle day. Here are three tough ones, maybe in order of increasing difficulty:

(edited to correct two errors in the statement of the problems, 8/13/03 11:00 pm)

(1) Say M is thinking of a number between 0 and 15; what's the smallest number of yes-no questions you'll have to ask M in order to know the number, if M is allowed to lie once?

(2) If you have 5 points on the surface of a sphere, prove that no matter where they are on the sphere, it will be possible to cut the sphere in half and have at least 4 on one half.

(3)

(i) P is told the product of two numbers A and B
(ii) S is told the sum of these two numbers A and B
(iii) A and B are numbers between 2 and 99

P says "I don't know what the two numbers are"
S says "I knew you didn't know them!"
P says "Oh, well, I know them now."
S says "Ah, well in that case, I know them too."

What are the two numbers?

this should be solvable without using a brute force approach

Riddles courtesy of Tania Lombrozo & Nelson Lai. I think I've solved the first one; not so much the last two.