Explain the following using Pigeonhole Principle is it is true:
1) If we choose 10 points in a $3 x 3$ inch square, there must be two points of the 10 which are at distance less than or equal to $sqrt(2)$ from each other.
2) Any collection of 12 possible integers must contain two numbers whose difference is divisible by 10.
3) In a $2 x 2$ foot square we select 17 points. Explain why two of these points must be at a distance less than or equal to $sqrt(2)/2$ from each other.
4) Choose 20 points in a $4 x 4$ inch square. Argue that two of these points must be at a distance less than equal to $sqrt(2)$ inches from each other.
5) A square with side length 4 is divided into 16 smaller squares of side length 1 in the obvious way. 19 points are chosen in the interior of the large square. Explain why two of them must have distance less than equal to $sqrt(2)$ from each other.
Im just learning Pigeonhole Principle and came upon these questions. But as simple as they seem, I must not understand PH principle because I don't know how to answer them. As you can tell, most of these questions are quite alike so you don't need to show how to do all of them. But few with detailed explanation would be appreciated so I can get the "feel" for it.