Here's a few more puzzles. #1 is similar in spirit to Johannes' puzzle, #2-#4 are tricky but can still be solved pretty elementarily, and for #5 the general idea is interesting, whereas fully solving it is pretty tedious and should be avoided.

EDIT: Added for clarification: always prove the correctness of your result.

**#1.** Assume you have a chocolate bar consisting, as usual, of a number of NxM

squares arranged in a rectangular pattern. Your task is to split the bar into

small 1x1 squares (always breaking along the lines between the squares) with a

minimum number of breaks. A break consists of a straight line cut through a

piece of the bar that goes from one side to the other. You can not break the

squares. You can only break one piece at a time. Stacking two pieces and

breaking them in one go counts as two breaks. What is the minimum amount of

breaks needed?

**#2.** Let N be the set of integers from 0 to n-1. How many ways are there to choose subsets A, B and C of N so that the union of the three sets is N, and the intersection is the empty set? Solutions that result as permutation of another solution are counted separately.

**#3.** There is a positive, finite amount N of spherical planets of radius R in space. We consider a point on the surface of a planet as non-exposed if it cannot be seen from any other planet. What is the total surface (summed over all planets) that is made up by non-exposed points?

**#3.5.** There's an infinite amount of light bulbs in a row, labeled 1, 2, ...

At the beginning, all of them are switched off. Now you toggle the state (i.e. off -> on and on -> off) of every light bulb, then you toggle every second one (i.e. those with index numbers dividible by 2), then every third one (i.e. those with index numbers dividible by 3), and so on. Once you're done with that (haha), which ones are switched on?

**#4.** You're given an acute-angled (this is not actually required, but then you can freely choose the base side) triangle with a base side. You're supposed to put N disjoint rectangles into this triangle so that the first is aligned with the base side and its edges touch the other two sides, the next is aligned on top of the previous rectangle and its edges touch these same two sides of the triangle, and so on, stacking these rectangles on top of one another. What is the optimal arrangement to cover as much of the area of the triangle as possible with these N rectangles?

**#5.** There are two persons, A and B. There are two secret positive integers whose sum is less than 100. A is told the product of the numbers, and B is told the sum. There is the following conversation:

A: I don't know the two numbers.

B: I knew that you didn't know.

A: Now I know them.

B: Now I know them too.

What are the two numbers?