Puzzles and Mathematics

We demonstrate the interaction between beautiful puzzles and interesting mathematics. Many interesting results in mathematics emerged from math puzzles and in the other direction interesting mathematics gives rise to beautiful puzzles. In the talk we will give several examples for puzzles and mathematical results and the interchange between them. Many simple and beautiful puzzles are still open mathematical problems.
Here is one: Is it possible to place an odd number of unit discs in the plane such that the area of all the points that belong to odd number of the discs is smaller than $\pi$?