If we color 10 pigeons with 9 colors, there will be two pigeons with the same color. This simple principle is extremely useful in mathematics, and can lead to amazing results that initially seem a lot less obvious. Here's an example: if there's a pigeon in every point of a plane, and we paint each of them red, green, or blue, there is always a rectangle with four pigeons of the same color! In this class, we'll show why this and some other applications of this principle in mathematics, including the problem of approximating irrational numbers.

If we remove opposite corners of a chessboard, we can't tile what's left with dominoes. If we color each of the points in a 2D plane with red, green or blue, we can always find two points of the same color at unit distance. If the vertices of a regular polygon all have integer coordinates, it must be a square. In this class, we'll see how to prove cool results like these using invariants - certain mathematical properties that are always the same, and monovariants, quantities that always increase or always decrease.

Do you like circles? In most high school geometry courses, the beauty of geometric diagrams is replaced by unfriendly formulas and computations. Often overlooked are the stunning and unpredictable results that can be found simply by drawing points, lines and circles on a piece of paper. And perhaps what makes pure Euclidean geometry so artistic - and so undertaught - is that it essentially has no real life applications; its only purpose is to be appreciated by how amazing it is. In this class, we'll start with the notion of the power of a point with respect to a circle, and use it to prove fascinating results such as the radical axis theorem, the existence of isodynamic points in any triangle and the Newton line in complete quadrilaterals. Most of the arguments will be purely geometrical, with a minimal amount of math involved.