Here’s a brief summary of what we discussed in **Session 1** of our Workshop on Extra-Curricular Mathematics.

We began by looking at the problem How Many Rectangles?

We looked at several different approaches to this problem.

The **categorization** method may seem tedious to some, but the method allows us to get our hands dirty and explore some of the many patterns that reside within this problem.

*Counting each kind of rectangle above, and the rest, leads to many interesting patterns.*

The **combinatorial method** is simple and elegant. Just think to yourself, “In order to define a **rectangle**, I need to **choose** two vertical sides and two horizontal sides. How many ways can that be done?” It’s a nice application of **nCr**, if you are comfortable with combinations.

Another interesting approach is to count rectangles by counting all the **possible diagonals**. This is an approach worth exploring, but be careful not to **overcount!**

We also talked about several related problems that would be fun to investigate, like counting **squares in a square**, **triangles in a triangle**, and **cubes in a cube**. Or what if we counted **oblique rectangles**, too?

We also discussed the famous **Cartesian City **problem: how many different paths are there from the bottom left to the top right, if you can only move up or right, one segment at a time? Another classic.

We took a break, had some coffee and cake, and worked on some great **AMC problems**. The Art of Problem Solving website has all the old AMC problems here, as well as tons of other problems from contests around the world.

We finished up by talking about the famous handshake problem. We discussed three classic approaches, and a few extensions.