# Day 3

The warm up on Day 3 was slightly different than usual. Instead of doing one of the usual logic puzzles, the mathematicians had to figure out how to draw a perpendicular lines using a compass. Then they moved on to figuring out how to draw a pentagon using a compass, building on what they learned in art.

In Geometry, the young mathematicians saw the “Chinese Proof” of the Pythagorean Theorem, and figured out how the diagram proved it.

In Graph Theory, we defined what a complete graph is (a graph with an edge between every pair of vertices), and then the mathematicians found a formula for the number of edges in a complete graph with with n vertices.

We figured out that you can represent the number of edges in Kn with the series:

0+1+2+3+…+n-1
This doesn’t seem like the most elegant formula however. Can you think of a way to write the formula that doesn’t need to use that pesky ellipses?

Then we used the Pigeon Hole Principle to prove that every graph has at least two vertices of the same degree. The Pigeon Hole Principle states that if you have pigeons, and n-1 pigeon holes, then at least one pigeon hole will have two or more pigeons.
For those that were wondering, pigeon holes look like this:

Can you think of any other scenarios where the Pigeon Hole Principle would apply?

During activities we split off in to groups, and some of us went to the waterfall!

To keep safe, we used the buddy system to keep an eye on each other. Since there was seven of us, there were two pairs of two buddies and one group of three buddies. How would you represent this with a graph? Would it be a connected graph? Would it be a simple graph?

In art the mathematicians continued the work they did yesterday and started cutting and assembling their polyhedron outlines.

In computer science the two groups embarked on different missions. Group A continued coding the drawings they designed themselves. The drawings ranged from bears to pizza to emojis to robots, all very cool things. Group B started coding their own computer game, which when completed will involve trying to catch a dropping ball with a block on the bottom of the screen.

That’s all for today. Check back again for more exciting mathematical adventures!

-Jessica