Mathematical Theory Track

**MATHEMATICAL THEORY**

The Math Theory track requires a minimum number of three participants. It is designed to engage students to think outside the box. During the seven week course, students spend their afternoons pondering questions with their peers, trying to prove or disprove a theory.

*In this math section, the thing that's distinctive is that
I tell them problems whether they are solved or not,
and I leave them on their own to figure it out.
So, they've really got to think like researchers, and they've got to figure it out or not,and then I verify whether the proof is a proof or not.*

–*Steven Rudich*

Some of the problems that the students will explore include:

**The Elephant Problem**

There is a group of elephants, all whom fight each other.

In each fight, one elephant dominates the other.

Prove that Elephant A will always dominate Elephant B who will
always dominate Elepahant C , ... , *etc*.

**The Pirate Problem**

On a pirate ship, there is a group of pirates, and within the pirates, there is a hierarchy. The highest ranking pirate decides how to divide the loot among the other pirates and himself. If he gives the other pirates too much, then he does not enjoy the wealth of the loot. However, if he keeps too much, and less than half of the crew agrees on his plan, then he walks the plank and dies, and the process begins again.

**Pascal's Triangle**

Dr. Rudich also has the students explore *Pascal's Triangle* and
study it on a deeper level. The students explore sums of cubes using
Pascal's Triangle, polygonal numbers and other patterns within the triangle.

**Cake Cutting Problem**

Another problem assigned to the students, as a continuous project, is the
cake cutting problem. The basis behind the problem is to find a fair and
envy-free way to cut cake among *N* number of people. The problem is
easy between two people, one person cuts the cake and the other chooses
which piece to have. There is also a proof for three people.

However, there is yet to be a proof for four people!

If anyone can solve this problem, Manuel Blum (Bruce Nelson University Professor at Carnegie Mellon,) will give the lucky individual a Ph.D.

Mathematical Theory is challenging and exciting. It is at the core of computational thinking. Students participating in the program have found it interesting, rewarding, and a great adventure in thinking!

– Mike, age 17

*I really like it because it's a lot different from what we do in high school. We explore problems that we wouldn't even begin to think about in school.
– Josh, age 16 *

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