## Monday, August 25, 2014

### Day 1

Things got off to a great start today, as our mathematicians started with a coin division problem that led to some interesting revelations about strategy and rationality. The problem, famously known as the Pirate game, asks how five pirates can distribute a treasure chest of gold amongst themselves using a set voting system.

If you liked the Pirate game, you might also want to consider another famous game in the discipline of Game Theory.

An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical items. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of \$100 per suitcase (he is unable to find out directly the price of the items), and in order to determine an honest appraised value of the antiques the manager separates both travelers so they can't confer, and asks them to write down the amount of their value at no less than \$2 and no larger than \$100. He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: \$2 extra will be paid to the traveler who wrote down the lower value and a \$2 deduction will be taken from the person who wrote down the higher amount. The challenge is: what strategy should both travelers follow to decide the value they should write down?
-Kaushik Basu

If you were a traveler faced with this situation, what value would you submit to the airline? Would this value change if you were playing this game against a perfectly rational computer?

While we were checking all of our mathematicians in, everyone received a t-shirt with a spiral pattern drawn on it. Luckily, our Proofs and Investigations module did some sequence and series work that might shed some light on the nature of the spiral pattern. How do you think that spiral was constructed? This spiral is often referred to as a “Fibonacci Spiral”, after the Fibonacci sequence which goes 1, 1, 2, 3, 5, 8, 13… how do you think the spiral and sequence are related?

In our Problem Solving module, we introduced a handshake problem to get to know one another while doing some good math. The problem we proposed asked how many handshakes would occur if everyone in the room shook hands with everyone else exactly once. Unfortunately for me, I got a little confused when I tried to duplicate the problem at a riddling party I attended. Instead of calculating the number of handshakes from a given number of people, I tried to find the number of people given the number of handshakes. In the riddling party, everyone shook hands with everyone else exactly once, and there were 4,656 handshakes that occurred. Can you help me find out the number of people that were in the party?

 The Handshake Problem: Students found many ways to approach the discovery of a general rule.

Stopping by our Computer Science Module, I saw our mathematicians exploring how to give instructions to computers in the form of code. I’m glad that I came into the class, because I have been having a lot of trouble trying to code my personal robot to go about doing my personal routine. If you had a robot that looked exactly like you, how would you instruct it to go about your daily morning routine? For example, putting on a pair of pants might go something like:

1.  Locate pair of pants.
2. Orient pair of pants so that the zipper is facing away from robot and leg holes are pointing down.
3. Place the right leg into the rightmost leg hole.
4. Place the left leg into the leftmost leg hole.
5. Shimmy pants up so that the belt loops are at the waist level.
6. Zip up and button pants.

Something that might be very simple to you might be pretty complicated for a robot. You may end up with a large number of instructions just to make sure your robot doesn’t go out the door with it’s pants backwards or on top of it’s head.

All in all, we had some great breakthroughs working with our mathematicians. I’ll leave you with one last math moment, from our mathematical artifacts module. We made figures out of Sonobe units today, and as I stacked three Sonobe unit cubes on top of one another, I wondered how many Sonobe units it would take to build a hollow rectangular prism with the same dimensions as my three stacked cubes.

Stay tuned for more blog posts on our progress as the week goes on! I'm sure we'll find some more hidden problems to share with you all on this blog.

-Justin Shin