## Thursday, August 28, 2014

### Day 4

Today was a day filled with shapes and colors of all sorts! In our Problem Solving module, we explored the volume and surface area of several Menger cubes. Using interlocking colored cubes to help us visualize the first step of creating a Menger sponge, we used patterns in the rise and fall of the volume and surface area to determine that a true Menger sponge has infinite Surface area but no volume!
 Mathematicians discuss how to calculate the Surface Area and Volume of level one Menger cubes.
Afterwards, we tried to build our very own Menger cube with some business cards one of our resident mathematicians found. Working together, we managed to construct some cubes of our own. We also got to continue our discussion on infinity we started yesterday in our Proofs and Investigations module.
 Mathematicians begin forming the starting cubes for a Menger Sponge.
A Menger cube is made by removing segments of a cube in an orderly fashion. Another fractal that works on a similar principle is a Sierpinski triangle. I have the first two steps shown below.
 Image Cre

Can you describe what happens in each step? If the area of the first triangle is 1 unit squared, what is the area of the step 3 figure?

In our Computer Science module, we continued working with the Processing computer language. Learning about if-then statements and Boolean operations added more tools to use for our final project, which will be presented tomorrow. We all use “if  ___ then”, “and”, “or”, and “not” in our daily conversations along with using them to help us code. My own Proofs teacher, Professor Ethan Bloch, proposed this question that tested my ability to understand logical operators:

If Susan likes fish, then she likes onions. If Susan does not like garlic, then she does not like onions. If she likes garlic, then she likes guavas. She likes fish or she likes cilantro. She does not like guavas. Can you tell if Susan like cilantro?

For our Puzzles and Logic segment today, we tried our hand at solving bridge-crossing problems. We used a game tree diagram to analyze the problem once we figured out some solutions. Good thing we went over a method of analyzing games, as I stumbled upon this unfinished tic-tac-toe puzzle:
It’s X’s turn. Can you draw a game tree describing all possible end states for this Tic-Tac-Toe board? How many different end states can this game result in?
 A game tree for a river crossing problem.
For our Proofs and Investigations module, we played around with one of my favorite environments for math problems, chessboards! As we attempted to tile chessboards with 2 x 1 domino pieces, I was reminded of a famous problem known to many mathematicians and chess players.

Can I place eight queens on a standard chessboard so that none of the queens are attacking one another?

When checking your solution, remember to check for diagonal attacks from each queen!

We have only one day left, and our resident mathematicians are busy preparing their last problem sets. Get ready, tomorrow we will be finishing up our final computer science projects as well as exploring some new problems in our modules.

-Justin Shin