## Wednesday, August 27, 2014

### Day 3

We are halfway through our time together, and we have built up our mathematical muscles with our Puzzles and Logic, Problem Solving, Computer Science, Proofs and Investigations, and Mathematical Artifacts modules. In fact, when I went to the doctor today, he told me that every day I go to Bard Math CAMP, the Math Lobe in my brain grows three times larger, and the Writing Lobe in my brain grows one and a half times larger. (I guess from writing all of these blog posts!)

If I started Bard Math CAMP with my Writing Lobe four times larger than my Math Lobe, on what day of camp will my Writing Lobe and Math Lobe be the same size? How many times larger will my Math Lobe be than my Writing Lobe at the end of the fifth (and last!) day of camp? How many times larger will my Math Lobe be than my Writing Lobe after n days?

The work we did today with ratios, and our work two days ago with Sequences might help with this one.

Our problem solving class worked on some good ratio problems today, first exploring fuel consumption on a car trip, and then examining a case of matching and mismatched socks. If you enjoyed our work with Ratios today, Frances (The lead mathematician in our Problem Solving module) gave me an extra problem to chew on, adapted from the New York Math League.

Mr. Zilch’s pocket just became empty.  Nine minutes ago, his pocket was exactly half full.  He was putting coins into it at a rate that would have filled his empty pocket in 36 minutes if it did not have a hole in it.  If his pocket were full at the moment this hole occurred, and he didn’t add any coins, how many minutes would it take for the pocket to empty?
 One of our mathematicians cooked up this handy diagram to help us think about fuel use on a car trip.
We took another field trip today in Proofs and Investigations, but instead of traveling in a time machine, we all took a visit to Hilbert’s Hotel, the only infinite hotel in the world, to help Dr. Hilbert assign room numbers to his guests. Dr. Hilbert’s Hotel is a popular resting place for mathematicians, as they know that they can trust Hilbert to give them a room. But the infinite number of rooms isn’t the only thing strange about Hilbert’s Hotel, Hilbert’s Hotel also has a special Elevator. When you enter the elevator and press a floor number, for every minute that passes, the elevator will close half of the remaining distance. For instance, if I went into this elevator and pressed floor number 40, after the first minute it would take me to floor 20, and after another minute I would be taken to floor 30, and after another minute I would be taken to floor 35, and so on and so forth.

If a mathematician enters the elevator and presses floor number 128, how many minutes will have passed when he hits floor 124? The mathematician still hasn’t come out of the elevator, and it’s been about six hours… what’s taking him so long? Will he ever leave the elevator? Why or why not?

If you would like to learn more about infinity, here is an interesting video made ViHart, a mathematician youtuber.

For our computer science module, we learned about loops and how to use the setup function in the Processing computer language. Unfortunately, I wasn’t paying attention, and I need help interpreting a code that Yulia (Our lead mathematician in our Computer Science module) gave me.

int x;
int y;

void setup(){
size(500, 500);
x = 30;
y = 30;

}

void draw(){
background(0);
rect(x, y, 30, 30);
x = x + 1;
y = y + 1;
}

I can see that we are drawing a rectangle, but when I run this code, what direction will the rectangle move?

For out Mathematical Artifact of the day, we discovered made our own tessellation patterns!

Can you tell me how many different regular polygons can be tessellated on a flat surface? How do you know that you haven’t missed a tesselateable polygon?

Finally, for our Puzzle and Logic module, we tried to use probability to boost our odds at a guessing game. By analyzing the different possible combinations of red and blue dots, we could determine a strategy that gives us a correct guess about 75% of the time, greater than the 50% success rate of random guessing. If you liked this probability exercise, try this one, posed by Martin Gardener in his monthly column for Scientific American:

1. Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
2. Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Are the probabilities the same?

… Are you sure?

Martin Gardner is famous for his column that explored a variety of fascinating mathematical puzzles. If you'd like some more, he has collected some of his most well known problems in this book.

As we pass the halfway point of our program, I can only look forward to what the next two days will bring. Together, we’ve already explored Graph Theory, Ratios, Path Counting, Loops, Modular Origami, Sequences, Variables, and many other facets of mathematics. Until tomorrow!

-Justin Shin