# The greatest common divisor and probability

The greatest common divisor of two numbers is the largest number that goes evenly into both. For example, the greatest common divisor of 30 and 45 would be 15. On the other hand the greatest common divisor of 30 and 41 would only be 1.

In the first portion of this meeting, we gave a lesson on an efficient way to calculate the greatest common divisor of two numbers: The Euclidean algorithm.

In the investigation portion of this meeting, we asked, given two random numbers, what is most likely to be the greatest common divisor of the two numbers? Using dozens of pairs of random numbers, and dividing the work among our participants, we discovered that the number 1 was far more likely to be the greatest common divisor than any other number. But why is it true?

Check out our handouts!
GCD handout
GCD worksheet

# Sending coded messages

In this discussion we explored Caesar ciphers and keyword substitution ciphers. We borrowed a handout from another math circle. We especially had fun decoding the Alice in Wonderland quotation on page 6! (Although the cipher was a little harder than just an ordinary keyword cipher.)

# Recognizing patterns

A “string” is just a sequence of letters or symbols. It could be a word, a sentence, a book, a code, a number, or anything. In computing it is important to be able to recognize patterns in strings. For example if your phone receives a signal from the internet, it needs to know whether it is intended for your web browser or your email reader.

An “automaton” is a machine that decides whether a given string has a certain pattern. Thus an automaton is like a very simple computer, and they can be very fun to build. What kinds of patterns can an automaton detect?

Build your own automaton at automatonsimulator.com! Then check out our handouts.
Automata exercises
Regular expression exercises

# Falling Chickens

In this circle we investigated something called the “falling sand pile”. But sand is boring, so instead we studied falling chickens! The primary rule of falling chickens is

When 4 chickens land in someone’s backyard, they “break out” and go to the neighbors’ yards instead.

# Patterns in data

Suppose you look up an index of the lengths of the world’s rivers. What digit do the lengths start with? Are there the same number of rivers whose length starts with a 1 as there are rivers whose length starts with a 2?

The digits of random data reveal to some very surprising patterns. So find your favorite data set and then check out our worksheets!

Handout 1 Worksheet to explore the first digits of your numbers
Handout 2 Counting numbers up to a power of ten
Handout 3 Exploring why you see what you do

# The logic of calculators

How can a calculator use the flow of electricity to add two numbers together? In this activity we explored the building blocks of electrical circuits, and the mathematical logic that lies behind it.

# Parametric equations

In this activity we explored periodic parametric curves. We particularly studied the patterns that arise when you change the periods of the $x(t)$ and $y(t)$ coordinates. To start playing around with parametric curves, you will need geogebra. Then you can use the following link to download our geogebra worksheet!

Parametric curve worksheet

# The secret lives of permutations

How can a line of people sort themselves by height? Or by age? It depends on what operations they are allowed to use. One might consider using “swaps” where two people in the line switch position with each other. One might also consider using “cycles” where someone at the end of the line moves to the beginning of the line.

This leads to an investigation of “permutations”, which is a fancy word for rearrangements of a set of objects. Some sets of permutations allow the people in the line to sort themselves in any order they choose, while other sets of permutations don’t.

Try working on our activity!

# Bart vs Lisa vs Fractions

Bart and Lisa compete in a two-day spelling contest. On the first day, Lisa gets a higher percentage right than Bart. And on the second day, Lisa gets a higher percentage right than Bart. So why did Bart win?? Continue reading Bart vs Lisa vs Fractions