All posts by Samuel Coskey

Recursion and the four numbers game

Recursion means doing a process again and again.

In the lesson portion of this meeting, we introduced the idea of recursion with the Pass the Candy game. In this game everyone starts with a pile of candy, and passes half of it to the left. When you repeat the process, everyone eventually ends up with the same amount of candy. If you can write down the recursive equation that simulate the game, then you can use math or spreadsheets to explain why this happens.

In the investigation portion of this meeting, we explored the Four Numbers Game. In this game one starts with four numbers on the corners of a square. One then forms a new square inside by finding the difference between each pair of adjacent numbers. When this is repeated, one eventually comes to the four numbers 0,0,0,0. Does it always work this way? If so, why?

Check out our handout:
Four Numbers Game Handout

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

Continue reading The greatest common divisor and probability

The BMC is now recruiting for 2017–18

Please apply for the 2017–18 Boise Math Circle program! The BMC is open to middle and high school students with some experience in algebra. If you like solving problems and want to learn more about the creative side of math, please consider applying. If you have a friend that does, invite them to apply too!

We are excited to announce one small change to our program. If you are familiar with our sessions, we generally spend all 90 minutes on an in-depth exploration. This Fall, we will begin each session with a quick introduction to a major concept in math, and then proceed with a closely related exploration afterwards. This means participants will learn or review key concepts such as modular arithmetic, matrices, enumerations, logic, and more!

We will begin on Saturday morning, September 2, and meet 15 times during the school year. See our schedule of meeting dates.


Please help us recruit participants by letting your students know about the program. If you’d like attendance reports for your students (e.g., for extra credit), please send us an email. You’re of course welcome to join the sessions.

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! Then check out our handouts.
Automata exercises
Regular expression exercises

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 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!

Continue reading The secret lives of permutations