# Graphs for Bioinformatics, Part 1: de Bruijn Graphs, Hamiltonian Paths, and Eulerian Paths

Posted in Computational Biology

# The Context: Rosalind.info

To provide a bit of context for a discussion of Euler paths and Euler cycles: starting around December, a group of us in the Lab for Data Intensive Biology (DIB Lab) started working through the textbook Bioinformatics Algorithms: An Active Learning Approach and the associated website, Rosalind.info.

Rosalind.info is a site that is similar in style …

# Project Euler Problem 172

Posted in Mathematics

# Overview: Problem 172

How many 18-digit numbers $$n$$ (without leading zeros) are there such that no digit occurs more than three times in $$n$$?

Link to Project Euler Problem 172

# Background

Project Euler Problem 172 is your classic Project Euler problem: short, simple, and overwhelmingly complicated.

To nail this one, it's important to start simple - very simple. What I'll do is walk through the process of …

# Euler's Theorem, the Totient Function, and Calculating Totients By Hand

Posted in Mathematics

## Introduction

Today we're going to delve into a little bit of number theory.

In number theory, we are usually dealing with modular arithmetic - expressions of the form:

$$a \equiv b \mod m$$

or

$$f(x) \equiv 0 \mod m$$

The mod indicates we're doing modular arithmetic, which is (formally) an algebraic system called a ring, which consists of the integers 0 through m.

An analogy to modular arithmetic is the way that the sine and cosine function "wrap around," and

 \sin \left …

# Project Euler Problem 1

Posted in Mathematics