Tag: mathematics


Project Euler Problem 172

Posted in Mathematics

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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 …



Tags:    computer science    mathematics    factors    sequences    euler    project euler   


4x4 Rubik's Cube: Part 1: Representations

Posted in Rubiks Cube

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This is Part 1 of a 4-part blog post on the mathematics of the 4x4 Rubik's Cube, its relation to algorithms, and some curious properties of Rubik's Cubes.

You are currently reading Part 1 of this blog post: Part 1: Representations

See Part 2 of this blog post here: Part 2: Permutations

See Part 3 of this blog post here: Part 3: Factoring Permutations

See Part 4 of this blog post here: Part 4: Sequence Order

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Tags:    rubiks cube    mathematics    combinatorics    permutations    python    puzzles   


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

Posted in Mathematics

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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 …


Tags:    mathematics    factors    number theory    euler