Favorite Pi Approximations

What's your favorite \(\pi\) approximation?

Some of my favorite approximations of \(\pi\) come from Ramanujan-Sato series. These are mathematical series that generalize from a remarkable formula for \(\pi\) given by Srinivasa Ramanujan, an Indian mathematician:

Table of Contents

• Overview: Problem 172
• Background
  • Multiset Permutations
  • A Simple Problem (And Solution)
  • A Slightly More Complicated Problem
• Algorithm
  • Ignoring Numbers Starting with 0
• Code
  • Test Cases
  • Recursion
  • Pseudocode
  • Python Code
• Final Answer

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 …

Table of Contents

• Introduction
• Complete and Reduced Residue Systems
• Euler's Totient Function
• Calculating the Totient Function by Hand

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:

or

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

Table of Contents

• Overview: The Problem
• Polyominoes
  • Castle Rules for Polynominoes
• Don't Generate - Enumerate!

Overview: The Problem

In an earlier post, I mentioned my efforts on Project Euler problems and the wide variety of problems there that can offer some profound mathematical insights.

Given that the first post covered Project Euler problem 1, I thought it would be nice if the next problem cranked up the difficulty factor by an order of magnitude. Project Euler Problem 502 is a very hairy combinatorics problem that required me to learn about a wide variety of combinatorial enumeration techniques.

Let's start with the …

Table of Contents

• Overview: The Problem
• Why This Problem?
• Going Deeper: An Example

Overview: The Problem

Project Euler is a website that provides mathematically-oriented programming problems. There are many (over 500) and they are a rich source of profound mathematical insights.

I have been considering a writeup that goes deep into a particular problem, so why not do it with problem 1?

Problem 1 of Project Euler asks:

Find the sum of all the multiples of 3 or 5 below 1000.

It is a pretty simple task - one of the first things covered in a decent programming course is the …

The Lattice Paths Problem

I first came across the lattice paths problem in Project Euler problem 15. The question described a 2x2 square lattice, and illustrated the 6 ways of navigating from the top left corner to the bottom right corner by taking the minimum number of steps - 2 right …

Table of Contents

• Number Systems and Representations
• Continued Fraction Representations
• Convergents of Continued Fractions
• Example: Continued Fraction Coefficients of \(\sqrt{14}\)
• Example: Convergents of \(\sqrt{14}\)
• Approximating Square Roots

Continued Fractions

Let's start part 2 of our discussion of computing square roots by talking about continued fractions. When we first learn mathematics, we learn to count in the base 10 system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We can construct representations of all of the integers using these 10 digits, by arranging them in a different order. So, for example, saying 125 is equivalent to saying …

Table of Contents

• Newton's Method for Finding Roots of Equations

• Newton's Method for Finding Square Roots

• Newton's Method for Finding Square Roots: Program

• Accuracy

• Speed

• Next Steps

• References

Newton's Method for Finding Roots of Equations

Suppose we have a function \(f(x)\) and we want to compute values of \(x\) for which \(f(x)=0\). These values of \(x\) are called the roots of \(f(x)\).

We can compute the roots using Newton's Method, which utilizes the derivative of the function to iteratively compute the roots of the function.

Newton's method begins with an initial guess. It evaluates the derivative …