Project Euler problem 312 is one of the harder problems we have worked through. It combines graph theory, fractal geometry, combinatorial counting, and modular arithmetic, and the final answer requires computing C(C(C(10000))) mod 13^8 where C(n) is the number of Hamiltonian cycles on a Sierpiński graph of order n.

We ended up leaning on OEIS to get the recurrence in closed form, which we will confess to below. Full notes: Project Euler/312.

The Problem

A Sierpiński graph of order 1, denoted S_1, is an equilateral triangle - three vertices, three edges.

S_{n+1} is …

Project Euler problem 227 - "The Chase" - is a nice example of a problem where the brute force approach is completely reasonable to reach for first, completely infeasible to actually finish, and directly points you at the right mathematical reformulation.

This is what we ended up with. Our full working notes are on the wiki at Project Euler/227.

The Problem

An even number of players sit around a table. Two players sitting directly opposite each other each start with a die. Each round, both players roll:

• Roll a 1 → pass the die to your left neighbor
• Roll a 6 → pass …

Project Euler problem 198 is a number theory problem that turns out to be almost entirely about continued fractions. The problem hides this - the statement talks about "ambiguous" real numbers - but the ambiguity has a clean characterization in terms of continued fraction expansions, and once you see it, the problem gets a lot smaller.

Wiki notes: Project Euler/198.

The Problem

Define a best approximation to a real number x with denominator bound d as a rational r/s in reduced form with s ≤ d, such that any other rational p/q closer to x than r/s has q …

Project Euler problem 60 is a nice mid-difficulty combinatorial search problem. It has enough structure that pure brute force falls over, and enough structure that a handful of small observations gives you a very tractable search.

Our working notes: Project Euler/60.

The Problem

The primes 3, 7, 109, and 673 are "quite remarkable" - concatenating any two of them in either order gives another prime. So 7109, 1097, 3673, 6733, 109673, 673109, and so on, are all prime. That is 12 concatenations, all prime.

The sum of these four primes is 792, and the problem states this is the smallest …

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

• 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 …