Project Euler – Problem 15 Solution

Problem

Starting in the top left corner of a 2×2 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 20×20 grid?

Solution

let rec factorial(n:bigint) = if n <= 1I then 1I else n * factorial(n-1I)
let combo n k = factorial(n) / (factorial(k) * factorial(n-k))
let answer = combo 40I 20I

It took me a while to figure out that this problem is actually a simple combination problem – consider a X by Y grid, any route from the top left to the bottom right corner without backtracking must have travelled Right X number of times and Down Y number of times. In the case of the original example:

RRDD, RDRD, RDDR, DRRD, DRDR, DDRR

This also means that all routes have a total of X + Y steps, and the number of routes is equal to the number of ways you can pick X number of R moves out of X + Y, i.e.

image

where n = X + Y and k = X.

Liked this article? Support me on Patreon and get direct help from me via a private Slack channel or 1-2-1 mentoring.
Subscribe to my newsletter


Hi, I’m Yan. I’m an AWS Serverless Hero and the author of Production-Ready Serverless.

I specialise in rapidly transitioning teams to serverless and building production-ready services on AWS.

Are you struggling with serverless or need guidance on best practices? Do you want someone to review your architecture and help you avoid costly mistakes down the line? Whatever the case, I’m here to help.

Hire me.


Check out my new course, Complete Guide to AWS Step Functions. In this course, we’ll cover everything you need to know to use AWS Step Functions service effectively. Including basic concepts, HTTP and event triggers, activities, callbacks, nested workflows, design patterns and best practices.

Get Your Copy