Project Euler – Problem 28 Solution


Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:


It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?


let rec sumDiagonals n m total max =
    match n with
    | 1 -> sumDiagonals (n+1) m 1 1
    | _ when n > m -> total
    | _ when n % 2 = 0 -> sumDiagonals (n+1) m total max
    | _ ->
        let newValues = [1..4] |> (fun x -> max + x * (n-1))
        let newMax = newValues |> List.max
        let newTotal = total + (newValues |> List.sum)
        sumDiagonals (n+1) m newTotal newMax

let answer = sumDiagonals 1 1001 0 0

As you’ve probably noticed already, there’s a pattern in the sequence 1, 3, 5, 7, 9, 13, 17, 21, 25, … generated by the spiral in the problem brief. As the dimension goes up by 2 at a time, all odd numbers, 4 new numbers are added to the sequence:

Dimension (d) Numbers on the edge of the square Max number from last dimension
1 1
3 3, 5, 7, 9 1
5 13, 17, 21, 25 9

Looking at the above table, it quickly becomes clear that the new numbers that are added to the diagonals when you expand the dimension by 2 can be calculated as:

max number from last dimension + (d-1) * i (where i is a number from 1 to 4)

My solution simply wraps this logic in a recursive function whilst keeping a running total of all the numbers that have been generated.

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