#### Problem

Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.

It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 62%.

If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?

#### Solution

let hasDivisor(n) =

let upperBound = int(sqrt(double(n)))

[2..upperBound] |> Seq.exists (fun x -> n % x = 0)

let isPrime(n) = if n = 1 then false else not(hasDivisor(n))

// define function that returns the number on the corners of a spiral of length n

let getCornerNumbers n =

match n with

| 1 -> [1]

| _ when n % 2 = 0 -> []

| _ -> [3..-1..0] |> List.map (fun n’ -> n*n – n’*(n-1))

let answer =

let mutable cornerNumbers, primeNumbers, size = 0, 0, 1

let mutable continueLoop = true

while continueLoop do

// get the numbers that appear at the corners of a spiral of the given size

let newNumbers = getCornerNumbers size

// increment the totals

cornerNumbers <- cornerNumbers + newNumbers.Length
primeNumbers <- primeNumbers + (newNumbers |> List.filter isPrime |> List.length)

let ratio = double(primeNumbers) / double(cornerNumbers)

if ratio < 0.1 && size > 1 then continueLoop <- false else size <- size + 2
size
[/code]
**UPDATE:** Having stumbled upon some very good algorithms for generating prime number sequence here, I decided to revisit my solution here and using the *PGSimple3* algorithm the new code now runs in seconds!

// generate all prime numbers under <= this max let max = 100000 // initialise the list with 2 which is the only even number in the sequence let mutable primeNumbers = [2] // only check the prime numbers which are <= the square root of the number n let hasDivisor n = primeNumbers |> Seq.takeWhile (fun n’ -> n’ <= int(sqrt(double(n)))) |> Seq.exists (fun n’ -> n % n’ = 0)

// only check odd numbers <= max let potentialPrimes = Seq.unfold (fun n -> if n > max then None else Some(n, n+2)) 3

// populate the prime numbers list

for n in potentialPrimes do

if not(hasDivisor n) then primeNumbers <- primeNumbers @ [n]
// use the same hasDivisor function instead of the prime numbers list as it offers
// far greater coverage as the number n is square rooted so this function can
// provide a valid test up to max*max
let isPrime n = if n = 1 then false else not(hasDivisor(n))
// define function that returns the number on the corners of a spiral of length n
let getCornerNumbers n =
match n with
| 1 -> [1]

| _ when n % 2 = 0 -> []

| _ -> [3..-1..0] |> List.map (fun n’ -> n*n – n’*(n-1))

let answer =

let mutable cornerNumbers, primeNumbers, size = 0, 0, 1

let mutable continueLoop = true

while continueLoop do

// get the numbers that appear at the corners of a spiral of the given size

let newNumbers = getCornerNumbers size

// increment the totals

cornerNumbers <- cornerNumbers + newNumbers.Length
primeNumbers <- primeNumbers + (newNumbers |> List.filter isPrime |> List.length)

let ratio = double(primeNumbers) / double(cornerNumbers)

if ratio < 0.1 && size > 1 then continueLoop <- false else size <- size + 2
size
[/code]

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.

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.

Further reading

Here is a complete list of all my posts on serverless and AWS Lambda. In the meantime, here are a few of my most popular blog posts.

- Lambda optimization tip – enable HTTP keep-alive
- You are thinking about serverless costs all wrong
- Many faced threats to Serverless security
- We can do better than percentile latencies
- I’m afraid you’re thinking about AWS Lambda cold starts all wrong
- Yubl’s road to Serverless
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- AWS Lambda – compare coldstart time with different languages, memory and code sizes
- Guys, we’re doing pagination wrong