Project Euler – Problem 77 Solution

Problem

It is possible to write ten as the sum of primes in exactly five different ways:

7 + 3

5 + 5

5 + 3 + 2

3 + 3 + 2 + 2

2 + 2 + 2 + 2 + 2

What is the first value which can be written as the sum of primes in over five thousand different ways?

Solution

// generate all prime numbers under <= this max
let max = 1000
let mutable primeNumbers = &#91;2&#93;

// 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 @ &#91;n&#93;
let isPrime n = if n = 1 then false else not(hasDivisor(n))

// implement the coin change algorithm
let rec count n m (coins:int list) =
    if n = 0 then 1
    else if n < 0 then 0
    else if (m <= 0 && n >= 1) then 0
    else (count n (m-1) coins) + (count (n-coins.[m-1]) m coins)

let answer =
    let tuple =
        Seq.unfold (fun state -> Some(state, state+1)) 10
        |> Seq.map (fun n -> (n, primeNumbers |> Seq.filter (fun n' -> n' < n) |> Seq.cache))
        |> Seq.filter (fun (n, l) -> count n (Seq.length l) (Seq.toList l) > 5000)
        |> Seq.head

    fst tuple

Yet another twist to problem 31 and problem 76, this time the coin change algorithm will be supplied with only prime numbers less than n.

 

Learn to build Production-Ready Serverless applications

Want to learn how to build Serverless applications and follow best practices? Subscribe to my newsletter and join over 3,000 AWS & Serverless enthusiasts who have signed up already.
As a BONUS, you will receive early access and discount for my new AppSync course.

Leave a Comment

Your email address will not be published.