Project Euler – Problem 49 Solution

Yan Cui

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Problem

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.

There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.

What 12-digit number do you form by concatenating the three terms in this sequence?

Solution

open System.Collections

let rec distribute e = function
    | [] -> [[e]]
    | x::xs' as xs -> (e::xs)::[for xs in distribute e xs' -> x::xs]

let rec permute = function
    | [] -> [[]]
    | e::xs -> List.collect (distribute e) (permute xs)

let rec comb n l =
    match n, l with
    | 0, _ -> [[]]
    | _, [] -> []
    | k, (x::xs) -> List.map ((@) [x]) (comb (k-1) xs) @ comb k xs

let max = 9999

// define a cache for holding records of which number is a prime
let cache = new BitArray(max+1, true)

// using prime sieve to fill out the cache
[2..max]
    |> List.iter (fun n ->
        if cache.[n] then
            [2..max]
            |> Seq.takeWhile (fun m -> n * m <= max)
            |> Seq.iter (fun m -> cache.[n * m]  List.filter (fun n -> cache.[n])

// define function to get the 4-digit prime permutations of a number
let getPrimePermutations n =
    let digitsStr = n.ToString().ToCharArray() |> Array.map string
    Array.toList digitsStr
    |> permute
    |> Seq.distinct
    |> Seq.map (fun chars -> int(chars |> List.reduce (+)))
    |> Seq.filter (fun x -> x >= 1000 && cache.[x])
    |> Seq.sort
    |> Seq.toList

let answer =
    primeNumbers
    |> List.map getPrimePermutations
    |> List.filter (fun l -> l |> List.length >= 3)
    |> Seq.distinct
    |> Seq.toList
    |> List.map (fun l -> comb 3 l |> List.filter (fun l' -> l'.[1] - l'.[0] = l'.[2] - l'.[1]))
    |> List.filter (fun l -> l |> List.length > 0)

The above solution returns two lists:

val answer : int list list list = [[[1487; 4817; 8147]]; [[2969; 6299; 9629]]]

The first of course, correlates to the example given in the brief, the other, is the base for your answer!

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