Project Euler – Problem 43 Solution

Problem

The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.

Let d₁ be the 1^st digit, d₂ be the 2^nd digit, and so on. In this way, we note the following:

• d₂d₃d₄=406 is divisible by 2
• d₃d₄d₅=063 is divisible by 3
• d₄d₅d₆=635 is divisible by 5
• d₅d₆d₇=357 is divisible by 7
• d₆d₇d₈=572 is divisible by 11
• d₇d₈d₉=728 is divisible by 13
• d₈d₉d₁₀=289 is divisible by 17

Find the sum of all 0 to 9 pandigital numbers with this property.

Solution

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)

// generate the 0 to 9 pandigitals
let numbers = permute [0..9] |> List.map (fun l -> l |> List.map string |> List.reduce (+))

// the corresponding prime divisors and digits
let primes = [2; 3; 5; 7; 11; 13; 17]
let ns = [2..10] |> Seq.windowed 3 |> Seq.toList

// returns the number retrieved from taking the digits at the supplied positions
let d ns (numberStr:string) = 
    int(ns |> Array.map (fun n -> numberStr.[n-1].ToString()) |> Array.reduce (+))

// predicate which identifies pandigital numbers with the desired property
let predicate number = List.forall2 (fun n p -> (d n number) % p = 0) ns primes

let answer = numbers |> List.filter predicate |> List.sumBy int64

One thing you might notice in the creation of the permutations of all 0 to 9 pandigital numbers is the use of (+) in the List.reduce function call. This is actually just the shortened form of:

List.reduce (fun acc item –> acc + item)

Another List function of interest here is the List.forAll2 function, which tests if all corresponding elements in both lists satisfy the given predicate pairwise. In this case, I’m using it to pair up the digits list ([[2;3;4]; [3;4;5];…]) with the primes list ([2;3;5;..17]) to test if a given number has the desired property as described in the problem brief.

We arrive at the answer by simply adding up all the pandigital numbers which matches the predicate condition.