Project Euler – Problem 33 Solution

Problem

The fraction ⁴⁹/₉₈ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that ⁴⁹/₉₈ = ⁴/₈, which is correct, is obtained by cancelling the 9s.

We shall consider fractions like, ³⁰/₅₀ = ³/₅, to be trivial examples.

There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.

If the product of these four fractions is given in its lowest common terms, find the value of the denominator.

Solution

// define function to check if two numbers has the cancelling behaviour
let isCancelling a b =
    let aStr, bStr = a.ToString(), b.ToString()
    if aStr.[0] = bStr.[0] then double(aStr.[1].ToString())/double(bStr.[1].ToString()) = double(a)/double(b)
    else if aStr.[0] = bStr.[1] then double(aStr.[1].ToString())/double(bStr.[0].ToString()) = double(a)/double(b)
    else if aStr.[1] = bStr.[0] then double(aStr.[0].ToString())/double(bStr.[1].ToString()) = double(a)/double(b)
    else if aStr.[1] = bStr.[1] then double(aStr.[0].ToString())/double(bStr.[0].ToString()) = double(a)/double(b)
    else false

// define function to find the greatest common dividor
let gcd a b = [2..min a b] |> List.rev |> Seq.filter (fun x -> a % x = 0 && b % x = 0) |> Seq.head

// define function that returns numbers >= n which are not multiples of 10
// and have two distinct digits (i.e. 11, 22 only have one distinct digits)
let numbers n = [n..99] |> List.filter (fun x -> x % 10 <> 0 && (x % 10 <> x/10))

let answer =
    // first work out the product of all 4 fractions, in fractions form (num, denom)
    let fraction =
        numbers 10
        |> List.collect (fun x -> numbers (x+1) |> List.map (fun y -> (x, y)))
        |> List.filter (fun (x, y) -> isCancelling x y)
        |> List.reduce (fun (num, denom) (x, y) -> (num*x, denom*y))
    // then define the denominator by the greatest common divisor
    (snd fraction) / gcd (fst fraction) (snd fraction)