FSharp.it

Today’s exercise is Project Euler Problem 9, that says:

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
a² + b² = c²

For example, 3² + 4² = 9 + 16 = 25 = 5².

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

Therefore this time we have to find a triplet (a, b, c), but a brute-force approach would require a billion combinations, since the three numbers can range from 1 to 1000.

However, we can reduce the number of tests exploiting a little algebra.

First of all, c = 1000 – a – b, so only two variables are in play, leading us to a million choices.

Then we go down to half a million, thanks to the constraint that a < b.

According to these premises, here is the F# code:

#light
let sq x = x * x

let ab =
    { for a in 1 .. 1000
      for b in a .. 1000 -> a,b } |> Seq.filter (fun (a,b) -> sq a + sq b = sq (1000 - a - b) ) |> Seq.hd

let answer =
    let a = fst ab
    let b = snd ab
    let c = 1000 - a - b
    a * b * c

At line 5 we enumerate all items of a collection created with list comprehension, i.e. the numbers between 1 and 1000.

Inside this loop we have another loop, but in this case we are interested in all numbers bigger than the current item of the outer collection, i.e. all numbers between a and 1000.

Then, we filter all couples (a, b) keeping only those for which we have: a² + b² = (1000 – a – b)².

According to the text of the exercise, there exists exactly one Pythagorean triplet that satisfies the equation a + b + c = 1000, so we can apply the Seq.hd (head) function to take the first (and only) element of the generated sequence.

At lines 9, 10 and 11 we evaluate a, b and c and finally we multiply them to get the answer.