Project Euler Problem 9 introduces the interesting mathematical concept of Pythagorean Triple. Let’s have a look at the question:

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a² + b² = c²For example, 3² + 4² = 9 + 16 = 25 = 5².

There exists exactly one Pythagorean triplet for which a + b + c = 1000.

Find the product abc.

As usual, I first show my solution and then comment on the approach:

#light // generate triplets using Euclid's Formula let pythagorean_triplets top = [ for m in 1 .. top do for n in 1 .. m-1 do let a = m*m-n*n let b = 2*m*n let c = m*m+n*n yield [a;b;c] ] // multiply all the values of a list let multiply_list list = List.fold (fun acc elem -> acc*elem) 1 list // find a triplet where the sum of values // is equal to a given number let find_triplet_with_sum sum = pythagorean_triplets sum |> List.find (fun [a;b;c] -> a+b+c=sum) let problem_9 () = find_triplet_with_sum 1000 |> multiply_list

Let’s go through the solution step by step:

- The first function,
`pythagorean_triplets`

uses Euclid’s Formula to enumerate all possible Pytagorean Triplets up to a given threshold. The formula can be summarized like this:

a = m² – n², b = 2mn, c = m² + n², where m and n are positive integers with m > n.It’s interesting to note that each item of the list is itself a

*list*of three numbers. I could have generated a tuple, but since in the end I need to multiply all the 3 values together, the list was more straightfoward to use. `multiply_list`

is just a convenience function to multiply all the elements of a list together`find_triplet_with_sum`

does the heavy work of generating the actual triplets picking the first one where the sum is equal to a given value.- At the end I just put the pieces together and solve the problem.

One caveat: Euclid’s Formula does not generate *all* Pytagorean Triplets(there are other formulas that do). I can say that I have been lucky that it generated the one requested by the Project’s Euler problem. The code above might not work if the goal is changed to find a triplet with a different product.

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