solve45.hs

-- https://projecteuler.net/problem=45
-- Run with: 'ghc solve45.hs && ./solve45' or 'runhaskell solve45.hs'
-- using Haskell with GHC 8.0.2
-- by Zack Sargent

{- Prompt:

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

  Triangle     Tn=n(n+1)/2    1, 3, 6, 10, 15, ...
  Pentagonal   Pn=n(3n−1)/2   1, 5, 12, 22, 35, ...
  Hexagonal    Hn=n(2n−1)     1, 6, 15, 28, 45, ...

It can be verified that T285 = P165 = H143 = 40755.
Find the next triangle number that is also pentagonal and hexagonal.

-} 
{- NOTE: This is a quick and dirty solution. 
   From the fourm, the use of Diophantine equations could significantly optimize this process. -}

-- Note: All hexagonal numbers have to be triangle numbers
-- so we don't need to check for triangle numbers.
rule n = toInteger (round n)
pentagonalNum n = rule ((n * (3*n - 1))/ 2)
hexagonalNum  n = rule  (n * (2*n - 1))

find x a b
  | x == pentagonalNum a && x == hexagonalNum b = x
  | x >  pentagonalNum a = find x (a + 1) b
  | x >  hexagonalNum  b = find x a (b + 1)
  | x `mod` 2 == 0       = find (x + 1) a b
  | otherwise            = find (x + 2) a b

findAbove x = find x 5 5

main = print (findAbove 1533700000)
-- -> 1533776805