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(3n1)/2		1, 5, 12, 22, 35, ...
Hexagonal		Hn=n(2n1)		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.

+-#integer_sqrt
def integer_sqrt(n)
  return Math.sqrt(n).to_i if n < 1000000
  guesses=[n/2]
  guesses.push((guesses.last+n/guesses.last)/2)
  guesses.push((guesses.last+n/guesses.last)/2) until guesses[-1] == guesses[-2]
  guesses.last
end

+-#triangles
def triangles(n)
  (n*(n+1))/2
end

+-#pentagonals
def pentagonals(n)
  (n*(3*n-1))/2
end

+-#is_pentagonal?
def is_pentagonal?(n)
  m=(1+integer_sqrt(1+24*n))/6
  pentagonals(m)==n
end

+-#hexagonals
def hexagonals(n)
  n*(2*n-1)
end

+-#is_hexagonal?
def is_hexagonal?(n)
  m=(1+integer_sqrt(1+8*n))/4
  hexagonals(m)==n
end

def p45
  i=285
  while true
    t=triangles(i+=1)
    if(is_pentagonal?(t) and is_hexagonal?(t))
      puts t
      return
    end
  end
end