euler0094.py

#!/usr/bin/python
# -*- coding: utf-8 -*-
"""Almost equilateral triangles

   It is easily proved that no equilateral triangle exists with integral length
   sides and integral area.  However, the *almost equilateral triangle* 5-5-6
   has an area of 12 square units.

   We shall define an *almost equilateral triangle* to be a triangle for which
   two sides are equal and the third differs by no more than one unit.

   Find the sum of the perimeters of all *almost equilateral triangles* with
   integral side lengths and area and whose perimeters do not exceed one
   billion (1,000,000,000)."""

from math     import sqrt, ceil
from eulerlib import perfectSqrt

__tags__ = ['triangles']

def solve():
    accum = 0

    # We must find all triangles with integral sides x,x,y where x = y+d,
    # d∈{1,-1}, such that the area A = yh/2 = y/2 √(x^2 - (y/2)^2) is an
    # integer.  This last expression equals y/4 √(3x^2 + 2xd - d^2); let k =
    # 3x^2 + 2xd - d^2.  If x is even, then both y and k are odd, and so A is
    # not an integer; therefore, x must be odd, and y and k must be even (and
    # y/2 must be an integer).  Moreover, in order for A to be an integer, k
    # must be a perfect square, and as k is also even, it must be a multiple of
    # 4, and so h = √(k/4) is an integer.  Hence, (y/2, h, x) is a Pythagorean
    # triple, and as gcd(x, y/2) = gcd(2y/2 ± 1, y/2) = 1, it must be
    # primitive.  Thus, there exist coprime positive integers m,n such that m-n
    # is positive & odd that satisfy:
    #     y/2 or h   = m^2 - n^2
    #     h   or y/2 = 2mn
    #     z          = m^2 + n^2

    # When y/2 = m^2-n^2 and h = 2mn, substituting into x = y+d and rearranging
    # yields 3n^2 = m^2 + d; it is thus necessary that (m^2+d)/3 is an integer
    # & a perfect square, and as m^2 mod 3 must be either 0 or 1, d must be -1.
    # It can be shown that any positive integer m>1 such that n = √((m^2-1)/3)
    # is an integer is sufficient to produce an integral almost equilateral
    # triangle; this is left as an exercise for the reader.  The triangle will
    # have perimeter 4m^2, and so only the values of m less than or equal to
    # √25e7 = 5000√10 will produce a triangle with perimeter not exceeding a
    # billion.

    for m in xrange(2, int(ceil(5000 * sqrt(10)))):
        n = m*m - 1
        if n % 3 == 0:
            n = perfectSqrt(n // 3)
            if n is not None:
                accum += 4 * m * m

    # When y/2 = 2mn and h = m^2-n^2, substituting into x = y+d and rearranging
    # yields n^2 - 4mn + m^2 - d = 0; it is thus necessary that n = 2m ±
    # √(3m^2+d) is an integer less than m, and this latter requirement
    # restricts us to only considering n = 2m - √(3m^2+d).  Moreover, as 3m^2+d
    # must be a perfect square, modular arithmetic tells us that d must be 1.
    # It can be shown that any positive integer m>1 such that n = 2m -
    # √(3m^2+1) is an integer is sufficient to produce an integral almost
    # equilateral triangle; this is left as an exercise for the reader.  The
    # triangle will have perimeter 12mn + 2, and so only the values of m less
    # than or equal to √((1e9+2)/(23 - 12√3)) will produce a triangle with
    # perimeter not exceeding a billion.

    for m in xrange(2, int(ceil(sqrt(1000000002/(24 - 12 * sqrt(3)))))):
        n = perfectSqrt(3*m*m + 1)
        if n is not None:
            n = 2 * m - n
            accum += 12 * m * n + 2

    return accum

if __name__ == '__main__':
    print solve()