Problem 9

Problem 9

25 January 2002

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.

由a² + b² = c²知道，a, b, c可以构成三角形，我们令a<=b<=c,则a的取值范围为[1,333]

又有a² + b² = c²，则a²=(b+c)(c-b), a + b + c = 1000, 得到a²=(1000-a)*(c-b)

这样我们先可以得到符合条件的a, 在进一步得到b, c

def fun9():
  a = -1
  b = -1
  c = -1
 
  for a in range(1, 334):
    bc = 1000-a
    s = a**2
    if s%bc == 0:
      cb = s/bc
      if (bc+cb)%2 == 0:
        c = (bc+cb)/2
        b = bc - c
        if (c>0) and (b>0):
          print a, b, c, a*b*c
          return a*b*c

answer = 31875000