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  1. import math
  2.  
  3. def count_parallelograms():
  4.     count = 0
  5.     n = 500
  6.     # iterate through all distinct side lengths a, b
  7.     # use the constraint: a + b < n
  8.     for a in range(2, n):
  9.         for b in range(1, a): # b < a ensures distinctness
  10.             if a + b >= n:
  11.                 break
  12.  
  13.             # Condition: a and b must be relatively prime
  14.             if math.gcd(a, b) != 1:
  15.                 continue
  16.  
  17.             # Parallelogram Law: d1^2 + d2^2 = 2(a^2 + b^2)
  18.             S = 2 * (a * a + b * b)
  19.  
  20.             # iterate through possible integer values for diagonal d1
  21.             # constraints on d1:
  22.             # 1) triangle Inequality: |a - b| < d1 < a + b
  23.             # 2) to avoid double counting and ensure d2 exists: d1 <= sqrt(S/2)
  24.             lower_bound = a - b + 1
  25.             upper_bound = int(math.isqrt(S // 2))
  26.  
  27.             for d1 in range(lower_bound, upper_bound + 1):
  28.                 # check if d2 is an integer
  29.                 d2_sq = S - d1 * d1
  30.                 d2 = math.isqrt(d2_sq)
  31.  
  32.                 if d2 * d2 != d2_sq:
  33.                     continue
  34.  
  35.                 # we have integer diagonals d1, d2.
  36.                 # now check for Integer Area using Heron's Formula on triangle (a, b, d1)
  37.                 # semiperimeter s must be integer for area to be integer => Perimeter even
  38.                 if (a + b + d1) % 2 != 0:
  39.                     continue
  40.  
  41.                 s = (a + b + d1) // 2
  42.                 # area squared = s(s-a)(s-b)(s-d1)
  43.                 area_sq = s * (s - a) * (s - b) * (s - d1)
  44.  
  45.                 if area_sq > 0:
  46.                     root_area = math.isqrt(area_sq)
  47.                     if root_area * root_area == area_sq:
  48.                         # All conditions met: integer sides, diagonals, and area
  49.                         count += 1
  50.                         # optional: Print found parallelograms
  51.                         # print(f"Sides: ({a}, {b}), Diagonals: ({d1}, {d2}), Area: {root_area * 2}")
  52.  
  53.     return count
  54.  
  55. print(count_parallelograms())
Success #stdin #stdout 1.39s 14036KB
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https://ideone.com/zqys57
language:
Python 3 (python  3.12)
created:
1 day ago
visibility:
public

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