Solution to Crux Mathematicorum M379

crux-mathematicorum-M379.py

"""
Problem M379, Crux Mathematicorum

Proposed by John Grant McLoughlin, University of New Brunswkci,
Fredericton, NB.

The integers 27+C, 555+C, and 1371+C are all perfect squares, the square
roots of which form an arithmetic sequence. Determine all possible values
of C.
"""

from sympy.solvers import solve
from sympy import symbols

x, y, C = symbols('x y C')
system = [(x - y)**2 - (27 + C),
          x**2 - (555 + C),
          (x + y)**2 - (1371 + C)]
print solve(system)

# Expected output: [{C: 229, x: -28, y: -12}, {C: 229, x: 28, y: 12}]

We need to solve the following system of equations.

(x-y)^2 = 27 + C
x^2 = 555 + C
(x+y)^2 = 1371 + C

The reader can verify that

(x-y)^2 - 2x^2 + (x+y)^2 = 2y^2 and
(27 + C) - 2(555 + C) + (1371 + C) = 288.

Therefore, 2y^2 = 288, and so y = 12.

Furthermore,

(x+y)^2 - (x-y)^2 = 4xy and
(1371 + C) - (27 + C) = 1344,

hence 4xy = 1344, and so x = 1344 / 48 = 28. Since x^2 = 555 + C, we conclude that C = 28^2 - 555 = 229.