Prove that 3 times the sum of 3 squares is also the sum of 4 squares.
This simply expresses the identity
3(a² + b² + c²)
=(a + b + c)² + (b² − 2bc + c²) + (c² − 2ca + a²) + (a² − 2ab + b²);
=(a + b + c)² + (b − c)² + (c − a)² + (a − b)².
Q.E.D.
Numerical Examples (not thought out).
3(1² + 2² + 3²) = 6² + 1² + 2² + 1².
3(1² + 3² + 7²) = 11² + 4² + 6² + 2².