Pillow-Problems: Problem #14

Problem #13 | Problem #15
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

Problem:

Prove that 3 times the sum of 3 squares is also the sum of 4 squares.

Solution:

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.