[Math Lair] Pillow-Problems: Problem #14

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

Problem:

14. (39)

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

[2/12/81

Solution:

14. (4)

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 + (bc)² + (ca)² + (ab)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.