Pythagorean Quadruple

'Pythagorean Quadruple' printed from https://nrich.maths.org/

Answer: $9^2+6^2+2^2=11^2$

Systematically search square numbers

largest square	middle square	$121-a^2-b^2$	comment
100 ( = 121 $-$ 21)	16	5	not square
	9	12	not square
81 ( = 121 $-$ 40)	36	4	yes!
	25	15	no
64	49	8	no
	36	21	no
49	36	36	not distinct
cannot be any smaller

Only possibility is 81, 36 and 4

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.