Pythagorean Quadruple
The sum of three square numbers equals $121$. What can those numbers be...
Age
14 to 16
Challenge level
Problem
Find distinct positive whole numbers $a$, $b$, and $c$, such that $a^2+b^2+c^2 = 11^2$.
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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