Perfectly Square
Problem
Perfectly Square printable sheet
We are first given that: $$x_1 = 2^2 + 3^2 + 6^2$$ $$x_2 = 3^2 + 4^2 + 12^2$$ $$x_3 = 4^2 + 5^2 + 20^2$$ Then show that $x_n$ is always a perfect square.
Getting Started
Try a few examples with numbers to convince yourself this works.
You are told that the answer will always be a perfect square - so the factorisation should be easier than you might first think.
Student Solutions
Look at the end for a neat short cut
The solution below is based upon the one submitted by Anna of Parkside school. I liked the explanation of how Anna arrrived at the factorisation.
The youngest person to send in a solution was Sairah of Kings Park School, Lurgan and Barinder sent an excellent solution with lots of clear explanation.
Just three of the large number of solutions to this problem. Well done to you all!
You just need to establish that $ (n)^2 +(n+1)^2 + [n(n + 1)]^2 $ is always a square of something
Can you follow that through, expanding and simplifying until you get: $ n^4 + 2n^3 + 3n^2 + 2n + 1$?
I'll leave you to decide what 'square' that is, but you might think of something inspired just from looking at the algebra, or maybe calculate the first line from the problem (it should come to $49$) and take it from there .
Teachers' Resources
Some experimentation with some integers might help.
The factorisation is not straight forward but the first and last terms are easy to establish, then the middle terms follow quite quickly.