Adding a square to a cube
If you take a number and add its square to its cube, how often will you get a perfect square?
Age
14 to 16
Challenge level
Problem
Find all $9$ integer values of $n$ between $1$ and $100$ for which $n^2+n^3$ is a square number.
This problem is adapted from the World Mathematics Championships
Student Solutions
Answer: $n$ could be $3, 8, 15, 24, 35, 48, 63, 80$ or $99$
$n^2+n^3=n^2(1+n)$
$(ab)^2=ab\times ab=a\times a\times b\times b=a^2\times b^2$,
So if $(1+n)$ is square then $n^2(1+n)$ will be square
For example, if $1+n = 2^2 = 4$, then $n=3$, and $n^2(1+n)=3^2\times2^2=\left(3\times2\right)^2=6^2$. So $n^2+n^3$ is a square number (you can also check that $n^2+n^3=36$.)
$1+n$ could be $4, 9, 16, 25, 36, 49, 64, 81$ or $100$
$n$ could be $3, 8, 15, 24, 35, 48, 63, 80$ or $99$
Those are 9 possible values of $n$, so they must be the 9 values we were looking for.