[Number Theory]
Problem: For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$.

Solution: Since the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order, then

$$
\begin{aligned}
2(n+1)^{2}-3 n^{2} & =2 \\
2 n^{2}+4 n+2-3 n^{2} & =2 \\
-n^{2}+4 n & =0 \\
-n(n-4) & =0
\end{aligned}
$$

and so $n=0$ or $n=4$.

If $n=0$, the list becomes $0, m^{2}, 2$. This means that $m^{2}=1$ and so $m= \pm 1$.

If $n=4$, we have $3 n^{2}=3 \cdot 16=48$ and $2(n+1)^{2}=2 \cdot 25=50$ giving the list $48, m^{2}, 50$. This means that $m^{2}=49$ and so $m= \pm 7$.

Thus, the possible values for $m$ are $1,-1,7,-7$.

Final answer: 1,-1,7,-7