smc

We are trying to find the value of $N$ such that $$1+2+3\cdots +(N-1)+N=\frac{N(N+1)}{2}=2016.$$ Noticing that $\frac{63\cdot 64}{2}=2016,$ we have $N=63,$ so our answer is $\boxed{9}.$ Notice that we were attempting to solve $\frac{N(N+1)}{2}=2016\Rightarrow N(N+1)=2016\cdot 2=4032$. Approximating $N(N+1)\approx N^2$, we were looking for a perfect square that is close to, but less than, $4032$. Since $63^2=3969$, we see that $N=63$ is a likely candidate. Multiplying $63\cdot 64$ confirms that our assumption is correct.