How many natural numbers sum to 666 ?

The sum of first $N$ natural number is $\dfrac {N(N+1}2)$ . Therefore,

$\begin{aligned} \frac {N(N+1)}2 & = 666 \\ N(N+1) & = 1332 & \small \blue{\text{Since }N < \sqrt{1332} < N+1} \\ \implies N & = \lfloor \sqrt{1332} \rfloor & \small \blue{\text{where }\lfloor \cdot \rfloor \text{ denotes the floor function.}} \\ & = \lfloor 36.496... \rfloor \\ & = \boxed{36} \end{aligned}$

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Reference : Floor function

(N * (N + 1)) / 2 = 666; N*(N+1) = 1332; N^2 + N - 1332 = 0; (N + 37)(N-36) = 0; N = 36