Question

The measures of the interior angles of a convex hexagon form an increasing arithmetic sequence. How many such sequences are possible if the hexagon is not equiangular and all of the angle degree measures are positive integers less than 360 degrees?

Answer 1

If we let $n$ be the common difference of the arithmetic sequence, then we can let the largest angle measure be $180 - 3n$. The sum of the interior angles in a hexagon is $180(6-2) = 180 \cdot 4 = 720$, so we have the equation \[180 - 3n + 180 - 2n + 180 - n + 180 + n + 180 + 2n = 720.\] Simplifying the left side gives $700 + 7n = 720$ or $n = 20$. However, we note that $180 - 3n > 0$, so $n < 60$. For $n > 60$, the angles are too acute for the hexagon to be convex. Thus, there are $\boxed{59}$ possible values of $n$.