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?
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 $n$.
Let's start by considering the sum of the interior angles of a hexagon. The sum of the interior angles of any polygon can be found using the formula: (n-2) * 180, where n is the number of sides of the polygon.
For a hexagon, the sum of the interior angles would be (6-2) * 180 = 4 * 180 = 720.
Since we want the measures of the interior angles to form an increasing arithmetic sequence, we can denote the angles as a, a+d, a+2d, a+3d, a+4d, and a+5d, where a is the first angle and d is the common difference.
The sum of the interior angles of the hexagon can be expressed as the sum of this arithmetic sequence: 720 = 6a + 15d.
Now, let's analyze the possible values for a and d.
For a, it has to be a positive integer less than 360.
For d, the common difference, it has to be a positive integer such that a+5d is also less than 360.
Let's break down the possibilities step-by-step:
Case 1: a = 1
In this case, we have 6 + 15d = 720.
Simplifying the equation gives us 15d = 714.
Since d has to be a positive integer, there are no integer solutions for d.
Case 2: a = 2
In this case, we have 12 + 15d = 720.
Simplifying the equation gives us 15d = 708.
Again, there are no integer solutions for d.
Case 3: a = 3
In this case, we have 18 + 15d = 720.
Simplifying the equation gives us 15d = 702.
Still, there are no integer solutions for d.
We can continue this process, increasing the value of a by 1 each time, until a reaches 44 (since 45 + 5d = 720 would exceed the limit of 360 degrees).
After examining each case, we can conclude that there are no such sequences possible where the measures of the interior angles of a convex hexagon form an increasing arithmetic sequence, under the given conditions.