Find the total number of trapezoids so that it can form a closed ring

To solve the problem, we need to understand how the angles of the trapezoids contribute to forming a closed ring.

Each trapezoid has angles of \(100^\circ\) and \(80^\circ\) at its ends. When trapezoids are joined together, the interior angles at a joint must sum to \(360^\circ\) for a closed ring.

Let \(n\) represent the number of trapezoids used. Each trapezoid contributes two angles to the closed ring (one \(100^\circ\) and one \(80^\circ\)). Therefore, for \(n\) trapezoids, the total angle contribution from the \(100^\circ\) angles is \(100n\), and from the \(80^\circ\) angles, it is \(80n\).

We want to ensure that these angles sum to \(360^\circ\):

\[
100n + 80n = 360k
\]
where \(k\) is the number of joints formed between the trapezoids. This simplifies to:

\[
180n = 360k \implies n = 2k
\]

Thus, the number of trapezoids \(n\) must be an even number.

The smallest closed ring can be formed with \(k=1\), which gives \(n=2\) trapezoids. Increasing \(k\) increases \(n\). Possible values for \(n\) are \(2, 4, 6, 8,\ldots\).

To keep it simple, the total number of trapezoids that can form a closed ring can be represented as:

\[
n = 2m
\]
where \(m\) is a positive integer.

Thus, the solution allows for infinitely many trapezoids that can be used to form a closed ring, with each possible count being an even number (2, 4, 6, ...). If you're looking for specific finite solutions, we can list a few:

- **2 trapezoids**
- **4 trapezoids**
- **6 trapezoids**
- **8 trapezoids**, etc.

In summary, the answer is that the total number of trapezoids \(n\) must be an even number greater than or equal to \(2\) to form a closed ring.