[Example 5.5.2] There are four chains, each with three closed rings. Now, some rings need to be opened to connect the twelve rings into a circle. Opening a ring costs 2 points, and connecting a ring costs 3 points. Moreover, the total score should not exceed 15 points. Which rings should be opened?
The general solution approach for this problem is as follows:

Step 1: Open one ring from the first segment of 3 rings and connect it to an outer ring of the second segment of 3 rings, forming a 6-ring chain (Figure 5-14);

Step 2: Open one side ring from the 6-ring chain and connect it to an outer ring of the third segment of 3 rings, forming a 9-ring chain (Figure 5-15);

Step 3: Open one side ring from the 9-ring chain and connect it to an outer ring of the fourth segment of 3 rings, forming a 12-ring chain (Figure 5-16);

Step 4: Connect the ends of the 12-ring chain to form a circle (Figure 5-17).
At this point, since each step involves opening one ring and connecting one ring, i.e., each step is $2+3=5$ points, the total is $4 \times 5=20$ points, which does not meet the requirements of the problem.
Therefore, we need to consider how to complete the task by opening as few rings as possible.
We can reverse the process described above, changing "connect" to "disassemble." For example, if we disassemble the three rings pointed to by the arrows in the figure, we immediately get three 3-ring segments. Then, we can connect the three rings pointed to by the arrows. With this idea, we can reverse the thinking: open all three rings from one of the four 3-ring segments, and then use each ring to connect two 3-ring segments. The total points required in this case would be $3 \times 5=15$ points, meeting the requirements of the problem.