I am having some trouble working out how "active" topics are defined in advanced forums. What are the rules used to determine if a forum topic is considered active or not?

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Maybe you can navigate to the "Advanced Forum active topics" view to find some clues.

This is what I was able to glean from the view. I recently shared it with a client.

The "Active topics" tab will display all topics across all Forums. By default, the list is sorted from newest to oldest by the date the last post was made to the topic. There is no concept of an "inactive" topic. If a topic is created, it's considered active. The list on the Active topics tab will be paged. By default, 20 items per page.

Hope that helps!