Post formatting:
See the formatting tips here:
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Generating an accurate hyper-link to a post:
You would think that clicking the permalink button would be good enough to get a permanent link, but this method sometimes fails because of the order in which posts are displayed (a bug). There is a work around. If you click the permalink button to the first reply to this post (by Algebe) you will get the url:

atheistrepublic.com/forums/site-support/useful-forum-tips#comment-86902

To make a permanent link out of that you need to to make it look like this:

http://www.atheistrepublic.com/comment/86902

edited 10/30/18 to reflect changes in forum behavior
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Searching the forums:
Let's face it, the search function here is basically useless. However, google indexes AR and has an excellent search engine so we'll just use that. Remember that time Breezy told us that mutations only have a 50% chance of making it to the next generation? Yeah, but where the hell is it? Let's search for it on google, put this in the search box:

mutation 50% Breezy nyarlathotep site:atheistrepublic.com

First result! (edit 08/22/2019: it is now the 2nd result, with this thread being the first result)
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Sending a private message (pm):
Sadly, the private message system has a nasty bug. When you click on the button to send a pm to someone it opens a form for you to fill out and populates the "to" field with their current name. But to successfully send a pm you must populate this field with their original name. So if they have changed their name, this auto-populated name won't work. You need to know their original name.

Let's try sending Tin-man a pm. We click on the "send Tin-man a message" button and it presents us with a form pre-addressed to "Tin-man". And when you try to send a message with this it will fail saying your message needs a valid recipient. This is your clue that you are using the wrong name.

If you look at their profile you can sometimes get a clue to a name that might work: looking at Tin-man's profile linked above, we see the line that currently says: "FORUM REPLIES BY TINMAN2577 (887)". The idea is that TINMAN2577 might be the correct name so lets try it. We click the button again and this time delete the name that was auto-filled in and replace it with TINMAN2577. And it works! But this does not always work (perhaps they have changed their name several times?). A last resort is to contact a moderator, who have additional tools to look up this information:
CyberLN
Nyarlathotep
Currently those listed moderators have not changed their names so you just need to click send a message and it will work.

Also once you have successfully send someone a pm, you can just reply to that message over and over again without jumping through all these hoops each time.
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Finding new posts in a large thread:
So you read a thread last night, then wake up in the morning and see that it has 10 new posts. You click the "view 10 new" link and
it takes you to the first new post. But since posts are not listed chronologically, you are on your own to find the other 9 new posts. Most browsers come equipped with a find text feature (typically the shortcut is ctrl-F). Use this feature by searching for today's date and you will quickly find the new posts.
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I will be editing this thread periodically to add more information.