I understand that there are differences between the way Armor Class works in AD&D (first edition) and Basic D&D (the boxed sets, Rules Cyclopedia, etc.), but I don't have the books to check it out. My understanding is that one starts at AC 9 and the other starts at AC 10, but I don't know which is which. I'm going to be running a campaign using Labyrinth Lord and the Advanced Edition Companion with old modules from both versions of D&D. Will I need to adjust AC? Are there any other armor differences I should know about?
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Basic D&D has different Dex modifiers to AC, and different entries for armors worse than chain. The combination can result in up to a 2 point AC difference.
Note that AD&D2E has the same AC's as 1E, but has more armors; I've put the ones not in 1E PH in (parentheses).
For monsters, you won't need to adjust at all. For characters, if AC is 6 or higher, you may want to adjust it.
Edit Note: I added the LL data in for completeness.
That is correct, Basic D&D starts off at 9 and AD&D starts off at 10. But the differences are slight. For example no armor is 9 in BD&D and 10 in AD&D, leather is 7 in BD&D vs 8 in AD&D, but chain is 5 in both, and plate is 3 in both.
The difference is inconsequential and represent at most a -1 to hit at the lower armor classes. If you really concerned about it then subtract 1 from AC when it lower than 5 if using an AD&D module. But likely you will be looking up the monster stat in the Labyrinth Lord rulebook anyway. So it is a wash.
That's basically it. Technically the Dexterity adjustments to armour class table is slightly different between 1e and Basic as well, but that's fairly minor - I probably wouldn't worry about that as DM, unless I was helping convert PCs between the two systems.
Note 1st edition also had a more complex system of attack modifiers depending on weapon-vs-armour type that further modified the to-hit rolls for extra realism, but I don't think you'd experience any major problems from not using it.