On Reader Mail, Love of THAC0

Jason Kirby writes in

"I think that THAC0 is awesome, but I can't exactly say why or properly articulate to my friends why it is better."

And then he follows with:

"Here is my dirty little secret, I am not actually sure its better, or if I am just wrapped up in some odd nostalgia."

There's some very skippable math here. The meat is past the reference. I encourage you to skip down instead of skipping out if your eyes glaze over.

Put simply, there are 5 terms in the equation:  
  • D20 roll
  • Class combat proficiency (which is defined as 'level' below for shorthand)
  • Armor Class (assumed in all cases to be descending), 
  • miscellaneous modifiers
  • baseline success (i.e. 20 hits AC 0).
Delta goes into depth about it, but what's relevant here, is that the equation is flexible. The inequality can have any one of 32 different formats
Ascending AC is D20 + Mods + Level  ≥ (20 - AC)*;
Descending AC is D20 + Mods ≥ (20 - Level)** - AC;
Delta suggests
Target 20 which is D20 + level + AC + mods ≥ 20;
This looks complicated, but it is the conversion to Ascending Armor Class from Descending Armor Class. If you are wearing no armor (AC 10) 20-10 = 10 AC in d20.
If you are wearing chain (AC 5) 20-5 = 15 AC in d20.
 If you are wearing plate mail + 4 with a Shield (AC -3) 20 - (-)3 = 23 AC in d20.
Note that in Pathfinder (half) Plate (+8) with a magic +4 and a Shield (+1) is equal to +13. 10+13 = 23.
** And (20 - Level) is the conversion for "THAC0"; where level is the combat ability of the character.
Now Delta's extensive analysis is quite correct. However, we all know that life is not about maximizing efficiency  and that there are always unforeseen consequences to any sort of change.

And, as pointed out by Jason Kirby, ThAC0 is Bounded. The lowest AC possible is -10, and the system is constructed to make that nearly impossible to get. In my several long term ThAC0 campaigns the lowest AC I ever saw was a -8 and he died to a death beetle's death wail two rooms in the adventure. Fighters usually sat around -2 to -4 (Full Plate and Shield and Ring of Protection).

Everyone knows 3e changed it up. In the last pathfinder campaign I played, it was fairly routine to have Armor Classes over 30 for the fighters and barbarians. (Plate (8) + Enchantment (2) + Shield (2) + Enchantment (2) + Deflection (4) + Dodge (1) + Natural Armor (2), for starters. From there it's a hop, skip, and a jump to morale, divine, and other bonuses)

So the 3e system is easier for people to grok (even though the equation is identical), but leads to the problems with unbounded accuracy.

Jason Kirby asks:
"Anyway, as I said I am not sure that I know how to articulate why THAC0 is a good system, but if you are willing to help I would appreciate it."
Why is ThAC0 a good system? Because it implies bounded accuracy. Because it reduces the number of calculations when rolling. Because the calculations are easy for the players and the DM. Remember, in OD&D, everyone was ThAC0 19 till level 4. They players just have to add their bonuses to their roll and report the number. I rolled a 12 with a +1 sword! 13! How could that be difficult?

But that isn't really addressing the question.

What is best? What's the best skill resolution system? What's the best saving throw system? What's the best to hit calculation? We talk a lot about the features of differing systems and how they affect play at the table and what the consequences of the choices of each are. But what's the best choice?

Why, the one you like to use of course!


  1. Exactly right! THAC0 may be the red-headed step-child of the old system, but I always found it elegant and easy to use.

  2. THACO is conceptually appealing because it applies the most frequent modifiers (armor & level) to the target, rather than to the roll. It feels more natural to have a clear sense of your target value, rather than to have a vague sense of the target value but more modifiers to help you reach it.

    I'm a teacher, and at the end of the semester everyone wants to know what he or she needs to get on the final exam. A system which gives them a bunch of extra credit on their exam score actually makes this harder to calculate, which annoys them more. They all want to know that they need exactly "82.3%" as their target.

    We want games to work the same way. "What do I need to roll here?" Fix the target first, by rolling all the modifiers into it, then shoot for it. It's ironic that many of the same gamers who think THACO is too complicated will spend 30 seconds performing the "what natural result do I need to roll for a hit" calculations in their heads before rolling, which amounts to the same thing!

    Note that, with the inevitability of dozens of situational modifiers (and more frequent stat-based bonuses) after 3e, the mental-math "cost" of having to calculate targets and modifiers was already being paid before every roll anyway, and so having one more factor (BAB) to fold into that set of modifiers was no longer a big deal.

  3. "Why is ThAC0 a good system? Because it implies bounded accuracy."

    Wouldn't it be even better to have bounded accuracy explicit? For example, you could get a similar effect with ascending AC by declaring that the highest AC is 20. (You would need to take care with modifier inflation too, but that's possible). I've been thinking that any future game I run will probably use an approach like this (currently I'm using descending AC where the best AC in the world is 2).

    As a point of order, is -10 ever explicitly declared the best AC in AD&D, or is that just a convention? I can't be bothered to hunt through my AD&D books right now, but I'm pretty sure I've seen ACs of -15 or whatever in play back in my AD&D days (I totally admit that it could have been "cheating").

    1. Yeah, this. I have yet to be convinced that the connection between THAC0 and bounded accuracy is anything more than a coincidence. The reason 3e went off the charts was because of cheaper magic items, enchantments and other bonuses (I don't even know what the fuck 'deflection' or 'natural armour' are). The exact same inflation could have happened just as easily with descending AC.

  4. Back in the 90's when I played 2E AD&D, we always just used:

    THAC0 - Modified Roll = AC Hit. Tell DM what AC you hit, DM tells you if hit or miss.

    (I never understood some folks' complaints about having to do a simple subtraction calculation to use THAC0. Maybe they were trying to use some other form of the calculation than we did? I'm not even that good at math past simple algebra, although I come from a background of lots of RPG systems, many of which have WAY MORE math than any version of D&D, although I started with Basic D&D then 1E AD&D.)

    1. That's how we played 2e also. "I hit AC 5, does it take damage?"

      However, I like d20's ascending system because it makes that same style of play easier. You can (and should) pre-compute your base attack and any standard modifiers into a "to hit" bonus. Roll d20, add your bonus, say "I hit , does it take damage?". Then, if you forgot to add something, someone else at the table says, "don't forget your +1 for Bless", and you just adjust the number up by one.

      I do agree that THAC0 is better than the old tables, though when the night is wearing on and everyone is getting tired, the tables work well enough. :)

  5. Thank you so much for addressing my question. You actually managed to make sense of the e-mail I sent you.

    Its true I really do enjoy Thac0, and just about all other values are equal, so I will use Thac0.

    Also, thank you to everyone who commented, some really great insights here.

    1. It is a perennial issue and I am glad to be of help.


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