Most readers are probably well aware there is no general “skill check” system in the main body of the rules. There is no universal list of skills that applies to everyone, and there isn't a rule saying that any time somebody tries something difficult, you make a Dexterity check or Strength check. But here's the funny thing: if you dig all the way to the back of Moldvay Basic, there actually is a rule for exactly this. Under “There's Always a Chance,” Moldvay suggests that the DM can have a player roll 1d20 against the appropriate ability score, with a bonus or penalty based on the difficulty. So the ability check isn't some later invention imposed on B/X from the outside. It's right there in the 1981 rules (although it seems like an afterthought). The difference is that it is presented as one of the DM's tools for dealing with situations the rules don't otherwise cover, not as the universal resolution mechanic that it eventually became.
The door rules are a good example of the other way B/X handles this. A normal dungeon door is often stuck or locked, and an unlocked but stuck door has to be forced open. Everybody gets a 1d6 roll, with a 1 or 2 succeeding, and the character's Strength adjustment modifies the target number. So a character with a good Strength score really does have a better chance of forcing the door. In other words, B/X uses ability modifiers to modify a d6-based chance when the rules specifically say to do so. What it doesn't say is that every generic 1-in-6 chance should automatically become a 2-in-6 chance for somebody with a +1 ability modifier. Secret doors, for example, are simply 1-in-6 for ordinary characters and 2-in-6 for elves. There is no instruction to add an Intelligence or Wisdom modifier to that roll. So as a DM, I think you can certainly decide that a particular 1-in-6 roll should be modified by an ability score, but I wouldn't claim that this is a general B/X rule. The precedent is there, but the universal rule isn't.
And then we get to the thief, which is where things get potentially out of whack. The Thief class was a later addition to the original pre-B/X rules, and that class alone uses a percentile system for special abilities. Suppose the party finds a locked door and the fighter says, “I'll pick the lock.” What happens? If I simply say, “Make a Dexterity check,” I've created a pretty ridiculous result. A first-level B/X thief has only a 15% chance to open a lock, while a character with a Dexterity of 13 would have a 65% chance if I'm using the Moldvay “roll under your ability score” rule with no difficulty modifier rule. The fighter is suddenly more than four times as good at the thief's special skill. So I can't just assume that every action that sounds appropriate to an ability score should be resolved with an ability check. The thief's Open Locks percentage is a specific procedure for a specialized activity. If you want the thief to matter, you have to respect that procedure. A non-thief might be allowed to attempt something clever with a lock--perhaps by dismantling the door, breaking the hinges, finding another way around it, or using some piece of equipment--but “I pick the lock” isn't automatically the same thing as “I make a Dexterity check.”
What's especially interesting, though, is that the fighter doesn't really have a corresponding list of things that only fighters can do. Everybody can force open doors. Everybody gets the benefits of Strength when doing so. Everybody can search for secret doors, everybody can listen at doors, and everybody can attempt plenty of ordinary physical actions. The fighter's special territory is overwhelmingly combat: better fighting ability, better weapons and armor, and the advantages that come from being the class built around hitting things. There isn't necessarily a noncombat action that belongs exclusively to the fighter. That makes me think the better way to look at B/X is not “every class has its own list of things it can attempt.” Instead, there are three different situations. Sometimes the rules give everybody a specific chance, like forcing a door. Sometimes they give a particular class a specialized procedure, like the thief's Open Locks percentage. And sometimes the rules don't say anything at all, in which case the DM has to make a ruling--possibly using Moldvay's ability-check rule.
And that, I think, is the real lesson. B/X doesn't assume that every uncertain action needs to be converted into a standardized check. Sometimes the answer is obvious and you don't roll. Sometimes the rules give you a specific roll. Sometimes the player's description of what he is doing changes the situation enough that the DM can simply make a ruling. And sometimes, yes, you roll under an ability score because Moldvay specifically gave you permission to do that. But I think the order matters: first decide what the character is actually doing, then decide whether a roll is necessary, and only then decide what kind of roll makes sense. If you start with “Which ability score do I roll against?” you can very easily end up making the character sheet more important than the circumstances of the situation--and, as the poor first-level thief demonstrates, you can accidentally make the wrong character the expert.
Of course, this is also one of the chief arguments against playing older editions in the first place. When the 3E team did their reboot a quarter-century ago, this must have been on their minds. There is a perfectly reasonable argument that it would be better to have one universal mechanic that handles everything consistently. I’ve made this argument myself. It is certainly easier to teach: whatever the character wants to do, find the appropriate ability, determine the difficulty, and roll the same kind of check. I'm not really arguing against that here. I'm not trying to prove that B/X's approach is better, because it’s not. I am asking, what does B/X actually do, and how do we run it without so many house rules that we’re effectively replacing it with a different game? If we're going to play an older edition on its own terms, then its rules gaps, its wonky procedures, and its unspoken assumptions are part of the game we're choosing to play. We can decide that we prefer a universal mechanic--and I do--but that's a different discussion from figuring out how the original game expects us to handle the situations it doesn't spell out.
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