Just to illustrate something, here are a pair of systems that you can use as a basis for resolution. They share two key characteristics - neither is math dependent but both are heavily expectations dependent.
Math is, I hope, self explanatory, but expectations are another matter. What that means is that they depend on the entire table sharing a reasonably common set of expectations regarding what can or cannot be done. This tends to demand that the game be strong rooted in the real world, or be of a genre that the table knows well. Most fantastic settings are problematic in this regard because the limits of magic or other supernatural elements can be much harder to intuit. Not to say this is impossible to overcome - sufficient familiarity of flexibility can find a way to deal with this, but it's a hurdle all the same.
As an example, consider the difference between running a game about normal kids at boarding school vs a Harry Potter game. Because the limits of what magic can and cannot do are so loosely defined within the Harry Potter world, you run the risk of a disconnect whenever players decide to address a situation with magic.
This idea of expectations is an implicit part of a lot of lighter games. The reason they can get away with leaving out rules for a of of situations is that the table already understands those issues to sufficient resolution to allow play. More detailed rules are a useful way to bring together a group that does not have a shared understanding, but they're certainly not always necessary.
(I am, by the way, really really interested in the idea of expectations. I think it's a keystone of gaming, but that's another topic.)
Anyway, expectations are important to both of these systems since they hinge on some of the ideas I've discussed in skills, specifically that if a character tries something they can succeed (modulo delay, sloppy work and so on) but not every character can try every thing. This may require a healthy "no" (another topic for another time) occasionally, but the closer the table's expectations, the less often that should happen.
System 1: Oracle Dice
Requires: 4 Fudge Dice
Process: When the character is faced with something that demands rolling, have a shared understanding of what will happen if nothing goes right or wrong. This will usually be success, but the situation may complicate it. Once that understanding is in place, roll 4 fudge dice.
If they all come up blank, things proceed according to the default established before the roll, simple as that. However, each [-] that comes up is something that goes wrong (Dropper a wrench), and every [+] that comes up is something that goes well ("this is UNIX! I KNow this!"). These unexpected twists are narrated by the player by default (and their implications interpreted by the GM) but the player may hand that responsibility off to the GM if he is so inclined.
Optional Rule #1: Players less interested in narration and twists may allow a [+] and a [-] to cancel out.
Optional Rule #2: Differing levels of player skill or situational complication can be represented by setting rather than rolling some dice. For example, consider the following skill ladder:
0 - No skill, no chance, shouldn't even roll.
J - Secondary skill. Not something the character can normally do, but something they *might* be able to do, like Han Solo overriding a security lock. Set one die at [-] then roll the other 3.
Q - Skilled - This is what you do. Roll the dice normally.
K - Extra awesomeness. Set one of the dice at [+] then roll the other 3.
A - Egregious awesomeness. Set 2 dice at [+] and roll the other 2.
Similarly, the GM may set some number of dice before the roll to reflect the situation being particularly favorable or unfavorable. For ease of use, this unrolled dice cancel each other out, so a secondary skill ([-]) character with just the right tool ([+]) will roll 4 dice with none pre-set. If there are more than 2 fixes plusses, reconsider calling for a roll. If there are more than 2 fixed minuses, no such caution is called for, so long as everyone knows things are about to get ugly.
System #2: Throughline
Requires: At least 4 Fudge dice, but a big pile of them is cooler
Process: Before play begins, roll the fudge dice and leave them in the middle of the table. Play proceeds until the time comes to roll the dice (assuming a situation where the character's actions remain within the sphere of expectations) and the player picks one of the dice from the middle of the table.
If it's a +, things go well, the situation turns in the character's favor.
If it's a -, things go to hell. Character still succeeds, but it sucks in some way. There's a price. Things go wrong. Whatever. Fail forward.
If it's a blank, then roll it. Resolve a + or - normally. If it comes up blank again, the player has two options: accept a boring success, or escalate. A boring success is just that - success, but no particular direction with it. Escalation means that before the situation is resolved, the stakes are raised. More is on the line, success and failure get bigger. And the die is rolled again, repeating the process.
(Really, when you can, you should always, escalate, but the path to escalation is not always obvious. The boring success options is mostly for situations where that's not practical.)
After resolution. roll the die and put it back in the middle of the table with the others.
Option #1: If you have enough fudge dice, you can build a physical chain of outcomes as the dice get used. This is kind of cool, but not actually useful.
(That said, if you're a fan of McKee or Laws, then the jump from this to narrative up and down beats is a pretty easy one to make.)
Option #2: This assumes a community pool - it's entirely possible to make the pools personal (giving each player their own "arc") but that adds extra bookeeping.
They're both very simple systems, but I hope thy illustrate ways you can change how you think about skill rolls.