This is going to be a short post, but I wanted to talk today about some other methods that could be used for determining success failure aside from dice. Now some of these are going to be obvious. Who am I kidding, most of them are going to be. I'm not that original. Anyhow, if I missed one, then please add it in the comments!
Cards are an obvious choice that could be used in place of dice. A deck is effectively a random number generator that goes from 1-13 (Jack, Queen, King for 11-13). Some games already use cards for things like initiative, but there are a lot of ways you could do it. I've often wondered how hard it'd be to make, or simply use, an existing card game - like poker for example - to do out a mass battle. The back and forth nature of chips from hand to hand could very nicely match up with the ebb and flow of battle.
Somewhere on this blog is a game system that uses a single coin to determine all things. Whatever a character does they have a 50/50 shot of success or failure, and it just goes from there. Coins are effectively D-2s though, so this isn't all that much of a chance now is it?
Paper Rock Scissors
White Wolf LARPS use this to handle dispute resolution instead of dice. How it works is each player wagers a trait, then you play PRS to see who wins. If you lose, you can wager a different trait to retest and the game then becomes one of strategy and desire. After all, if you burn through all your physical traits on that one check, what are you going to use on the next challenge?
Odds & Evens
By the same token, one could easily do something similar with the game odds and evens. Course, this is basically just a heavy mutation of PRS so...
I only know of one game system right now that does this (Amber), but there are ways to make games work so that there is no 'random' chance of things happening. Everything is determined by stat levels and what the GM gives you for your set up. If I have a swords of 5 and you have a swords of 6, you're going to beat me every time. Unless I cheat of course. There could be other ways of doing this without dice too, especially if you borrowed the trait wagering from the Paper Rock Scissors entry above.
Finally, for this entry, it is possible to use puzzles to determine success/failure/time. When a player goes to do something, you give them some sort of mental puzzle to solve. Scale the difficulty based on the task being done and the character's intelligence, but then go for it. See how long it takes the player to solve the problem (or give them a set number of tries) and go from there.