Different level of dice

With those faces, A is twice as fast (on average) at matching symbols 1 and 2, while B is twice as fast at matching symbol 3.

Matching one symbol will lead to lots of ties because more than one player is likely to match on the first roll. If there are only two symbols to match, I think C will get creamed fairly regularly because it's still mostly about how the first roll works, but now the odds start to matter. Matching six symbols will get tedious as players roll their last die or two repeatedly. So there's some sweet spot in between, and I'd recommend playtesting rather than trying to use probability theory.

Just time yourself as each player rolling against a representative sample of cards, and do that enough times to get a reasonable feel for what's typical.

Don't get confused by the fact that he uses numbers. I almost made that mistake and did math on average etc.

He means, he has 3 different symbols.
Symbol 1
Symbol 2
Symbol 3

Now the question remains, how many times does each die have to roll to have each symbol rolled at least once.

When all 3 symbols are rolled, the card does damage.

Also, I think that when the number of rolls are known. Seeing as how it is one on one. The health should be inverted.

I don't know the actual rolls. But lets say that the average rolls for A is 3, for B is 4 and for C is 5. Then health will be 3 for A, 4 for B and 5 for C.

Who can write a simulator? Because calculating exactly with division by 0 is rather not doable.

That is right. I mistook the symbols for being to be a complete set.

Even one symbol rolled could have a meaning already.
This makes things very complicated.

More information is needed.

You cant do what you want to do, at least not with a six sided die. You hit on the solution (and the problem) in your example, assuming a 0 means you dont hit any of the symbols. The solution is that you need such a non-matching symbol to change the ability of the die to quickly match (adding nulls makes it slower), as in C. The problem is that you cannot alter the probabilities any other way, for example changing the distribution of matching symbols as in A and B, as this causes one die to be better in one situation, but worse in another. Assuming the card you have to match is random, then your probability changes will always average out to zero.

The best die will have equal probabilities for all symbols and no nulls. A properly tiered die will have the same amount of nulls for each symbol. So in order to get a tiering of increasingly slower dice for all possible cards, you need dice that have enough faces to allow for the needed nulls. This total number of faces would be

n = (# of symbols)(# of tiers)

so in the case of 3 symbols (1,2,3) and 3 tiers (A,B,C) you need to use nine-sided dice.

A: 1,1,1,2,2,2,3,3,3
B: 0,1,1,0,2,2,0,3,3
C: 0,0,1,0,0,2,0,0,3

Whether or not you can save matching rolls or whatnot doesnt really change things for the dice themselves, it just shortens or lengthens the total time that the problem will take to solve, on average. If you have to match all the symbols in one roll and in order, then you are looking at lottery type conditions and your players will die of old age. I think it is safe to assume you arent doing that.

Even then, is this a fun mechanic? The player has no real control, they are just rolling the dice as fast as they can and hoping they get matches. If they lose, they hate the dice. If they win, its just pure luck which isnt rewarding in itself, and no matter how you set up the mechanics, its going to take a bit of time and just feel tedius.

He did mention rolling 6 dice per player.

At least you can see all possible combinations for each die for that round. And the chance to get a certain combination.

The question remains: " is that it for that round?". Or can players take dice with them to the next round?

You don't need precisely symbols-times-tiers because you could in principle have different tiering for different symbols. An example with d8:

A: 1, 1, 1, 2, 2, 2, 3, 3
B: 0, 1, 1, 0, 2, 2, 3, 3
C: 0, 0, 1, 0, 0, 2, 3, 3

Here symbol 3 is the one that "levels the playing field," so whenever a card has a 3 on it player type C has a better (or at least less bad) chance.

The 0 could either be obviously useless (e.g., blank), or it might have some defensive use (e.g., three "shields" blocks a point of damage). Now types B and C have some strategizing to do when deciding what to reroll.

Yeah thats true FrankM, and a smart way to keep the die size down while retaining some symbols.

Most games have randomness (certainly games involving dice!).. you can't get rid of luck totally.

So the first question is, which element is worrying you? The speed rolling will give some people a natural advantage and actually reduce luck, in favor of the "skills" of (a) fast rolling and (b) fast evaluation of the rolls. But that may be its own negative.

(My 5-year-old will never, ever beat me at Tenzi, even without the stacking part.. but she can occasionally beat me at Quixx, and I know she enjoys it more)

The randomness "problem" is mostly an issue of variance -- the swings of luck. Some things are more prone to swinginess than others. A die with 1 2 3 4 5 6 on it will give an average roll of 3.5. A die with 1 1 1 6 6 6 on it will also give an average roll of 3.5. The latter one has higher variance.. you're much more likely to get a very good or very bad result.

So if you have an issue with luck in the game, maybe look to reduce variance.

I'll give you a simple example that might help with your exact issue.

If one card requires a 6 6, and one card requires a 5 6, which is easier to get?

Answer: the 5 6. There are two ways to get it (5 6 and 6 5), while there's only one way to roll a 6 6. It's twice as likely on a single roll.

And for your game, I assume that getting a single match is helpful as well (you then reroll the other die, yes?). Same thing, the 5 6 is more likely to give you a single match, as each die has two possible good values rather than one.

Don't know if this is helpful, but keeping your card values in the "easier" category will reduce the variance.

I don't like this asap mechanic either.
If it is only a one time roll of 6d6, and you combine the best combination possible for a certain damage output or effect by choices.
Then you have a much more potential mechanic to use.

ASAP 1 damage or a one time 10 damage. Which do players prefer?

One extra die might work.

Here are some ideas about how to aid rolling/placement abilities (X and Y represents some number):

Choose the outcome of X die before rolling. Adjust by (exactly, up down or both) one, two or three. Flip the dice. Rolls of X also count as rolls of Y. Rolls of X count as 2 rolls of X. Swap dice from a common pre-rolled pool. Players get to roll X dice once before the rolling session. Players acquire a chit that is a permanent roll result.

Here are some ways to limit abilities:

odd results, even results, results that are of the number X, Result on dice of a color (assuming different color dice), when the total of all dice are less than/greater than X, if no dice are X, if at least 1 dice is X, Can or cannot cross the one to six, to activate this ability set as side one die from your pool.

You could also consult Roll for the galaxy, Sagrada, Castles of Burgundy for ideas that I do not remember off the top of my head.