I hope this is the right place to put this.

Certain parts of probability and combinatorial mathematics makes a lot of sense to me. But some I think there should be a short cuts I can do (aside from brute forcing) - but I'm not sure what they are. And in certain cases, I think there should be a way for some software to automate the manual process I go through in order to figure out - for example - all the potential cards/tiles I could make that are a combination of the various configurations of elements I have designed.

So I have a two part question - one specific, one generic.

First: How would you go about figuring out all of the possible 6-sided dice that can have any non-negative value on the faces (including 0), but that the sum of all the faces is 12? For purposes of this exercise the elements are non-exclusive and the order of the elements is irrelevant (ie 12,0,0,0,0,0 and 0,12,0,0,0,0 are identical for this)

Since as numbers become decided, the options become fewer (but not a simple removal from a list like the formula #insetA*(#insetA-1)...for each slot [as it were] would be) I don't think the formulas I know can account for that.

For this problem the first side has a potential 13 values, if it is 12 though, each other side as only 1 possible value. If its 11, also there is only 1 possible set of remaining sides. If its 10 though, there are 2 sets of remaining sides. 9 has 3 sets of remaining sides, and 8 has 5 and so on.

Is there a formula to discover this and problems like it (I could likely generate a formula for this problem specifically, but that wouldn't be helpful for future problems.)

Second: Has anyone come across a tool that allows you to generate the full set of outcomes for such a problem? Knowing how many are available is interesting for balance reasons - expansion possibilities and stuff like that - but actually generating them (to avoid the tedious process of hand building a full set of every potential) would be incredibly useful.

anydice.com looks awesome - and is great when I'm taking the crazy dice I made and am seeing what the results are when they are rolled against each other!

I did the brute force thing, on that problem already - but the whole time I was doing it - i was SURE there had to be a way to solve it instead of just working it out. It's sometimes complicated to make sure that you are getting every combination when you do it by hand. If I did it correctly - there were only 49 solutions to that problem.

Here's another one that that maybe is solvable that I HAVEN'T done by hand due to the sheer size of the solution.

Imagine a Hex Tile. Each side of a hex tile can either have a road, or no road. Each road can form a path from its side to any other side or dead end (not connect to another side), but the roads cannot cross each other - though multiple roads can go to a particular other side.

So for example, if the sides are numbered clockwise 1,2,3,4,5,6 A path from 1 to 4 would prevent a path going from 2 to 5, but not one from 2 to 4, or from 5 to 4.

Even if I ignore the exclusionary element of the problem - but keep that each individual path is unique - I run into issues with the solution - is it a simple 3^6? Or is it 3^6 + 3^5 + 3^4 + 3^3 + 3^2 + 3^1? Or am I off the mark entirely.

One Solution I think I know is if the options are a road that goes to the center (thus touching all other center roads), or a road that dead ends, or no road, it should be a simple 6^3 possible tiles right?