TuggyNE

2013-01-18, 01:51 AM

This is based on a D&D question but isn't terribly system-specific

Let's start with a rough approximation: 1d4 becomes 1d6, 5d8 becomes 5d12, and so on. However, that's not quite right! If, instead of multiplying by 1.5, we merely double, it's more obvious; twice 1d4 is definitely not the same as 1d8, because it has a different mean, different standard deviation, different minimum, and so on.

So how about adding half dice? For example, 3d4 becomes 3d4+3d2, 11d6 becomes 11d6+11d3, and so on. Well, obviously those are rather tedious, since in many cases you have to roll the regular die size and carefully count the results, then halve them. Also, it's still not quite correct; it has too high a minimum and too high a mean, and its standard deviation looks a bit low, though I can't say for sure.

Then suppose we attempt to adjust for this by rolling the very minimum number of half-sized dice and folding the others back into the regular die size: 7d6 becomes 10d6+1d3, 9d8 becomes 13d8+1d4, and so on. Checking on AnyDice shows that this is very close, but still a bit too high.

Speaking of AnyDice, there's an algorithm in there for multiplying dice, but it seems to merely multiply the results. In any case, it seems to return results with slightly too low a mean, and with gaps between possible numbers.

A theoretical solution for empowering 3d4 should have a minimum of 4.5, a maximum of 18, and a mean of 11.25. Here's a summary of die schemes I've tried (http://anydice.com/program/1c32/graph), including two I didn't mention, and none of which are quite right. If any brave soul wants to calculate the desired standard deviation, or can think of a better approximation that isn't entirely impractical to perform, I'd be endlessly grateful.

Let's start with a rough approximation: 1d4 becomes 1d6, 5d8 becomes 5d12, and so on. However, that's not quite right! If, instead of multiplying by 1.5, we merely double, it's more obvious; twice 1d4 is definitely not the same as 1d8, because it has a different mean, different standard deviation, different minimum, and so on.

So how about adding half dice? For example, 3d4 becomes 3d4+3d2, 11d6 becomes 11d6+11d3, and so on. Well, obviously those are rather tedious, since in many cases you have to roll the regular die size and carefully count the results, then halve them. Also, it's still not quite correct; it has too high a minimum and too high a mean, and its standard deviation looks a bit low, though I can't say for sure.

Then suppose we attempt to adjust for this by rolling the very minimum number of half-sized dice and folding the others back into the regular die size: 7d6 becomes 10d6+1d3, 9d8 becomes 13d8+1d4, and so on. Checking on AnyDice shows that this is very close, but still a bit too high.

Speaking of AnyDice, there's an algorithm in there for multiplying dice, but it seems to merely multiply the results. In any case, it seems to return results with slightly too low a mean, and with gaps between possible numbers.

A theoretical solution for empowering 3d4 should have a minimum of 4.5, a maximum of 18, and a mean of 11.25. Here's a summary of die schemes I've tried (http://anydice.com/program/1c32/graph), including two I didn't mention, and none of which are quite right. If any brave soul wants to calculate the desired standard deviation, or can think of a better approximation that isn't entirely impractical to perform, I'd be endlessly grateful.