Rather, I'm talking about the die mechanic that says when a player rolls the maximum value on a die, they get to roll again. I've been contemplating this rolling mechanic for my own nefarious purposes, and decided to do a bit of mathematical analysis.
There are a number of different types of exploding dice and I'm focusing this analysis on one specific mechanic: maximum roll results in extra roll, and extra rolls can also explode. The scientific name for this is probably something like rollus maximus explodus infinitus.
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rollus maximus explodus infinitus in its native form |
Many gamers experienced in die rolling will be familiar with the standard averages for non-exploding dice, or rollus boringus. Over time, perfect dice in perfect conditions will result in:
d4: 2.5 d6: 3.5
d8: 4.5 d10: 5.5
d12: 6.5 d20: 10.5
d8: 4.5 d10: 5.5
d12: 6.5 d20: 10.5
In math, we prefer artifical constructs called equations to express reality, so let's define A as the average and d as the size of the die. A(d) = (d+1) / 2.
Now let's define E as an average for rollus maximus explodus infinitus. This is tricky, because it's an infinite series that involves probabilities. Scared? Neither was I, so I guess we're both giant nerds. Let's look at an exploding d4 empirically.
Original die:
- Chance of a 1: 25%.
- Chance of a 2: 25%.
- Chance of a 3: 25%.
- Chance of a 4: 25%. Plus this one explodes so we have to delve deeper.
Of the 25% of d4s that explode:
- Chance of a 1: 25%, or 6.25% in terms of the first roll (25% of 25%).
- Chance of a 2: 25%, or 6.25% in terms of the first roll (25% of 25%).
- Chance of a 3: 25%, or 6.25% in terms of the first roll (25% of 25%).
- Chance of a 4: 25%, or 6.25% in terms of the first roll (25% of 25%). Plus this one explodes so we have to delve deeper.
The third die:
- Chance of a 1: 25%, or 6.25% in terms of the second roll (25% of 25%), or 1.5625% in terms of the first roll (25% of 25% of 25%)
- .... ad infinitum
I have a 25% chance of getting an extra die, which has a 25% chance of getting an extra die, which... a ha! This is suspiciously similar to what mathemologists call an infinite geometric series. We can use this to our advantage as there is a standard form for these types of series, which I shamelessly stole from my college textbooks, which is more use than they got when I was actually in college.
E(d) = (d2 + d) / (2d – 2)
That's hard to calculate without one of those fancy RPN calculators I used to play games on while in my Medieval Studies elective, so here's the cheat sheet:
d4: 3 1/3
d6: 4.2
d6: 4.2
d8: 36/7, or about 5.1
d10: 55/9, or about 6.1
d10: 55/9, or about 6.1
d12: 78/11, or about 7.1
d20: 210/19, or about 11.1
d20: 210/19, or about 11.1
Those numbers are pretty easy to remember. But, if you want something even easier, and don't play a weird sci-fi game that has decimal points in the hit points, you can approximate all of this by just assuming E is around 0.5 higher than A.
E = 1 + (d/2)
Really, this is only slightly easier than C3H6N6O6 - the main compound in C-4 - so it's your call on which one is more fun.
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