## Caltrops

,I love four-sided dice (which I will refer to from here on as d4s, in keeping with standard notation). I also love clean, simple dice mechanics in TTRPGs. Many of these use d6s, Fate uses d3s in the shape of d6s, some use only a percentile set or a single d20. I’m certainly not about to say that there aren’t any d4-based systems out there. But I have not encountered one on my own time, and my love of these pointy little bits has had me thinking about potential workings for a while now. And while I don’t have anything resembling a system here, I had some interesting thoughts and had my computer roll a few tens of millions of digital dice for me, and I’d like to lay out a few initial thoughts that may, some day, turn into something.

The TL;DR is this: players can, for any resolution^{1}, roll two, three, or four d4s. If every die has the same value, *regardless of what this value is*, that counts as a special. Otherwise, the values are summed with 1s and 2s treated as negative (so, -1, -2, +3, +4). And that’s it, roll complete! What is a special, exactly? Well, I don’t really know. My initial thought was that the all-of-a-kind roll would be a critical success. After seeing the maths, and thinking about what I would opt to do in any given situation. Which led me to believe that the all-of-a-kind roll should certainly be *special* in some way, but likely a more interesting and dynamic way than just ‘you score very big’. This could be a trigger for something special on your character sheet related to whatever thing you are rolling for, or it could be a cue for the GM to pause the action and shift course. It should certainly always be something *positive*, but I don’t think the traditional crit mentality quite fits.

I’ll get into the numbers in more detail in a minute, but the key takeaways are:

- The expected value for any given roll in this system is that of the number of dice rolled (so, if you roll 3d4 with these rules an infinite number of times, you’ll average right out to 3, &c.)
^{2} - The percent chance of special rolls (all-of-a-kind) is roughly 25% for 2d4, just over 6% for 3d4, and 1.5% for 4d4.
- The highest roll you can get for 2d4 is 7, the lowest is -3; the highest for 3d4 is 11, the lowest is -5; and the highest for 4d4 is 15, the lowest is -7
- For 2d4, there’s a 25% chance of ending up with 2, all other non-crit values are equal. For 3d4 the likely results that jump out are 0, 1, 5, and 6. 4d4 has a lot of possibilities and is trickier to describe, but the most likely results other than 4 are -1, 3, 5, and 9.

Ignoring specials for a minute, we see a clear advantage to rolling more dice. Generally speaking, we will trend toward getting higher values, and the likeliest values for us to get on a given roll are better. When we factor in specials, rolling two dice becomes a lot more attractive; specials come up 25% of the time! Which is a very cool way to shift the balance, in my mind, but it’s also why it needs to be something other than just ‘BIG SMASH’. Make it too strong, and it basically becomes the universal choice. Making it more dynamic or narrative seems like a likely way to make the decision meaningful for players. Another possibility is a potential cooldown mechanic where rolling two specials in an encounter would force that character to cut out; that would likely leave the 3d4 option unused, however, as players would roll 2d4 until hitting a special, and then switch directly to 4d4.

I wrote a quick and dirty Lua^{3} script to let me roll a few tens of millions of virtual dice and run the numbers. The resultant percentage table is below. My initial script only returned the number of specials, positives, negatives, and zeroes. Upon seeing the steep declination toward 0% specials on rolls of more than 4 dice, I decided I was only going to do further testing on 2, 3, and 4. I’ve included the percentages of specials for 5, 6, 7, and 8 dice just to show the trend.

# d4s | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|

Special | 25 | 6.3 | 1.6 | 0.4 | 0.1 | 0.025 | 0.006 |

-7 | 0 | 0 | 1.6 | ||||

-6 | 0 | 0 | 2.3 | ||||

-5 | 0 | 4.7 | 1.6 | ||||

-4 | 0 | 4.7 | 0 | ||||

-3 | 12.5 | 0 | 1.6 | ||||

-2 | 0 | 0 | 6.3 | ||||

-1 | 0 | 4.7 | 9.4 | ||||

0 | 0 | 14.1 | 6.3 | ||||

1 | 12.5 | 14.1 | 1.6 | ||||

2 | 25 | 4.7 | 2.3 | ||||

3 | 12.5 | 0 | 9.4 | ||||

4 | 0 | 4.7 | 14.1 | ||||

5 | 0 | 14.1 | 9.4 | ||||

6 | 0 | 14.1 | 2.3 | ||||

7 | 12.5 | 4.7 | 1.6 | ||||

8 | 0 | 0 | 6.3 | ||||

9 | 0 | 0 | 9.4 | ||||

10 | 0 | 4.7 | 6.3 | ||||

11 | 0 | 4.7 | 1.6 | ||||

12 | 0 | 0 | 0 | ||||

13 | 0 | 0 | 1.6 | ||||

14 | 0 | 0 | 2.3 | ||||

15 | 0 | 0 | 1.6 |

One final (for now) takeaway after having stared at these numbers in multiple forms. I mentioned the use of *special* instead of *critical* because of a traditional *critical* making a roll of 2d4s too powerful; you’ll get that hit 25% of the time. There’s another truth to 2d4 rolls, however, and that is that the chance of *negative* rolls is the lowest: 12.5% of 2d4 rolls are negative, 14.1% of 3d4 rolls are negative, and 22.8% of 4d4 rolls are negative. Every negative 2d4 roll is -3, however, and the chance of getting -3 or lower for 3d4 is 9.4% and for 4d4 is 7.1%. This raises a question as to what is a better motivator. You’re more likely to get a negative with more dice, and it’s possible to get a *worse* negative, but the trend is toward a *better* negative (the above numbers didn’t reflect zero; the likeliest non-positive result for 3d4 is, in fact, zero). It’s worth running through how this plays out and deciding whether negative *values* matter, or simply the fact that a negative was, in fact, rolled. My instinct says stay with values, but that doesn’t take into account the *feeling* of how the dice are treating you.

Clearly there are a lot of ‘what ifs’ to work through, and there’s a lot more involved in practical testing than just rolling millions and millions of dice. But I do think I’m on to something interesting here, something simple, but with slightly-less-than-simple decision determinations.

- This is obviously a tentative thought, and I can imagine there may be instances when a GM would want to veto a certain number of dice. I can also imagine a scenario where players might have a set pool of dice to pull from, making the decision more impactful, though
*fewer*seems to potentially be more beneficial than*more*, so this probably won’t bear out. A potentially cleaner way to mix this up would be to directly tie die counts to stats/abilities. ↩︎ - This holds true for other-sided dice as well, albeit in multiples that correspond to the number of sides (as compared to four). So 3d6 averages right out to 4.5, &c. The influence of the crit/special rule reduces as sidedness increases. ↩︎
- One of my half-hearted goals for 2020 is to become more proficient in Lua. Aside from just being a capable scripting language that I largely find myself comfortable with, it’s the basis of the Pico-8, the upcoming Playdate console, and the Love 2D gaming engine, all things that I would like to dabble in. ↩︎