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How do I mathematically determine a baseline difficulty number when converting a d100 "roll under" system to a d20 "roll over" system?

The Basic RP system (and variants) uses a d100 skill-based resolution system, where success or failure is determined by rolling under a character's skill, which is expressed as 0-100%. These systems often have a critical threshold at the extremes (01 and 100, for example.)

How would I convert a percentile-based RPG system to a D20 system? D20 systems often use a target number and a "roll greater than or equal to" for determining success.

My initial thought is to use a base difficulty number for D20 challenges ("10" is the average d20 roll, but I think D&D 5E actually uses a base difficulty of "8", as given by spell save DC and other tertiary computations.) From that point, I would modify the difficulty up or down depending on how challenging the task is (for example, a very easy task is -20% easier (d100) or 4 points less (d20)).

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    \$\begingroup\$ At least one variant of the Basic RP system, Pendragon/Paladin, is a d20 system. It's still "roll under", but "roll as high as possible while rolling under". Rolling over your skill + modifiers means you tried something but didn't manage to pull it off, rolling equal means you performed at the peak of your performance (a critical success) and if competing with someone, highest successful roll wins. (Not an answer, just mentioning a d20 BRP system exists) \$\endgroup\$
    – From
    Jan 19, 2023 at 4:42
  • \$\begingroup\$ What's the end goal? Are you just trying to play a d100 game using d20 dice? Are you trying to port some rules of a d100 game into a d20 game? You mention BRP and D&D 5e, are they just examples, or are they the actual systems you are converting between? \$\endgroup\$
    – Ruse
    Jan 20, 2023 at 1:03

2 Answers 2

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The Arithmetic

Let's say x is your target number (to roll under) in d100, and we're looking for y, your target number (to roll over) in d20. Then

  • round x-1 down to the nearest multiple of 5,
  • divide by 5, and
  • subtract that number from 20 to get y.

For example, if 41 is your d100 roll-under number, we'd round 40 (x-1) to 40, divide by 5 (getting 8), and subtract from 20 to get 12 as your d20 target number.

Another example: if 55 is your d100 roll-under number, round 54 (x-1) up to 55, divide by 5 (getting 11), and subtract from 20 to get 9 as the d20 target number.

(The x-1 is there rather than x for reasons of -under and -over. Poke me if you want more clarity on that. Or ignore it, and you'll only be off by 1 in 1/5 of cases.)

The Impacts

  • you'll get ties five times more often.
  • the percentages aren't quite-exactly the same in many cases, but they're only off by a percent or two. But that's unavoidable when rolling a d20 instead of d100.
  • modifiers to the roll are five times as impactful in d20; put another way: there's much more granularity in d100.

By the way, 5E's basing spell DCs off of 8 bakes in the presumption that they will also benefit from the caster's proficiency bonus, which bottoms out at +2.

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  • \$\begingroup\$ I don't get what you are trying to tell below the horizontal line. "8 + something" is used for setting DC, like saves, not for rolls. If it was meant to make character with "human average" make the test half of the time if there are no other bonuses on the side of the caster, it should actually be 12. \$\endgroup\$
    – Mołot
    Jan 19, 2023 at 0:57
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    \$\begingroup\$ @Mołot it's kind of a commentary on OP's last paragraph. They note some incongruity ("'10' is the average d20 roll, but I think D&D 5E actually uses a base difficulty of '8'") in how 5e's spell saves are calculated, and I point out that that 10% gap is filled in by baseline proficiency. Does that make sense? \$\endgroup\$
    – nitsua60
    Jan 19, 2023 at 1:01
  • \$\begingroup\$ Kinda. But 8 as base assumes that DC will be set with proficiency, and the roll not, opposite to what you wrote. \$\endgroup\$
    – Mołot
    Jan 19, 2023 at 1:03
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    \$\begingroup\$ @Mołot <headdesk> Yes, of course. I was looking at the wrong side of the die when I wrote that sentence. Better now? (And thanks for pointing it out.) \$\endgroup\$
    – nitsua60
    Jan 19, 2023 at 1:09
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    \$\begingroup\$ Thanks! Yes, it's better. And I wasn't sure you are wrong, I was expecting you are right and I just don't see something :) But now it's clear, and consistent with how I see it, so I have no further objections or questions. \$\endgroup\$
    – Mołot
    Jan 19, 2023 at 1:13
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Converting from d100 to d20

Just divide by five. That's all it takes.

OK, let me unpack that a little. First, of course, 5 × 20 = 100. So take a d20 and multiply all the numbers on its sides by five. You'll get a 20-sided die with the sides numbered 5, 10, 15, 20, 25, etc. all the way up 90, 95, and 100.

You could use that die in your "roll target % or less on d100"* mechanic just fine. Well, mostly just fine. Of course if your mechanic had e.g. a "critical failure" on a natural 1, it would never happen with this die because the lowest it can roll is 5, while a natural 100 would be five times more likely than with a real d100. And of course if your success percentage for a roll wasn't a multiple of 5%, then using the relabeled d20 would skew the probability a little bit (effectively rounding it down to the nearest multiple of 5%). But other than such details, in general it would work fine. And being off by 5% isn't really a big deal most of the time.

And of course, instead of multiplying the numbers on the die by five, you can just use a normal d20 but divide the target numbers by five instead.** So, for example, "roll 80 or less on d100" becomes "roll 16 or less on d20", giving an 80% success probability either way.

Converting from roll-under to roll-over

Just replace "1 or lower" with "20 or higher", and "2 or lower" with "19 or higher", and so on. Mathematically, that means replacing "X or lower" with "21−X or higher".

That's assuming that you want to always count hitting the target exactly as a success. If you're using a literal "roll under" or "roll over" mechanic, where hitting the target exactly is a failure, then you'll have to adjust the target (or the die roll) up or down by one.


Here's a summary of all these equivalent rolling mechanics in a convenient table, because I know RPG players love tables. :D

Success d100 ≤ 5×X z100 < 5×X d20 ≤ X d20 < X+1 d20 ≥ 21−X d20 > 20−X
0% d100 ≤ 0 z100 < 0 d20 ≤ 0 d20 < 1 d20 ≥ 21 d20 > 20
5% d100 ≤ 5 z100 < 5 d20 ≤ 1 d20 < 2 d20 ≥ 20 d20 > 19
10% d100 ≤ 10 z100 < 10 d20 ≤ 2 d20 < 3 d20 ≥ 19 d20 > 18
15% d100 ≤ 15 z100 < 15 d20 ≤ 3 d20 < 4 d20 ≥ 18 d20 > 17
20% d100 ≤ 20 z100 < 20 d20 ≤ 4 d20 < 5 d20 ≥ 17 d20 > 16
25% d100 ≤ 25 z100 < 25 d20 ≤ 5 d20 < 6 d20 ≥ 16 d20 > 15
30% d100 ≤ 30 z100 < 30 d20 ≤ 6 d20 < 7 d20 ≥ 15 d20 > 14
35% d100 ≤ 35 z100 < 35 d20 ≤ 7 d20 < 8 d20 ≥ 14 d20 > 13
40% d100 ≤ 40 z100 < 40 d20 ≤ 8 d20 < 9 d20 ≥ 13 d20 > 12
45% d100 ≤ 45 z100 < 45 d20 ≤ 9 d20 < 10 d20 ≥ 12 d20 > 11
50% d100 ≤ 50 z100 < 50 d20 ≤ 10 d20 < 11 d20 ≥ 11 d20 > 10
55% d100 ≤ 55 z100 < 55 d20 ≤ 11 d20 < 12 d20 ≥ 10 d20 > 9
60% d100 ≤ 60 z100 < 60 d20 ≤ 12 d20 < 13 d20 ≥ 9 d20 > 8
65% d100 ≤ 65 z100 < 65 d20 ≤ 13 d20 < 14 d20 ≥ 8 d20 > 7
70% d100 ≤ 70 z100 < 70 d20 ≤ 14 d20 < 15 d20 ≥ 7 d20 > 6
75% d100 ≤ 75 z100 < 75 d20 ≤ 15 d20 < 16 d20 ≥ 6 d20 > 5
80% d100 ≤ 80 z100 < 80 d20 ≤ 16 d20 < 17 d20 ≥ 5 d20 > 4
85% d100 ≤ 85 z100 < 85 d20 ≤ 17 d20 < 18 d20 ≥ 4 d20 > 3
90% d100 ≤ 90 z100 < 90 d20 ≤ 18 d20 < 19 d20 ≥ 3 d20 > 2
95% d100 ≤ 95 z100 < 95 d20 ≤ 19 d20 < 20 d20 ≥ 2 d20 > 1
100% d100 ≤ 100 z100 < 100 d20 ≤ 20 d20 < 21 d20 ≥ 1 d20 > 0

But what about those < 5% criticals etc.?

Each side on a d20 has a 5% chance of being rolled, so all probabilities in any mechanic based on a single d20 roll necessarily come in increments of 5%. If you want to have something happen with a probability of less than 5%, you're going to need to use multiple rolls.

For example, a naïve way to get a 1% critical failure chance would be something like this:

On a natural 1, roll another d20. If it rolls less than 5 (i.e. from 1 to 4), you suffer a critical failure.

Of course, that feels like a rather arbitrary and boring mechanic. You might be able to "spice it up" e.g. by giving the player some ways to affect the second "confirmation" roll (or the target number for it) at some cost (either expending limited resources or risking future consequences).

In general, for these kinds of "special" outcomes, a straightforward mathematical conversion isn't likely to be the best approach. Rather, you should (also) think about what the mechanic is trying to accomplish in the broader context of the gameplay and how to best achieve a similar sense of (e.g.) danger and excitement in your new system.


*) There are two common ways to roll a d100 using two d10, one giving results from 1 to 100 and the other (sometimes called "z100") from 0 to 99, and two associated ways to do a percentage roll. Basically, both "roll X or less on d100" and "roll less than X on z100" give an X% chance of success. I'm assuming the former convention (i.e. rolls range from 1 to 100 and you roll equal or under to succeed), but of course they're mathematically equivalent.

**) Simple but handy mental math trick: to divide by 5, first divide by 10 and then multiply by 2.

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