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tl;dr- Not only is there no simple formula to generate Skill DCs, but it's actually mathematically impossible for Incredible difficulty. I consider this bad game design.
My metric for whether a table like this is well-designed is predictability. It's better game design if you can generate a table on your own, instead of always having to reference the table for values. Three well-designed tables by this metric are BAB in 3.PF, XP/encounter in PF 1e, and XP/level in 3.5. Two poorly designed tables by this metric are the Skill DC table in the playtest and XP/level in PF 1e.
The BAB tables are particularly famous for this sort of design, because the underlying formulas are frequently used as names for the progressions, and the formulas are even explicitly given in the bestiary. Poor BAB is 1/2*HD, Average BAB is 3/4*HD, and Good BAB is your full HD, with the first two being rounded down in d20 system tradition.
The advancement and XP tables are interesting. In 3.5, you always needed 1000*[current level] XP to level up. But they also instituted a rule where CR+2 doubles the XP, so they had to have different XP values depending on your current level¹. The advancement table is definitely predictable, while the XP table only vaguely so. Meanwhile, PF 1e standardized XP rewards to not depend on your current level, making it very predictable. But while they still name average numbers of encounters per level up (for example, 20 encounters/level for medium speed), the advancement table doesn't strictly follow it. The values are always close to the expected values, but it's still unpredictable, so you have to reference the table.
Thus, we get to the Skill DC table. Because it frequently increases by 1 or 2, I figured it would operate on a similar formula to BAB and base saves. Pick some highly composite level, plug it and level 0 into the two-point form of a line, and round down when not an integer value. Except there are two key levels which render such a formula mathematically impossible. For every difficulty except Easy, the DC increases by 2 from level 0 to level 1. And for Incredible difficulty, the DC increases by 3 from level 6 to level 7. I assert the following (proof given in footnotes):
- For the Medium through Ultimate progressions, the approximate true step size is between 1 and 2.² (Cf. Medium BAB having a step size of 0.75)
- For a progression generated by rounding (m*Lv + b), with m between 1 and 2, the step at any given level will always be 1 or 2, whether you round up, down, or to the nearest integer.³
- For a progression generated as floor(m*Lv + b), with m between 1 and 2, the step from level 0 to 1 must be 1.⁴
In other words, it's impossible to fit a BAB-style equation to any of the non-Easy progressions, because of the jump at level 1. And it's impossible to fit any equation like that to Incredible, even if you change the rounding pattern, because of the jump at level 7. (And I've checked. Changing the rounding method doesn't help fit the other three progressions)
There actually are some approximation which get incredibly close and aren't too nasty to use. 11+5/4*Lv for Medium, 13+11/8*Lv for Hard, 14+3/2*Lv for Incredible, and 16+13/8*Lv for Ultimate. (Okay, so Medium and Incredible are a bit easier than the other two) But as it stands, the only way to get the numbers under RAW is to either look them up or memorize the entire table.
This sort of design also leads to a more intuitive meaning of DC progressions. For example, the Easy progression being 7+Lv means that any increases in your skill check above the +1/Lv go directly to your likelihood of success. Meanwhile, Incredible being approximately 14+3/2*Lv means that in addition to the +1/Lv, you have to find another +1 every other level through things like stat boosts and feats just to keep up with the DCs.
1. a_n = 1000*n does not fit the recursion relation b_n = 2*b_(n-2). Therefore, a_n and b_n must use different tables.
2. If you take the slope of any progression using levels 0 and 23, the value is between 1 and 2
3. This one's actually more difficult to prove. Two assumptions which make it easier to understand why this is the case, even if it isn't a rigorous proof: Suppose m equals x/y, and that 1/y is the smallest fraction possible. For a step of 3 to happen under these conditions, the narrowest margin would be n+(y-1)/y to n+2. The difference between these values is (2y+1)/y, which is greater than 2, contradicting the assumption that m is between 1 and 2. A similar argument can be made that the step size cannot be 0.
4. Before rounding, the difference in values is m. Because we assume integral values of b, the only fractional part is the fractional part of m. Thus, when rounding down, we get b+floor(m) = b+1.
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However, I'll go a bit further than you and say that the whole scheme needs to be tossed. Get rid of the endless modifiers. Get rid of the easy/medium/hard/etc levels. Skills and checks need to be streamlined and simplified, not made more complicated.
These things are meant to help adjudicate situations in the game, not become the game themselves.
Uuuuh... have you tried the simplified 10.2 I wrote a while ago?
It's not perfect, but it's a good approximation.
DC begin at 12+level for Medium tasks, and increase by +2 for each additional difficulty. Easy tasks DC are reduced by 5 instead.Medium and Incredible tiers DCs increase by an additional +1 and +2 respectively every 5 levels.
Hard and Ultimate tiers DCs increase by an additional +1 and +2 respectively every 4 levels.
Thus, a "hard" level 14 DC is equal to 12 (base) +14 (level) +2 (hard) +3 (12/4 adjustment) = 31.
*EDIT: You meant to calculate the DC. Gotcha. I feel less dumb now, something clicked.
Yep. My issue is that the DCs are all effectively arbitrary numbers, rather than something a GM could calculate on the fly. Imagine if 3/4 BAB didn't advance at a random 5 levels, as opposed to all the ones that equal 1 mod 4. Or as a tangible example, you need to look back at the advancement table to find XP to next level in PF 1e, but not in D&D 3.5, because there's random variability beyond 30/20/13 times XP/encounter.
I've checked, and there's no simple formula where you round a linear function for any of the progressions, regardless of rounding method. But it's more impossible to produce one for the Incredible progression because of that +3 from level 6 to level 7.