# Beyond d20 Limits of '1' and '20'

### Playing the Game

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Because attack and saving throw checks (more checks in PF2) always succeed on a d20 roll of'20' and always fail on '1', beyond a certain point further improvements to attacks, armor class, saves, and save DCs have no affect. I propose two solutions to operate beyond these limits using a lookup table and percentile dice OR an additional dice when '1' or '20's are rolled.

When the difference between a DC and a d20 roll bonus is '2' or less there is always a 95% change of success (i.e. AC=10 and attack=+8), and when the difference between the DC and bonus is '20' or greater there is always a 5% chance of success (i.e. AC=20 and attack=+0), with a linear gradient of success between '2' and '20'. This piecewise success function can be well approximated with a Conjugate Error Function or Erfc. I would present the plot comparing these functions (Erfc[(x-11)/10])/2, and Piecewise[{{0.95, x<2}, {(21-x)/20., 2<=x<=20}, {0.05,
x>20.0}}]), but I don't know how to place figures in forum posts. This Erfc function allows d20-like success rates without discontinuities and the limits imposed by piecewise functions. Probabilities of success could be determined by subtracting the bonus from the DC and comparing to the table below, followed by a percentile roll for success.

Success Probability Table:

DC minus bonus, percent chance of success
-8, 100
-7, 99
-6, 99
-5, 99
-4, 98
-3, 98
-2, 97
-1, 96
0, 94
1, 92
2, 90
3, 87
4, 84
5, 80
6, 76
7, 71
8, 66
9, 61
10, 56
11, 50
12, 44
13, 39
14, 34
15, 29
16, 24
17, 20
18, 16
19, 13
20, 10
21, 8
22, 6
23, 4
24, 3
25, 2
26, 2
27, 1
28, 1
29, 1
30, 0

These values were rounded to the nearest whole percent numbers, resulting in some outcomes having 0 or 100 percent chance of success, indicating tasks that there is less than 0.5 percent change of success or greater than 99.5 percent chance of success, that is, tasks that will be consistently failed or completed practically every time.

For example, a DC 20 check with a bonus of 10 would have a 56% success chance (compared to 55% change on a d20), if the bonus was 5 then the success chance would be 29% (30% on d20), if the bonus was 0 then the success chance would be 10% (5% on d20), if the bonus was -5 then the success chance would be 2% (5% on d20), and if the bonus was -10 then the success chance would be 0% (5% on d20).

ALTERNATIVELY
The d20 system could also be maintained if, when a '1' or '20' is rolled, an additional d10 is then rolled to subtract (for '1' rolls) or add (for '20' rolls) 0-9 to the original value instead of guaranteeing failure or success. This method is less mathematically justifiable but doesn't require an Erfc function or lookup table, is easy and fast to remember and implement, and my players prefer it.

Why this was important to me:
The normal rules work well most of the time, but not when PCs deal with things that are MUCH lower or higher CR than themselves. I often have my PCs contend with armies, where they fight large numbers of MUCH lower CR opponents. At one point the PCs had reassembled the Shield of Aroden, and gave it to the party tank. I realized that her armor class was already so high that the Shield of Aroden wouldn't actually help her at all against the low ranking soldiers she was fighting, and I saw that as a problem that needed to be addressed.

Additional rolls, lookup tables, no thanks. anything that adds time and complexity is always bad for a game.

I understand your concern with lots of weak opponents but there is a better way of dealing with this I personally houseruled.

For PF1 I created a new monster template called minion. The idea was that when dealing with more than 5 identicle creatures, they would gain the minion template. increasing in power based in its size. minions shared hp pools and each successful attack would effectively reduce the number of minions instead of tracking hps individually. It gets a little more complex when dealing with area spells and such and the rules are too complex to repeat here but the result was the ability to have PCs engage armies of monsters and still feel challenged and have risks rather than rolling dice for the sake of it.

Quijenoth wrote:

The idea was that when dealing with more than 5 identical creatures, they would gain the minion template. increasing in power based in its size. minions shared hp pools and each successful attack would effectively reduce the number of minions instead of tracking hps individually.

Could you post the rules for your Minion template? I also tried the Mob template from 3rd edition D&D (Dungeon Master's Guide 2, I think), but it made groups far too powerful and wasn't built to handle magic effects very well.

My players have liked just rolling a single extra d10 when they roll a '1' or '20' and the situation calls for it. I think a lookup table would be relatively easy to implement, but the fact that it only really helps in fringe situations may make it more trouble than it is worth most of the time. I am mostly reporting the lookup table because there are no Conjugate Error Function dice, which would have been very mathematically faithfully to the regular system.

Sent you a PM. The rules are a 3 page PDF so too much to post here.