## Re: The 2d10 Variant.

What you are talking about is probability. Yet nobody is using math.

*sigh*

So, a simple way to break down a given random number generator is mean and standard deviation. If you then do a CDF (cumulative distribution function) of the results, translated based on mean and scaled based on standard deviation, the graphs look quite similar. :) The differences mainly occur at the far end (at the < 5% of events scale).

The standard deviation of a 1dX is sqrt((X^2-1)/12).

The standard deviation of YdX is sqrt(Y * (X^2-1)/12 )

So we have:
E(1d20) = 10.5
E(2d10) = 11
SD(1d20) =~ 5.77
SD(2d10) =~ 4.06

The lower SD on 2d10 is an expression of the "tighter" distribution.

The rough conversion between (1d20+N) to (2d10+M) is:
(N - 10.5) = 1.4*(M-11)

(1.4 is 5.77/4.06).

Do you want modifiers to the d20 and difficulty to be roughly 1.4 times as important? Because you did that, regardless of your goal one way or other.

...

So that describes what happens most of the time. The next problem is dealing with the "tails".

The critical distribution above:
20/x2 -> 20/x3.5 (calculate as if x4, but deal only half damage on the final hit)
19-20/x2 -> 19-20/x3
18-20/x2-> 18-20/x2
20/x3 -> 20/x5
20/x4 -> 20/x6
is very different than the standard D&D criticals.

P(2d10 >= 20) = 0.01
P(2d10 >= 19) = 0.03
P(2d10 >= 18) = 0.06
P(2d10 >= 17) = 0.10
P(2d10 >= 16) = 0.15
P(2d10 >= 15) = 0.21
P(2d10 >= 14) = 0.28
P(2d10 >= 13) = 0.36
P(2d10 >= 12) = 0.45
For (21-C) >= 12, we get:
P(2d10 >= 21-C) = (C+1)*(C)/200

In comparison, the P(1d20 >= (21-C)) = C/20

The ratio works out to be:
P(1d20 >= (21-C)) / P(2d10 >= (21-C)) = 10/(C+1)

That is the ratio of 1d20 crit chance to 2d10 crit chance. Notice how it is rather large -- at C=1 (ie, 20s only), you need crits to be 5 times larger to be just as good!

However, we need to balance both keen/imp crit and normal. Given the shape of the curve, I'd recommend that keen/imp crit should do something slightly different.

I'd vote for "increase crit multiplier by A, and increase crit width by B".

Using that, we can now figure out what has to happen to the base crit multiplier for keen weapons to be about as good and base weapons to be about as good, on the average. (they will probably have a worse variance).

Also, should we change the baseline crits? I'm thinking "yes".

Because 20x2 under d20 has to become 20x6 under 2d10 to be just as good. And 20x4? 20x16! That's getting silly.

First proposal:
20x2 (+5%) -> 19x3 (+6%)
20x3 (+10%) -> 19x5 (+12%)
20x4 (+15%) -> 18x4 (+18%)
19x2 (+10%) -> 18x3 (+12%)
18x2 (+15%) -> 16x2 (+15%)

That follows the general rule that "crits are larger and less common".

Now, let's try the "width boosted by 1" version of keen:
K20x2 (+10%) -> 18x3 (+12%)
K20x3 (+20%) -> 18x5 (+24%)
K20x4 (+30%) -> 17x4 (+30%)
K19x2 (+20%) -> 17x3 (+20%)
K18x2 (+30%) -> 15x2 (+21%)
Hmm. Works well, except for the keen scimitar. What if we special case that?
K20x2 (+10%) -> 18x3 (+12%)
K20x3 (+20%) -> 18x5 (+24%)
K20x4 (+30%) -> 17x4 (+30%)
K19x2 (+20%) -> 17x3 (+20%)
K18x2 (+30%) -> 16x3 (+30%)

There, it works reasonably well!

So, here is a model:
Code:
```Old     Standard        Keen
20x2    19x3            18x3
20x3    19x5            18x5
20x4    18x4            17x4
19x2    18x3            17x3
18x2    16x2            16x3```
Ie, keen adds 1 to the width of your crits, unless your weapon is an x2 weapon. In that case, it instead boosts your crit damage multiplier by 1.

...

Note that 'rolling to confirm on a 1d20' is both ugly and not required. Just roll 2d10 to confirm as well, it is cleaner.