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DSA 4.1 Character Sheet Formulas Documentation
Purpose
DSA 4.1 rules can be quite convoluted at times and as a consequence so are the formulas used. Roll20's inline roll maths abilities are quite limited and combining DSA 4.1 rules with Roll20 maths can cause some headaches and, of course, bugs.
Since most formulas have been touched by several parties, many bugs have already been fixed. In order to prevent newcomers to fall into the same traps as their predecessors and to make the formulas less difficult to understand, this document contains background information on some of them.
NB: The English terms used for this largely German system may not match any official ones due to the author's laziness. Feel free to submit a pull request for fixing these things.
Common Techniques
Number-to-Roll Conversion
In order to use drop/keep highest/lowest the input needs to consist of dice rolls such as {1d20, 2d10}kh1. Using raw numbers such as {10, 2d10}kh1 does not work: Cannot mix M and sum rolls in a roll group. But, one can add numbers to dice rolls and the result is a roll: {10 + 1d20, 2d10}kh1 works.
In order to use plain numbers, one just has to add zero 1-sided dice: {10 + 0d1, 2d10}kh1.
Stat Checks (Eigenschaftsproben)
{@{MU} - (?{Erleichterung (−) oder Erschwernis (+)|0}) - 1d20cs1cf20, @{MU} + 0d1}dh1
The stat value (@{MU}) is increased (easy) or decreased (difficult) by the modifier ((?{Erleichterung (−) oder Erschwernis (+)|0})). Then, the roll result (1d20cs1cf20) is subtracted. This would suffice for getting the correct result (automatic) success/failure, but the magnitude could be off in the case of negative modifiers (easy). One example: Modifier -7, MU stat 12, roll result 5. The check result would be +14, although the result is capped at the stat value, i.e. 12.
Therefore, the first part is grouped with the roll @{MU} + 0d1 and the highest value of both is dropped. In the above example, 14 would be higher than 12 and dropped, so that the result would be 12. When the check result is less than the stat value, the second part would be higher and dropped. The same is true for negative results (failures). Automatic successes (1) and failures (20) are handled by the roll template. Since stat values of 20 and higher are possible (and negative modifiers as well), the roll template currently does not show the resulting points in the case of automatic failures. The reasoning behind this is that the result would be non-negative, suggest success to the player and might confuse newbies. It should be noted that automatic success/failures in stat checks are an optional rule according to WdS, p. 7.
Speed (Geschwindigkeit/GS)
(@{GS_Basis} + @{GS_Mod} + (floor((@{GE_Basis} + @{GE_Mod} + 100)/111)-1) + (ceil((@{GE_Basis} + @{GE_Mod} + 100)/115)-1) - @{wound_Bauch} - (@{wound_RB}/2 - @{wound_LB}/2))
The GS stat is 8 by default and modified by certain advantages/disadvantages (which are not currently respected) and certain wounds. Additionally, the agility stat (GE) influences the speed: GE less than 11 reduces the GS by one, GE greater than 15 increases the GS by one. The above formula contains some magic values to accomplish the correct behaviour.
floor((@{GE_Basis} + @{GE_Mod} + 100)/111)-1
@{GE_Basis} and @{GE_Mod} represent the current agility without wound effects. The trick to get this term to be -1 for GE less than 11 lies within floor(). floor() always rounds down. So, 0.999 gets rounded to 0. GE less than 11 also means GE less than or equal to 10 and with GE = 10, the fraction 110/111 will be rounded down to 0. The -1 after the floor(...) makes the whole term become -1.
ceil((@{GE_Basis} + @{GE_Mod} + 100)/115)-1
Same situation as before, but this time using ceil() to always round up. Until GE = 15, the ceil(...) is 1 and the -1 afterwards reduces it to zero cancelling out any effect. Starting with GE = 16, the ceil(...) is 2 and the whole term +1.
Life Energy (Lebensenergie/LE), Stamina (Ausdauer/AU) and Astral Energy (Astralenergie/AE)
LE: round(@{KO_Basis} + (@{KK_Basis}/2) + 0.05)
AU: round((@{MU_Basis} + @{KO_Basis} + @{GE_Basis})/2 + 0.05)
AE: round((@{MU_Basis} + @{IN_Basis} + @{CH_Basis})/2 + 0.05)
These formulas are quite straightforward implementations of the official rules with one small change: Prior to rounding, 0.05 is added as a safeguard to ensure that .5 is correctly rounded up by making it 0.55. Not sure, if this is really necessary.
It is important to note that the base values of the stats are used here (_Basis). This prevents these base values to change upon temporary effects such as wounds, spells or disease. A side effect of this is that the "Mod" column on the "Grundwerte" tab must not be mistaken for the "Mod" column on the official PDF character sheet which takes boni/mali from the character generation. In its current form, this character sheet assumes that the value entered under Basis is the value of the current stat with all permanent modifications already factored in.
Magic Resistance (Magieresistenz/MR)
round((@{MU_Basis} + @{MU_Mod} + @{KL_Basis} + @{KL_Mod} + @{KO_Basis} + @{KO_Mod})/5)
Since stats are positive integers and are divided by 5, the first decimal place always is an even number (.0, .2, .4, .6 or .8). Therefore, no safeguard is needed. Temporary modifiers without wounds are applied. No source is yet known for this behaviour and therefore, this will likely change in a future release.
Initiative/Attack/Parry/Long Range Base Values (INI/AT/PA/FK-Basiswert)
Analog to Magic Resistance, but wound effects are honoured for Attack/Parry/Long Range base values. No source is yet known for the application of temporary modifiers and therefore, this will likely change in a future release.
3d20 Checks
Four cases have to be distinguished in order to determine the result of a roll:
- Automatic Success: At least two dice show a 1.
- Success: The sum of the points available for all checks is greater than or equal to the sum of the points needed to succeed all three 1d20 checks.
- Failure: The sum of the points available for all checks is less than the sum of the points needed to succeed all three 1d20 checks.
- Automatic Failure: At least two dice show a 20.
Three 1s or three 20s do not per se have any further influence on the roll, it is up to the GM to decide the effects in such cases.
The complete algorithm to determine the exact result of a roll cannot be adequately implemented using Roll20 rolls. Also, certain special skills affect the outcome as well such as Misfortune Magnet (Pechmagnet) or Fixed Matrix (Feste Matrix). Therefore, the currently used formula has some limitations.
[[ { [[{0d1 + @{skill} - (?{mod|0}), 0d1}kh1]] - {1d20cs1cf20 + [[{0d1 + (?{mod|0}) - @{skill}, 0d1}kh1]] - @{stat1}, 0d1}kh1 - {1d20cs1cf20 + [[{0d1 + (?{mod|0}) - @{skill}, 0d1}kh1]] - @{stat2}, 0d1}kh1 - {1d20cs1cf20 + [[{0d1 + (?{mod|0}) - @{skill}, 0d1}kh1]] - @{stat3}, 0d1}kh1, 0d1 + @{skill}}dh1]]
To break down the overall structure a bit, let's look at this in a very abstract way:
[[ {points after all rolls, max. points left}dh1 ]]
At the highest level, the points after all rolls are compared to the maximum points left after the rolls (usually the skill value). If the max. points left are higher than the points after all rolls, the former is discarded and the latter used as the roll result. Example: Skill value (= max. points left) is 7 and 3 points were consumed by high rolls, so that points after all rolls is 4. {4, 7}dh1 yields 4. If the check was modified by -7, i.e. 7 more points were available for countering high dice rolls, the points after all rolls would be 11. {11, 7}dh1 yields 7. This is the desired and correct way to calculate the points after all rolls as the points after all rolls cannot exceed the skill value.
In a less abstract way, the roll looks like this:
[[ { [[points to counter high rolls]] - {points needed by roll 1, 0d1}kh1 - {points needed by roll 2, 0d1}kh1 - {points needed by roll 3, 0d1}kh1, max. points left}dh1]]
Points to Counter High Rolls
{0d1 + @{skill} - (?{mod|0}), 0d1}kh1
Without any modifier (mod = 0), exactly @{skill} points can be used to counter high dice rolls. Negative modifiers increase these points, positive modifiers decrease these points. There are always at least 0 points left due to the kh1 against 0d1.
Points Needed by Roll n
{1d20cs1cf20 + [[{0d1 + (?{mod|0}) - @{skill}, 0d1}kh1]] - @{statn}, 0d1}kh1
The result of each roll is the amount of points needed by the roll. Each roll consumes at least 0 points as ensured by the kh1 against 0d1. The points consumed are calculated as the roll 1d20cs1cf20 plus the effective skill value and at least 0. This ensures that in the case of difficult checks with the modifier exceeding the skill value, the now effectively negative skill value is making each roll more difficult by its absolute value. Example: Skill value is 7, but the modifier is +9 (very difficult), so that the effective skill value becomes -2. Now, each roll has to be modified by +(abs(-2)) = +2. Another kh1 against 0d1 guarantees that positive effective skill values do not give another bonus; they already benefit the check by providing more points to be consumed by high rolls. Finally, the corresponding stat value is subtracted. If the result is negative or zero, the outer kh1 already mentioned will set this roll to 0, otherwise the result of this term is the required points.
Max. Points Left
0d1 + @{skill}
Number-to-roll conversion to cap the check result at the skill level.
Specializations
For specializations, whenever @{skill} is mentioned, a + 2 was added, because the effective skill value is higher when using the specialization.
Automatic Failure and Automatic Success
These two conditions cannot be caught with Roll20 rolls at present. Roll templates check for WasCrit and WasFumble, but this is just a boolean and does not give the number of 1s and 20s. Therefore, an info is printed in rolls with at least one 1 or one 20.
Bonus Damage from High Strength (TP/KK)
Weapons come with a pair of values for the calculation of increased/decreased damage based on your strength. The first value is the threshold, which describes the strength needed to use the weapon in a balanced way. The second is the strength step needed to determine at which strength values more/less damage can be dealt.
The calculation is not simple, therefore I want to start with pre-calculated values from WdS, p. 82:
- TP/KK = 12/2 (threshold: 12, step: 2)
- Strength 12 - 2 * 2 = 8: Damage - 2
- Strength 12 - 1 * 2 = 10: Damage - 1
- Strength 12 ± 0 * 2 = 12: Damage ± 0
- Strength 12 + 1 * 2 = 14: Damage + 1
- Strength 12 + 2 * 2 = 16: Damage + 2
- Strength 12 + 3 * 2 = 18: Damage + 3
While this may look pretty straightforward at first sight, the problem lies within the interval sizes:
- Damage - 2: 7 and 8
- Damage - 1: 9 and 10
- Damage ± 0: 11, 12 and 13
- Damage + 1: 14 and 15
- Damage + 2: 16 and 17
- Damage + 3: 18 and 19
The ± 0 interval has 2 * step - 1 values, while all others have step values. Therefore, the formula looks like this:
floor((abs(@{STR} - @{threshold}))/@{step})*((@{KK} - @{threshold} + 0.01)/abs(@{STR} - @{threshold} + 0.01))
The first part floor((abs(@{STR} - @{threshold}))/@{step}) is necessary to get the magnitude of the damage increase/decrease:
- Strength 8: floor(abs(8 - 12)/2) = 2
- Strength 9: floor(abs(9 - 12)/2) = 1
- Strength 10: floor(abs(10 - 12)/2) = 1
- Strength 11: floor(abs(11 - 12)/2) = 0
- Strength 12: floor(abs(12 - 12)/2) = 0
- Strength 13: floor(abs(13 - 12)/2) = 0
- Strength 14: floor(abs(14 - 12)/2) = 1
- Strength 15: floor(abs(15 - 12)/2) = 1
- Strength 16: floor(abs(16 - 12)/2) = 2
The second part ((@{KK} - @{threshold} + 0.01)/abs(@{STR} - @{threshold} + 0.01)) has the task to give the correct sign which is the reason why the same terms are in the numerator and denominator, just once without abs(...) and once with. The results are:
- Strength 8: (8 - 12 + 0.01) / abs(8 - 12 + 0.01) = -1
- Strength 9: (9 - 12 + 0.01) / abs(9 - 12 + 0.01) = -1
- Strength 10: (10 - 12 + 0.01) / abs(10 - 12 + 0.01) = -1
- Strength 11: (11 - 12 + 0.01) / abs(11 - 12 + 0.01) = -1
- Strength 12: (12 - 12 + 0.01) / abs(12 - 12 + 0.01) = 1
- Strength 13: (13 - 12 + 0.01) / abs(13 - 12 + 0.01) = 1
- Strength 14: (14 - 12 + 0.01) / abs(14 - 12 + 0.01) = 1
- Strength 15: (15 - 12 + 0.01) / abs(15 - 12 + 0.01) = 1
- Strength 16: (16 - 12 + 0.01) / abs(16 - 12 + 0.01) = 1
As can be seen for Strength 12, without the + 0.01 the denominator would become 0 causing a division by 0 error. Therefore, it's important to keep this.
NB: In the term "TP/KK", "TP" does not stand for "Threshold" (or anything close) and "KK" does not stand for "Step" (or anything close). "TP" stands for "Trefferpunkte" and roughly means "Damage points before armour". "KK" stands for "Körperkraft" and is the complete German name of for the strength stat.
Sources
WdS: Wege des Schwerts