Strife:Using Alternate Dice Sizes

Note that equations are built into the character and GM's sheets.

Notation

 * $$\lceil Ceiling\ function\ brackets \rceil$$: These brackets round any non-integer values to the higher integer (i.e., indicate rounding up).
 * $$Intended\ dice\ size$$ is the dice size that is intended to be used, for example, for standard Strife rolls this value would be 100.
 * $$Desired\ dice\ size$$ is the dice size that you want to roll with, for example, this would equal 20 if you want to roll using a d20.
 * $$Threshold\ with\ intended\ dice\ size$$ is the maximum value that you would succeed on when using the intended dice size (discounting luck), for example, with 36 hand-to-hand this would equal 36 if you were trying to punch.
 * $$Threshold\ with\ desired\ dice\ size$$ is the threshold you need to pass with the new desired dice size.
 * $$Rolled\ damage\ in\ intended\ dice\ size$$ is the damage converted to what it would have been if you rolled using the intended dice size.
 * $$Rolled\ damage\ in\ desired\ dice\ size$$ is the value you rolled in the dice size you wanted to roll damage with.

Generalised
When rolling with a different than intended dice size, roll thresholds need to be adjusted. The equation for this is:

$$Threshold\ with\ desired\ dice\ size=\left \lceil \frac{Desired\ dice\ size \times Threshold\ with\ intended\ dice\ size}{Intended\ dice\ size} \right \rceil$$

For example, if you had 36 hand-to-hand and you wanted to roll with a d20 instead of a d100 for a punch, then:

$$Threshold\ with\ desired\ dice\ size=\left \lceil \frac{20 \times 36}{100} \right \rceil=\lceil 7.2 \rceil=8$$

Meaning that you would have to get an 8 or below on the d20 to succeed the punch.

Going from using d100s to d20s
If you want to use a d20 instead of a d100, this equation simplifies to:

$$Threshold\ with\ desired\ dice\ size=\lceil (Threshold\ with\ intended\ dice\ size)/5 \rceil$$

This is useful for playing Strife in real life, rolling with real dice instead of a dice-bot or random number generator.

Generalised
Luck thresholds are also changed, instead of being a range of $$ Success\ and\ Failure\ Threshold \pm 10$$ i.e., $$ -10 \le Success\ and\ Failure\ Threshold \le 10 $$, they are defined as:

$$ Success\ and\ Failure\ Threshold \pm \left \lceil \frac{Desired\ dice\ size}{10} \right \rceil$$ or $$ Success\ and\ Failure\ Threshold-\left \lceil \frac{Desired\ dice\ size}{10} \right \rceil \le Luck\ Threshold \le Success\ and\ Failure\ Threshold+\left \lceil \frac{Desired\ dice\ size}{10} \right \rceil $$

Going from using d100s to d20s
If you want to use a d20 instead of a d100, luck thresholds are instead:$$ Success\ and\ Failure\ Threshold \pm 2$$ i.e., $$ -2 \le Success\ and\ Failure\ Threshold \le 2 $$.

Critical Success and Critical Failures
For all dice sizes Critical Successes and Critical Failures are the highest and lowest rollable values on the dice respectively, rolling a 1 for a Critical Success and a (dice size) for a Critical Failure.

Generalised
This is for "well-placed hit... on yourself", "well-placed hit... on a random target", "distracted", and "backfires".

Highest 5 thresholds are also changed, instead of the highest 5 values (discounting max value) they are the highest $$\lceil (Desired\ dice\ size)/20 \rceil$$, also discounting max value.

For example, using a d60 instead of a d100 is the highest $$\lceil 60/20 \rceil=3$$ values discounting max value. This means on a 57, 58, and 59 you critically fail with a highest 3 value, but on a 60 you get a Critical Failure.

Going from using d100s to d20s
Using a d20 instead of a d100 is the highest value discounting max value. This means on a 19 you critically fail with a highest value, but on a 20 you get a Critical Failure.

Generalised
This is for "Well-placed hit".

Lowest 10 thresholds are also changed, instead of the lowest 10 values (with 1 overlapping) they are the lowest $$\lceil (Desired\ dice\ size)/10 \rceil$$, also with 1 overlapping.

For example, using a d60 instead of a d100 is the lowest $$\lceil 60/10 \rceil=6$$ values with 1 overlapping. This means on a 6, 5, 4, 3, and 2 you get a Well-placed hit, but on a 1 you get a Critical Success.

Going from using d100s to d20s
Using a d20 instead of a d100 is the lowest 2 values with 1 overlapping. This means on a 2 you get a Well-placed hit, but on a 1 you get a Critical Success.

Damage dice
Sometimes the right damage dice size is not available when rolling with real dice, it's not every day someone can pull out a d67. So here is an equation to roll using a different resolution of damage dice:

$$Rolled\ damage\ in\ intended\ dice\ size=\left \lceil Rolled\ damage\ in\ desired\ dice\ size \times \frac{Intended\ dice\ size}{Desired\ dice\ size} \right \rceil $$

Moving to a larger dice size instead of moving to a lower dice size
This is made in mind of rolling with a smaller dice size for in real life play, but you can move up dice sizes with all equations still applying the same. Success and Failure thresholds are the same too, but it's reversed, and they are ranges instead of single values. This means that on a certain success and failure threshold, the same threshold on the new dice would be the outputted equations number and all lower values that do not merge with other thresholds. An mathematical visualisation of this is:

$$\left \lceil \frac{Desired\ dice\ size \times Threshold\ with\ intended\ dice\ size}{Intended\ dice\ size} -\frac{Desired\ dice\ size}{Intended\ dice\ size}\right \rceil< Threshold\ with\ desired\ dice\ size \le \left \lceil \frac{Desired\ dice\ size \times Threshold\ with\ intended\ dice\ size}{Intended\ dice\ size} \right \rceil$$

Which can be simplified to:

$$\left \lceil \frac{Desired\ dice\ size \times (Threshold\ with\ intended\ dice\ size-1)}{Intended\ dice\ size}\right \rceil< Threshold\ with\ desired\ dice\ size \le \left \lceil \frac{Desired\ dice\ size \times Threshold\ with\ intended\ dice\ size}{Intended\ dice\ size} \right \rceil$$

For example, if you moved from a d20 to a d100, then the general equation is $$Threshold\ with\ desired\ dice\ size=5 \times Threshold\ with\ intended\ dice\ size$$. This means that for a 1 threshold on the d20 it'd instead be 1-5 on the d100, for a 2 on the d20 it'd instead be 6-10, for a 3 on the d20 it'd instead be a 11-15, etc...

Advantage and Disadvantage
Advantage and Disadvantage rolls are the exact same, except instead of rolling with the intended dice you roll the Advantage and Disadvantage rolls with the desired dice, where the roll thresholds are the exact same for a standard roll, or the rolled damage is then used in the Damage dice equation for a damage roll.

Standard roll example: if you move from a d100 to a d20 you use the Success and Failure thresholds equation normally, and then roll the Advantage or Disadvantage with the d20, attempting to get on or below the $$Threshold\ with\ desired\ dice\ size$$.

Damage dice example: if you move from using a d67 to a d20 damage dice, you roll for Advantage or Disadvantage damage with the d20 and the highest or lowest value respectively is then used as the $$Rolled\ damage\ in\ desired\ dice\ size$$ for the Damage dice equation.

Character sheets for d20 rolling
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