Miscellaneous Applications

There are two applications to recapitulate interesting computations from the OpenD6 rules.

These are concrete confirmation of the math underlying the published tables.

die_simplification

Creates the Die Code Simplification table from first principles.

This script emits the published wild-die table:

Die Code

5D

Wild Die

\(n\)

\(\lfloor 0.5 + 3.5(n-5) \rfloor\)

\(\lfloor 0.5 + 3.5(n-1) \rfloor\)

Alternate

There might be slightly more accurate approach, but it doesn’t seem any more playable.

Any given die roll of \(n\) die has two components: Ordinary Die and Wild Die.

  • Ordinary Die. Average is \((n-1) \times 3.5\).

  • Wild Die. There are three possible outcomes.

    • \(\frac{1}{6}\) Critical Failure. This can reduce the total.

      The simplification however, doesn’t change the total. Die value is 1. Contribution to the total is about 0.17.

    • \(\frac{1}{6}\) Critical Success. Roll this Wild Die again, accumulating the values. There can be multiple re-rolls of the wild die.

      • \(\frac{5}{6}\) the re-roll is not 6. Expected value is \(\frac{1}{6} + (6 + 3\times\frac{5}{6}) = \frac{17}{12} \approx 1.42\).

      • \(\frac{1}{6} \times \frac{5}{6}\) the first re-roll is 6, the second is not 6. Expected value is \(\frac{1}{6}^2 \times (12 + 3\times\frac{5}{6}) = \frac{29}{72} \approx 0.4\).

      • \((\frac{1}{6})^3 \frac{5}{6}\) three re-rolls. Expected value is \(\frac{1}{6}^3 \times (18 + 3\times\frac{5}{6}) = \frac{41}{432} \approx 0.095\).

      • etc. \(\sum 6\times\frac{1}{6}^n = \frac{6}{5}\) for \(n\) re-rolls. Total is \(\frac{1}{6} (\frac{6}{5} + 3\times\frac{5}{6}) = \frac{27}{10} = 2.7\).

    • \(\frac{4}{6}\) Nothing Special. Die average is 3.5. Contribution to the total is about 2.3.

    Total: \(\frac{31}{6} \approx 5.16\). This is too small a difference to justify the more complicated computation. Rolling \(n\) die seems to be \((n-1) \times 3.5 + \frac{31}{6}\).

Implementation

opend6_tools.die_simplification.die_code(start: int = 1, stop: int = 51, step: int = 1) → Iterator[tuple[str, str, str]][source]

Emits the sequence of die code simplifications as triples. (Die code, 5D, Wild Die).

Parameters:

  • start – starting Die code (default 1)

  • stop – ending Die code (default 51)

  • step – step vvalue (default 1)

opend6_tools.die_simplification.main(start: int = 1, stop: int = 51) → None[source]

Produce CSV-formatted table of Die Code simplifications.

spell_measure

Creates the Spell Measures table from first principles.

Two cases:

  • Very first range has steps: 1, 1.5, 2.5, 3.5, 5.

  • Remaining ranges and bands have steps: 1, 1.5, 2, 2.5, 4, 6.

Both are essentially this.

\[v = \lceil 5 \times \log_{10}(m) \rceil\]

However, the very first range and all the remaining ranges have slightly different decimal rounding rules.

  • The first five are ROUND_CEILING.

  • All the rest are ROUND_HALF_UP.

The whole table has 3 ranges: \(\times 10^0\), \(\times 10^1\), and \(\times 10^2\), in 6 bands: “”, “,000”, “ million”, “ billion”, “ quadrillion”, “ quintillion”.

Implementation

opend6_tools.spell_measure.value_measure() → Iterator[tuple[Decimal, str]][source]

Emits the sequence of game values and the source measures.

opend6_tools.spell_measure.main(rows: int = 35) → None[source]

Produce CSV-formatted table of Spell Measures.

The number of rows defines the format.

  • For 100 rows, the table is in a single pair of columns.

  • For less than 100 rows, the table is split into columns to meet the row constraint. The published table has 35 rows, therefore, this is the default.