## Sunday, August 27, 2017

### Goblin Hit Dice

Do you recognize this stat block?

FREQUENCY: Uncommon
NO. APPEARING: 40–400
ARMOR CLASS: 6
MOVE: 6"
HIT DICE: 1–7 Hit points
% IN LAIR: 40%
TREASURE TYPE: Individuals K, Lair C
DAMAGE/ATTACK: 1–6 or by weapon
SPECIAL ATTACKS: Nil
SPECIAL DEFENSES: Nil
MAGIC RESISTANCE: Standard
INTELLIGENCE: Average (low)
ALIGNMENT: Lawful evil
SIZE: S (4' tall)

There isn't much remarkable here. The hit dice entry has excited commentary in our group, though. 1–7 hit points. How do you roll that?

In OD&D a goblin was 1 - 1 hit points (i.e. 1d6 - 1). This can result in zero hit points.

Maybe you roll d8 and re-roll on 8.  There is evidence that Gygax used similar methods. For example, he allowed player characters to re-roll if they got one hit point. See this post on Gary's House Rules.

Our house style is to resolve with single rolls. We don't role-play; we roll-play. Roll-players have enough dice and they design games correctly for the dice they have.

There are non-uniform approximations, such as to roll d8 and replace an 8 with a 7. Of course, you could also roll d8-1, and replace a 0 with 1. The fact that there at least two ways to do this seems unaesthetic.

To measure the quality of a non-uniform approximation, let's use the mean absolute deviation of the approximation from the uniform distribution. Lower numbers are better. For "roll d8 and replace 8 with 7", the MAD from uniform is:

$$\frac{6 \cdot |\frac{1}{8} - \frac{1}{7}| + |\frac{1}{4} - \frac{1}{7}|}{7} \approx 0.031$$
One could use 2d4-1 to generate a 1–7. The MAD from uniform is:

$$\frac{2 \cdot |\frac{1}{16} - \frac{1}{7}| + 2 \cdot |\frac{2}{16} - \frac{1}{7}| + 2 \cdot |\frac{3}{16} - \frac{1}{7}| + |\frac{4}{16} - \frac{1}{7}|} {7} \approx 0.056$$
There is an idea I like which could be called "if high, then low". Roll a d8 and a d6 simultaneously. Use the d8 unless it rolls an 8, in which case use the d6.  The MAD from uniform is:

$$\frac{6 \cdot |\frac{1}{8} + \frac{1}{8} \cdot \frac{1}{6} - \frac{1}{7}| + |\frac{1}{8} - \frac{1}{7} |}{7} \approx 0.005$$
Another proposal. An encounter says there are 12 goblins, but if any of them are determined to have zero hit points, rule that they are already dead!

I analyzed cave D of "The Keep on the Borderlands" to get a hint of how to generate goblin hit points. This module was written by Gygax himself. There are 28 goblins, not counting the chieftain, females, and young. Here is a distribution of their hit points.

I'm not sure how Gygax randomly generated numbers with this distribution. If you have any ideas, shoot me an email.

We like to stick to the standard six shapes for dice: d4, d6, d8, d10, d12, d20. The Platonic solids and the bastard interloper known to science as the pentagonal trapezohedron. Looking over these new dice shapes makes us guffaw and chortle and a few other synonyms. We see useless bits of dice bag clutter which deceive people into rolling them by mistake. But goblins have made us reconsider one shape. Or possibly two shapes.

The orange die is a truncated sphere. The manufacturer supposedly found 7 maximally separated points on a sphere and then truncated the sphere at those points.

The other two dice are pentagonal prisms. Did the manufacturer, Gamescience, take an empirical approach to choosing the height of the prism so that the die rolls fair?

The p-values are 0.38, 0.23, and 0.08, suggesting the dice are fair enough for casual play. The Gamescience dice put the 6 and the 7 on the pentagons. 6s and 7s look to be over-represented in my Gamescience rolls, even if we showed the effect is not statistically significant for this sample size. It might be the truncated sphere is the better choice.

Here are suggestions from Gamescience on how to use your new 7-sided die: