In a gaming campaign or even just a
single adventure, many occasions arise
when a referee must devise a method for
rolling damage, or for generating a random
number in some non-standard fashion. The
rules of the game provide figures for all the
standard monsters, weapons, magical attacks,
or whatnot. But often a new creature,
trap, hazard, or other device will be created
in order to surprise and amuse the players.
If a new weapon or attack form is introduced,
this may require some different rule
for calculating damage or some other randomly
determined result. In most cases, this
result is arrived at by the roll of a die, or by
calculating the sum of several rolls of the
same type of die. This article delineates
some of the limitations of this standard
approach, and suggests some of the possible
alternatives.

Plotting the distribution of the roll of a
particular die or group of dice yields a
graphic illustration which indicates the
frequency of occurrence of that specific roll.
In other words, this type of graphing charts
the chance of a specific number (say, 3 on
1d6) appearing when considering all the
possible results (1 through 6 on 1d6) that
could be obtained. Figure 1a plots the roll
of a single die. This is a flat, or linear,
distribution: each result has the same
chance of occurring. Figure 1b, on the other
hand, represents a bell-curve distribution:
when several dice are rolled, each result no
longer has the same chance of occurring.
(With 2d6, results of 2 or 12 occur less
frequently than any results between those
two extremes.) The top of the curve indicates
the number that occurs most often.
(For 2d6, this number is 7.) Traveling from
that high point down both sides of the
curve, the frequency of each possible result
decreases symmetrically; the numbers that
correspond to each end of the curve have
the same frequency, but that frequency is
much less than that of the number at the
top of the curve.

Figure 1a and Figure 1b have one element
in common: symmetry. In contrast,
Figure 1c diagrams an asymmetric distribution.
If a dice roll using an asymmetric
distribution were used for damage calculations,
the long tail at the right of the figure
would represent a small chance for damage
far beyond the typical (average) amount.

The chance of an extraordinary event ?
in this case, abnormally high damage ? is
usually beneficial to the enjoyment of the
game. Player characters become more wary
of an adversary who can occasionally inflict
great damage, and overcoming the challenge
then proves to be more satisfying.
And, giving player characters the chance to
do extraordinary damage encourages them
to keep trying when they otherwise might
not do so, and helps to build a legacy of
memorable events in the campaign.

The extraordinary event must be, by
definition, rare. Low-probability events can
be found in Figures 1b and 1c, where the
height of the curve is low: the long tails on
both sides of the bell curve, and the long tail
on the high (right) end of the asymmetric
curve. Both curves have a long right tail; in
game terms, this means high damage with a
low probability. The sum of several rolls of
identical dice gives a distribution like that of
Figure 1b, the bell curve. Why, then, use
anything else? The problem with the bell
curve is that its average result is a higher
number than the average result on an asymmetric
curve with the same maximum
value. For example, the distribution of a
4d4 dice roll does not have very long tails,
but the average is 10. In a certain case, the
referee may want the maximum result to be
relatively high, but not at the expense of a
high average result. Therefore, a method
that renders a small probability for abnormally
high damage, yet that has a reasonably
low average damage level, must be
found.

A divided die roll is created by dividing
the result of one roll by another roll. For
instance, for a d20/d4 divided die roll,
simply roll a d20 and a d4, divide the number
on the d20 by the number on the d4,
and round the result to the nearest integer.
(For certain purposes, you may want to
round all fractions up, to avoid a result of
0.) The distribution of d20/d4 is plotted in
Figure 2. This diagram is an asymmetric
distribution just like the one in Figure 1c.
The most common number produced will
be between 1 and 6 inclusive, but there is a
slight chance that the damage could go as
high as 20. (Specifically, there is a 1-in-80
chance of this happening.) The average
result is approximately 5.6, which is almost
the same as the average for a roll of 1d10.
So, substituting a d20/d4 die roll for a 1d10
roll leaves the average result essentially the
same, but adds the chance of rolling values
from 11 to 20.

A few examples will help to illustrate the
possible uses of divided rolls. Some lowlevel
monsters in the AD&D® game (e.g., a
large spider) do 1 hit point of damage with
a successful hit. A similar creature, however,
could do d4/d6 of damage. Some of
the time, no damage will be done -2/5
and 1/3, for example, are less than 1/2, so
the result would be rounded down to 0.
(The attack was enough to be felt, but it
proved to be little more than a scratch.
Occasionally, though, the damage could be
as high as 4 points. The d4/d6 method
produces an average roll of about 1. A d10/
d20 roll is another alternative here. A much
greater chance of no damage exists, but the
maximum possible damage is 10 (when the
d10 roll is 10 and the d20 roll is 1).





Generally, a hit from a bare fist does 1-2
hit points of damage. This is an average of
1.5. Two divided rolls that can be used 
instead are d4/d4 and d8/d8. Which system
you should choose depends on how high you
want the maximum damage to be; this
figure is always the maximum of the first
die (the one before the slash). The higher
maximum damage of a d8/d8 roll is balanced,
however, by a greater chance of a
result of 0. Similarly, a dart does 1-2 points
of damage against large opponents. This is
commonly determined by ½d4. Replacing
the ½d4 roll with d8/d8 could result in the
dart merely scratching the monster?s hide
(no damage). Once in a while, however, a
dart strike can be as vicious as a good sword
blow. Those reluctant to change the rules
for all darts could introduce a new type of
dart ? perhaps one with a longer, thinner
point. Often the point bends or breaks ?
but if it hits solid, watch out.

A dagger does d4 (1-4) points of damage
against small or medium opponents. It
seems that a dagger-like weapon could be
more deadly than this. Any of three divided
rolls could be used instead of d4: the rolls of
d8/d4, d12/d6, and d20/dl2 would be appropriate.
These three choices provide a
good selection for potential damage. With a
d8/d4 roll, 8 points of damage is the maximum
? enough to kill any zero-level type,
as well as many first-level characters. However,
there is but a 1-in-32 chance of rolling
this result. The d20/d12 roll, on the other
hand, has a much greater maximum of 20
-- enough to fell the average third-level
lighter in one blow. But there is also about
a 13% chance that the blow will do no
damage. (In contrast, the chance of no
damage with d8/d4 is only 6%.)

Divided die rolls can be useful in many
situations not covered by the rules. For
instance, consider the problem of determining
damage caused by a rockslide that has
struck a group of adventurers. Rolling a
separate amount of damage for each party
member would be a logical procedure.
Some characters might get hit with really
big rocks, whereas others would suffer only
a hail of pebbles. The maximum damage
from a rockslide should be very high, while
the average damage should be moderate.
Perhaps it would wipe out a first-level party,
but in a fifth- to seventh-level party, only
the really unlucky would die. Let?s try a
divided roll of d100/d10: in the long run,
5% of the rolls will generate damage of 50
hit points or more. The average damage is
around 15, and over half of the time damage
will be less than 10. This sort of dice
roll would also be well suited to a monster
with a powerful breath weapon or other
magical attack form. Braving such a creature
would be an acceptable risk for even a
moderately low-level party ? yet the attack
has the potential to slay all but the strongest
characters. This is a situation guaranteed to
generate excitement.

Some game systems allow for the possibility
of a "critical hit" -- that is, an extraordinarily
rare blow that does an extremely
high amount of damage. Many of the
critical-hit systems (both official and unofficial)
used by referees call for an extraordinary
result on a roll of 20 on the attempt to
hit. Applying this sort of rule to the AD&D
game combat system, we can see that it
leads to some odd results. For instance, an
orc that attacks a human who has an armor
class of -1 will always score a critical hit if it
hits at all (because the orc needs a 20 to hit
AC -1). A frost giant attacking that same
human will achieve a critical hit only onetenth
of the time that the giant scores a hit
(because it hits on 11 or better, and 20 is
just one of ten numbers that indicate a hit).
Overall, both monsters have the same
chance for a critical hit ? a 1-in-20 chance.
But the ?to hit? roll already reflects the
defender?s armor class, the attacker?s level,
any magic defenses and weapons, dexterity
adjustments, and so on. It should not be
routinely burdened by carrying any more
information. For all these reasons, the rare
chance of exceptional damage -- if this is to
be a part of the combat system -- should be
carried within the damage roll itself.

Critical-hit systems also commonly have
a provision for automatic death, but divided
die rolls include no such provision. If death
is a desired possibility for every blow, regardless
of the attacker?s power or the defender
?s hit points, then another system
must be used. This point should also be
noted: critical-hit systems diminish the
value of gaining experience levels, since the
chance of dying from a critical hit is not
reduced at all as a character gains levels;
however, monsters, spells, and magical
weapons that can (according to the rules)
cause instant death become less special,
because lesser creatures or weapons can do
the same thing, just less often.

Determining the number of creatures
appearing in an encounter is another good
use of the divided die roll. For example,
d100/d4 could be used for creatures that
tend to lair in large numbers, but that hunt
or patrol in smaller groups. A d100/d4 roll
has an average result of about 26, and
about half of the time, it will be 20 or less.
Some monsters appear singly or in pairs.
Beholders, for example, are listed in the
Monster Manual as appearing singly only.
The logic behind this fact is that they are
too voracious for two to share a territory
(see DRAGON® Magazine, issue #76, p.6).
<Ecology of the Beholder>
Yet since the monsters are intelligent and
lawful, there seems to be no reason why a
small group of beholders cannot be occasionally
encountered. When a d4/d6 roll is
substituted for the number appearing, a
high-level party might come to regret its
casually considered frontal assault on a
beholder?s lair when the party discovers that
the resident beholder has invited two or
three of its friends over for a game of ?zap
the humans.? A result of 0 could be treated
in three ways: it could mean no encounter,
the 0 could be changed to 1 (though this
raises the average), or it could mean that
the party does not (yet) encounter the beholder,
but acquires evidence that the monster
is nearby.