Fire Optional: Roll two different colored dice. One color is for an overshoot and the other is for an under-shoot. To decide which number of use you take the higher of the two. Miss is in inches, shown by dice spots. If they tie then the rock lands at the specified range. This method is simple but effective. [CM, p. 12-13]Recall that both fireballs and giant rock-throwing refer back to these catapult rules for their resolution (throughout both the Chainmail Fantasy Supplement and Original Dungeons & Dragons Vol 1-3). For example, this same variation rule pops up again for the giants in Holmes blue-book D&D:
Giants can throw rocks like a catapult, range 200 feet with a 20 foot hit area. Each rock does 2 dice of damage to anything it hits. A giant can throw one rock every 5 melee rounds.I'll have more to say about the overall giant-throwing rules in the future, but I'll stick to my brief, main point for today -- That variation rule is crazy-ass broken! If you roll two dice for over/under variance and take the "higher" (in both versions above) then that's biased towards being as far away from your target as possible. As one example, notice that it's impossible to land exactly 1 inch away -- a (1,1) tie lands on the bulls-eye, but otherwise (1,x), lands on whatever that other number is (x>1). Here's a histogram of all the possible results:
There are several ways to calculate catapult (giant) fire. This one is adapted from CHAINMAIL. If figures are being used on a table, the giant estimates the range to his target and throws. The actual distance is then measured. Two six-sided dice of different colors are then rolled. One color is an overshoot and the other an undershoot. To decide which number to use, take the greater. The miss is in inches, shown by the die spots. If they tie the rock lands at the specified range. Anything within 2 inches of the impact is hit. If figures are not used, treat the thrown rock as an arrow or other missile on the combat table. [Holmes D&D, p. 26]
Have you ever seen a probability distribution look like that? No, that's just plain nuts. So my guess is there was a typo and the word "higher" should properly be replaced with "lower". (Or the original writer was just in a rush and didn't properly think it through. Or they actually played that way and didn't realize how batshit crazy it was.) Here's what the probability distribution looks like if you replace the word "higher" with "lower":
Ah, that's more like it. That's what you'd call a "triangular" distribution, with the maximum probability mass at the target (central distance x=0), and sloping off evenly to each side. Basically each of the tails have been inverted; it's now quite likely to land 1" off the target (in fact, the most likely absolute distance), and it's impossible to be as far as 6" away (maximum 5"). Note also that this is exactly equivalent to: roll 2d6, and every point away from 7 is the distance from your target (over or under). And if you added more dice to this model (like roll 3d6, vary by points away from 10 or 11, etc.), then the the histogram would look more like a properly bell-shaped curve; the limiting distribution being, as is the case in any partially-sane universe: the Normal Curve (as per the central limit theorem).
Excel spreadsheet of full data here.