I still recommend the Collins piece.

Math, history, and design of old-school D&D.

deltasdnd.blogspot.com

Excerpt:

"...Now, obviously, those last few were for humorous illustrations only, and I assume not many people would want to use those systems. But what criteria can we use to choose the "best" possible system? Let's consider the following as guiding principles (and we'll back each of them up with results from experiments in cognitive psychology as we proceed):

(1) Additions are easier than subtractions. Although mathematically equivalent (and using fundamentally the same operation in digital computing systems), most people find subtraction significantly harder than addition. For example, see the paper by

MacIntyre, University of Edinburgh, 2004, p. 2: "Addition tasks are clearly completed in a much more confident manner than the subtraction items, with over 80% of the study group with at most one error on the items. Subtraction items appear to have presented a much bigger challenge to the pupils, with over 50% having 3 or more of those questions wrong."

**(2) Round numbers are easier to compare than odd numbers.** In other words, when comparing which of two numbers is larger (the final, required step in any "to hit" algorithm) it will be easier if the second number is "20" than, say "27". This follows from the psychological finding that it's faster to compare single digits that are farther apart; see Sousa,

*How the Brain Learns Mathematics*, p. 21: "When two digits were far apart in values, such as 2 and 9, the adults responded quickly, and almost without error. But when the digits were closer in value, such as 5 and 6, the response time increased significantly, and the error rate rose dramatically..." In our case, setting the second digit to zero would maximize the opportunity for a large (and thus easy-to-discern) difference between the numbers.

**(3) Small numbers are easier to compare than large numbers.** This has also been borne out by a host of psychological experiments over the last several decades. Again from Sousa, p. 22: "The speed with which we compare two numbers depends not just on the distance between them but on their size as well. It takes far longer to decide that 9 is larger than 8 than to decide that 2 is larger than 1. For numbers of equal distance apart, larger numbers are more difficult to compare than smaller ones." Again, this is true for human computers only, not digital ones (ironically, the digital processor "compare" operation is really just an application of the same "subtract" circuitry).

Okay, so let's think about applying these principles to find the cognitively-justified best tabletop resolution algorithm. Applying principle #3 means that we'd generally prefer dealing with smaller numbers rather than larger. Before considering anything else, it's clear that it will be hardest for people to mentally operate in a d% percentile system, easier in a d20-scaled system, and easier still on a d6-scaled system. We should pick the easiest of these that gives the fidelity necessary to our simulation, and the d20-scale does seem like a nice medium..."