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First, we need to acknowledge that there is a haptic element to dice rolling that has a broad appeal. The sensation of rolling a number of dice is inherently pleasurable.
Second, though, we also need to acknowledge that for a proportion of gamers - but clearly not all of them - there are competing aesthetic priorities: on the one hand, people clearly love dice. The number of ludicrously successful Kickstarter campaigns doing nothing but producing a range of pretty dice is pretty powerful indication that pretty dice are an attractive element to which players respond. On the other hand, people also love a sense of narrative immersion in their games and few things break narrative immersion than a tabletop covered in dice.
Going back to the OP, then, I think some bloke is right that there must be a statistical sweet spot that represents the "right" number of dice that maximizes the haptic return whilst minimizing the aesthetic interference. My purely personal and unscientific take is that it's "about 5".
This has (almost) nothing to do with good game design, but you could begin a game design process by building mechanics around the roll of an optimally-pleasing number of dice (4-6) so all tests would be taken using the optimal number of dice. To be frank, I've had worse starting points for a game design process.
But the OP goes further than just an optimal number of dice. He posits that there is also an optimal number of times that dice should be rolled to resolve any given query. In his example (40k) he suggests that four rolls per player is too many and I would tend to agree. But the GW argument - well tested in the marketplace, although it's arguable whether that amounts to a test of game design so much as it does a test of good marketing - is that roll to hit, roll to wound, roll to save provides the right balance of statistics and agency. Would you rather roll twice, needing a 4+ on each, or just once needing a 5+? Statistically, we should prefer the 5+ roll (1/3 chance) to the two 4+ rolls (1/4 chance) but psychologically, people have a success bias. It is more psychologically rewarding to observe more successes on the first roll, even if at least half will be lost in the second, than it is to observe more successes overall in a single roll.
So, again, good design (fewer rolls) loses to good psychology (more rolls).
Like our haptic optimum, there must be a psychological optimum number of rolls per interaction that maximizes the experience of "success satisfaction" (and haptic feedback) whilst optimizing speed of play.
I would suggest that this is about 2.5: that is, normally two rolls (hit/wound) but with the occasional inclusion of a re-roll (hit/reroll/wound or hit/wound/reroll). I think if every roll allowed a re-roll it would be too many rolls and interfere with the speed of play too much. But occasional re-rolls both optimize success satisfaction and maximize haptic feedback with minimal play flow interruption.
So my personal opinion, purely based on rough guesses informed by the aesthetic, haptic and psychological experience of dice rolling, suggests that the best experience for players will come from rolling 4-6 dice a little more than twice per interaction.
Now, if you were to come back to good game design, I would look at the work I did on an unsuccessful fantasy skirmish game (which will one day be re-launched when I have a more saleable reputation!) called Skrapyard. Skrapyard is - if I do say so myself - a beautifully designed game and resolves all interactions with a single dice rolled a single time per player per interaction. It's elegant and mathematically quite beautiful (again, if I do say so myself). But it doesn't provide the elements of haptic and psychological feedback to which players unconsciously respond. I would argue that it does provide aesthetic feedback, because the dice rolled was always a d12 and the d12 is, subjectively speaking, the most beautiful of all the Platonic solids.
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