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Within DnD 5E, the number 20 is a central motif of system design. The basic roll is made on a d20. Ability scores cap (or at Peak at) the number 20. The level cap is 20. Potentially this could be coincidence; it is also possible that it was an aesthetic choice. It concerns me, however, whether diverting from this symmetry would have any measurable consequences to play itself.

Is there a mathematical consequence to the symmetry of these design elements?

Knowing this would help determine both the reason for this system and whether altering it would have a serious effect on the math, but the question is more concerned with the math itself, rather than cause or effect. The answer could of course be 'No, there is no mechanical reason for this,' but I'm seeking a reasonable investigation into whether or not any sources indicate that this occurrence exists with design-specific implications. I'm curious if I'm missing something- perhaps some rule which makes use of ratios between these aspects of the game; ratios which, of course would need to be adjusted if these scaled differently.

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    \$\begingroup\$ Meta query to others: should this question be targeted on designer quotes to avoid speculation (which is endless)? \$\endgroup\$ – doppelgreener Sep 3 '16 at 13:20
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    \$\begingroup\$ @doppelgreener absolutely \$\endgroup\$ – Wibbs Sep 3 '16 at 13:26
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    \$\begingroup\$ Part of your question is already answered (or challenged) here: Why the level cap at 20? \$\endgroup\$ – SirTechSpec Sep 3 '16 at 14:14
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    \$\begingroup\$ I removed the d20-system tag because a) it makes this question too broad and b) it makes it too incorrect (levels and ability scores aren't capped at 20 in d20 system!). As for "why all 20s in 5e," to make this not opinion-based answer must cite developer reasoning or otherwise be not pure supposition. Of course the question is still muddled; do you want to know a history-of-gaming why 20, or do you want to know if you can change something specific about the game without imbalancing it (seems like its' your premise)? \$\endgroup\$ – mxyzplk Sep 3 '16 at 14:20
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    \$\begingroup\$ More to the point, as mxy said, please clarify whether you're asking "how did things get to be this way?" or "what are the consequences of changing X?"; these are different questions that will have different answers. (And if it's the latter, please be more specific about what you're looking at changing, how, and why.) \$\endgroup\$ – SirTechSpec Sep 3 '16 at 14:43
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Mainly, it's inherited. The oldest editions (much like the modern one) determined traits with 3d6, even if it was weighted or extra unkept dice were used. Which is why the scale goes from 3-18, with an average of 10-11. The game mechanic favored high rolls for attacks and saves, however skill and ability rolls were based on rolling a d20 and rolling below your target number. So having a 12 Intelligence and trying to remember something was a 1-12 on a d20 to succeed. Racial traits were typically +1 not +2 because it had a varying scale of effects instead of "+/- 1 all trait stuff per 2 points". So an Elf could start with 19 Dex and still have that natural 20 chance to fail a task. Later on however, they did feature in ways to have a 20+ in any given trait, which had the most trouble around Strength. At level 18 Strength had a sub number on a percentile scale to determine its final bonuses.

On a reddit thread, When Did the d20 Come Into Play, they speculate that Dave Arneson suggested the use of the d20 to Gygax's game Chainmail (which has been since recreated) and created an alternate set of rules which helped develop the above system.

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I would argue that the icosahedron, being the regular polyhedron with the highest number of faces, meaning the easiest "big" die to create, is the natural choice for a game in which you want a high "luck" component in order to obtain a high variation.

About the other aspects, I think that for simplicity reasons when you want to create a "d20 system" you try to make it so that all the "high" numbers become 20. Since 20 has no particular property that makes it a better number than any other (if anything, 20/3 is not a integer, which often complicates stuff in-game) I think that "keeping it simple" is the only reason.

Specifically about the 5E, well, D&D has always been centered on a "20" system, so it's pretty much tradition at this point.

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    \$\begingroup\$ This could do with more citations over conjecture; we could all present our opinion but it doesn't help create definitive answers. \$\endgroup\$ – doppelgreener Sep 3 '16 at 13:21
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    \$\begingroup\$ 4e hard-capped at 30 while 3.5 simply changed its mechanics at 20 with no hard cap at all; some earlier editions had different level caps for different characters. Similarly, the idea of an ability score cap is not a common element across D&D editions--much less a cap at 20. What 20s are "always" or "traditionally" part of D&D aside from the d20? \$\endgroup\$ – BESW Sep 3 '16 at 14:02
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    \$\begingroup\$ I think you're on to something with the icosahedron, but the rest is conjecture. \$\endgroup\$ – SirTechSpec Sep 3 '16 at 14:27

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