[RPG] How to approximate rolls for potions of healing using only d6’s

anydicedicednd-5estatistics

I want to make tabletop potions of healing out of corked vials (test tubes) filled with dice. Each vial would be labeled and filled based on the potion it represents. For example, a vial for a potion of greater healing would contain four d4's and be labeled with a +4 bonus; when administering such a potion, the player would just dump the vial and total the dice plus the bonus, yielding the correct 4d4+4 result, without having to fiddle with their own dice. The vials are meant to speed up play, act as physical reminders that a player has a potion available, and look super cute (assume that these intentions are inviolable and that this craft project is serious business).

It's hard to find enough d4's to pull off this craft project, and test tubes are normally too small for standard 16mm dice anyway, so I've considered using miniature 12mm d6's instead, which are much easier to come by in blocks of large quantities for cheap. The problem is that d6's are slightly more swingy than d4's, and I don't want to grossly deviate from the math underlying potions.

How can I approximate the rolls for each potion of healing while avoiding swingy results? By swingy results I mean unexpectedly low or high totals or a distribution that violates conventions for how healing works in the game.

The following are the restrictions on a valid solution:

• Only d6 dice can be used. It's a physical constraint of the problem.
• Each vial must contain a constant number of dice to dump and roll for the result, without requiring additional dice that weren't in the vial.
• Basic mental math like addition and subtraction is fine.
• Rerolling below a minimum total or similar rules of thumb are fine if they are simple.

Answers telling me to use the average instead of rolling, to roll with online tools, to find tinier d4's or bigger test tubes, to buy a commercially available set of d4-filled vials, or the like aren't solutions. I promise you this question doesn't suffer from an XY problem. The restrictions are inherent to the nature of the craft project, a very serious and important craft project.

For bonus, corresponding AnyDice formulas would be nice but not vital.

Let's not overcomplicate it: a simple Nd6 + X formula will work just fine.

Specifically, the effects of Greater (4d4 + 4) and Superior (8d4 + 8) healing potions are quite closely replicated by rolling 2d6 + 7 and 4d6 + 14 respectively. These formulas yield exactly the same average number of hit points healed as the originals (14 for Greater and 28 for Superior potions), and the shapes of the distributions are quite close.

For normal healing potions, naïvely applying the same conversion formula would require us to roll 1d6 + 3.5. Rounding this up to 1d6 + 4 makes these modified potions slightly better (by 0.5 hit points, on average) than the originals, which your players probably won't mind. It also somewhat compensates for the extra "swinginess" (i.e. higher variance) of rolling a single d6 vs. rolling 2d4.

Here's a graph from an AnyDice script demonstrating these distributions:

In particular, rounding the healing amount for the standard potion upwards ensures that the modified potion always have the same or higher chance of healing at least N points as the original, for any N, which is probably desirable. The following graph, plotted from the same script as above, shows the cumulative probabilities of healing at least N points using all these formulas:

BTW, for Supreme healing potions, 5d6 + 28 will give a rather good approximation of the standard 10d4 + 20 formula. As with the standard potion, the average amount healed per potion will be half a hit point higher than with the original formula (45.5 points vs. 45 points), but at these levels that's not really noticeable. The AnyDice script linked above also includes these formulas, but they're commented out by default.

Just for completeness, here are all the original and d6-based formulas in a convenient table:

$$\begin{array}{l|rrrr|rrrr} & \text{Original} &&&& \text{d6-based} \\ \hline \text{Potion type} & \text{HP gained} & \text{Min} & \text{Avg} & \text{Max} & \text{HP gained} & \text{Min} & \text{Avg} & \text{Max} \\ \hline \text{Normal} & 2{\rm d}4+2 & 4 & 7 & 10 & 1{\rm d}6+4 & 5 & 7.5 & 10 \\ \hline \text{Greater} & 4{\rm d}4+4 & 8 & 14 & 20 & 2{\rm d}6+7 & 9 & 14 & 19 \\ \hline \text{Superior} & 8{\rm d}4+8 & 16 & 28 & 40 & 4{\rm d}6+14 & 18 & 28 & 38 \\ \hline \text{Supreme} & 10{\rm d}4+20 & 30 & 45 & 60 & 5{\rm d}6+28 & 33 & 45.5 & 58 \\ \hline \end{array}$$

A nice thing about this method, in my opinion, is that it should work really well with your test tube based physrep idea. All you need to do is put the appropriate number of six-sided dice (1, 2, 4 or 5) in each tube, and add a sticker with the fixed number of points to add (+4, +7, +14 or +28). No fancy math or rerolls or relabeling dice required. Simple!

(Alternatively, if you'd prefer your Greater and Superior healing potions to retain the property of being exactly twice and four times as good as a normal potion, you could adjust the fixed parts of their d6-based formulas up to +8 and +16 respectively. The Supreme potion doesn't follow this pattern even with the original formulas, but adjusting its additive constant up to something like a nice round +30 would probably be reasonable if you choose this approach.)