OK, so the bounty notice says:
The testing listed in this question hasn't actually been conducted by any of those attempting to answer the question, this question requires additional attention to determine whether or not this testing method is viable for determining a dies balance.
Well, clearly this calls for science!
Actually, I'm not sure what my testing would prove that these two videos on YouTube haven't already demonstrated, but whatever. After all, reproducibility is the essence of the scientific method. I've got dice, I've got salt and I've got water. Let's do this!
TL;DR: Yeah, it works. Bigger, rounder dice (d12 / d20) are easier to test than smaller, more angular ones (d8 / d6 / d4). And boy, are those fancy textured Chessex dice poorly balanced. The transparent ones are much better — even the ones with visible air bubbles inside. Also, I've got salt stains everywhere now.
Preparation and methodology
I started by just filling a plastic cup with water, tossing in a few dice, and adding salt and stirring until the dice would start to float. I used plain old table salt (NaCl) because that's what I had, but just about any water-soluble substance that makes the water denser should work.
This kind of worked, but I noticed a couple of problems:
It takes a lot of salt (like, a lot!), and a lot of stirring to make it dissolve. The former issue is unavoidable; the latter we can fix.
The water ends up full of air bubbles. This a problem, not just because it makes the water milky and hard to see through, nor because the bubbles actually reduce the mean density (which is what we want to increase), but mostly because the bubbles stick to the dice unevenly, ruining their balance even if they normally are balanced.
Also, I'm not sure if it was something in my salt, or if the cup I used was dirty or something, but my water ended up full of not just bubbles, but also some cloudy yellowish organic-looking gunk that made it look just plain nasty.
So, for my second test, I came up with an improved procedure:
Boil the water first. This gets rid of any dissolved air, and also helps with the next step.
Divide the boiled water into two parts. Mix one of the parts with as much salt as it will dissolve. It helps to do this while the water is still hot, since salt dissolves faster and easier in hot water. You'll still have to do quite a bit of stirring to get a really saturated brine.
Let both batches of boiled water cool down. (You can speed this up with a cold water bath.) If the water looks dirty, run it through a coffee filter to get rid of any gunk.
(Come to think of it, if you had a coffee maker, you might be able to do all this just by filling the filter with salt and running a cup or two of water through it. Alas, I don't have one around to test it.)
Put the dice you want to test in a small cup, pour enough of the salty water in so they float nicely, and then gradually add non-salty boiled water until they float just barely.
Check that there are no visible air bubbles stuck on the dice (especially inside the pips / numbers) that could upset the balance. If there are, try to get rid of them. (Dripping single drops of salt water from above on top of the floating dice seems to be a fairly effective way to shake the bubbles off.)
Poke the dice a couple of times, and see if they consistently return to the same orientation. If so, they're unbalanced.
If you're careful in step 4, you can get a layer of less salty water to float on top of the denser salt water in the bottom of the cup, and have the dice float in the middle without touching either the bottom or the surface. This is by far the most sensitive method, able to detect even tiny imbalances that are impossible to see otherwise.
You do need to still poke the dice and see if they consistently turn the same side up. Some dice — particularly d10's — can be multistable, e.g. preferring to always float with one (any) corner up. This does not necessarily indicate an imbalance, unless one orientation is clearly more stable than its opposite.
Also, you should do this all in the sink, or some place that you can easily wash, because you will spill salt water at some point, and it will leave salt stains when it dries. You'll also get salt on your hands, and from there to everything you touch. I've cleaned salt stains from my floor, my table, my jeans, my glasses, my camera and my keyboard. My hands also still feel dry and salty, even after washing them, and I think I've got a little bit of salt in my eye. Rubber gloves might be advisable, even though it's just salt.
One thing you'll notice is that different dice have different densities, and require very different salt concentrations to float. This is true even for dice within the same set; one consistent trend I noticed is that the d4's tend to sink while the d12's and the d20's float. Presumably, this is because the surface of the dice is denser than the interior, and so the bigger, rounder dice with a lower surface-to-volume ratio tend to float easier.
The bigger dice (d12 / d20) are also much easier to see bias in, because, being so close to round, they freely rotate even while bobbing on the surface. Dice with fewer sides (d4 / d6 / d8), on the other hand, tend to get "stuck" in a local equilibrium with one side on the surface, even if the side on the top isn't actually the lightest. To see any imbalance in those smaller dice, you have to fine-tune the salt concentration so that they just barely float, preferably without touching either the bottom or the surface.
(The d10's are somewhere in between the two extremes; having two "sides" with five faces on each side, they'll easily rotate between faces on the same side, but they won't easily flip from one side to the other on the surface.)
The dice I had available for testing included:
- A clear translucent 7-die polyhedral set from Chessex. (For some reason, I seem to have an extra d10 for this set; no idea where that came from.)
- A clear translucent 12 mm 36d6 set, also from Chessex.
- A small opaque dark green textured 7-die polyhedral set with gold markings. I believe these are also from Chessex, but I can't find a good match in their catalogue just now. They look vaguely like these dice, but without the purple, and they're smaller than usual (the d6 is only 8 mm wide).
- A bunch of random d6's from various board and card games, including Illuminati, Munchkin and a couple of generic d6's whose origin I don't remember.
The YouTube videos I watched suggested that opaque textured dice are the worst, and sure enough, the d12 and the d20 in the green mini-dice set showed a very obvious bias: the d12 invariably floated with the 6 side on top, while the d20 favored the corner surrounded by 10, 12, 15, 7 and 17, with a slight but easily noticeable bias towards the 12 side among them.
The two d10's in the same set also showed a somewhat noticeable imbalance, but in both cases it was towards a point on the "rim" between the two sides. This meant that, in order to see the bias clearly, I had to reduce the density of the salt water until they no longer always floated with one face on the surface.
Notably, one of the d10's in that set turned out to be a lot denser than the other — almost as dense as the d4! Based on this observation, I suspect that the reason (or at least one reason) why these textured dice are so badly balanced is because they're made by mixing different colored plastics in the mold, and those plastics have different densities. Since the mixing is necessarily uneven, to produce the desired swirls and speckles, this all but guarantees that the dice won't be precisely balanced.
The d4 in the set also seemed to favor one side, at least once I actually got it to float. The d6 and the d8, on the other hand, did not reveal such an obvious bias. This might, of course, just be a coincidence.
The clear transparent dice that I tested, also from Chessex, were much better balanced, even though some of them had visible air bubbles in them. One of the transparent d6's in the 36d6 set, with a fairly big air bubble near one corner, did noticeably favor that corner, as expected, but even then, the bias wasn't huge.
None of the dice in the clear 7-die set (well, 8-die, since I have that extra d10) showed any detectable imbalance that could not be attributed to external air bubbles, except for the d20 that had a visible air bubble inside it. Even then, the imbalance in the d20 was minuscule, and I had to use the layering technique to suspend the die in the middle of the water, away from the surface, to reliably observe it.
I also tested some of the random d6's, and was pleasantly surprised by their balance, at least compared to the grossly unbalanced green 7-die set. The Illuminati dice did slightly prefer to float with the 1-face (with the Illuminati logo) up, but again, I had to use the layering trick to really observe this. The two random generic d6's I had would each slightly favor one corner, but again, only when suspended in the middle of the water.
The Munchkin die was really dense. I could not get it to float at all. :-(
Floating dice in salt water really is a practical way to test their balance, although it does take some practice to get reliable results. For d12's and d20's, simply floating them on the surface of salt water is enough to reveal any obvious imbalance. For dice with fewer sides, or in order to detect more subtle imbalances, suspending the die between two layers of salt water with different densities may be required.
As the YouTube videos link above already showed, opaque textured dice seem to be particularly prone to poor balance. In some sense, this is hardly surprising: the presence of the texture clearly shows that such dice are not made of a homogeneous material. Still, the magnitude of the imbalance in some of the tested dice was startling.
Translucent dice seem to be much better in this regard, having a significantly more uniform density. The main cause of imbalance in such dice seems to be trapped air bubbles, which can be detected by eye.
Even the clear dice with bubbles inside them seemed much better balanced than the textured opaque dice. The miscellaneous opaque but non-textured plastic d6's also appeared to be quite surprisingly well-balanced in comparison.
It would be interesting to carry out a chi-square test on the various dice tested above, to see how much bias their observed imbalances actually cause when rolled. I'll try to add those results later.