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Recently a question has popped up in the comments of another question I've recently answered where a player has happened to roll three 18s and other high stats at a table with his dice, which could lead me to believe that he may be playing with a set of loaded or imbalanced dice.

Is the method presented in the youtube video How to check the balance of your d20 an accurate representation of a die's weighting and balance and could it be used to properly and reliably test whether dice are loaded?

The video provides the following instructions for testing whether or not a die is balanced or not:

Ingredients:
1/4 cup of hot tap water (our water is a little hard)
6 tablespoons of Epsom salt

  1. Put the water in a small jam jar.
  2. Dump 2 tablespoons of Epsom salt into the water; put the lid on it and shake it till it dissolves.
  3. Dump 2 more tablespoons of Epsom salt into the water; put the lid on it and shake it till it dissolves.
  4. Add the last 2 tablespoons of Epsom salt; microwave the water on high for 30 seconds.
  5. Put the lid on it and shake it till it dissolves (use a dish towel to hold this, it is hot at this point).
  6. Once dissolved, set the closed container in a cold water bath until it cools back down to a little cooler than the room temperature.
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    \$\begingroup\$ physically, the test described is properly designed to find an inconsistency in internal mass-distribution that one'd associate with the term "loaded." Are you asking how precise the method is, and if it's precise enough to answer the question about your player's dice? If so, please specify the level of unfairness you'd be willing to tolerate; without that spec the method in Jordan's answer can't be evaluated. \$\endgroup\$ – nitsua60 Nov 6 '15 at 18:49
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    \$\begingroup\$ The instructions don't appear to have anything to do with dice. Perhaps something was left out? \$\endgroup\$ – Robert Nov 7 '15 at 1:31
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    \$\begingroup\$ This question would probably be more appropriate on physics.se or similar, its connection to RPGs is extremely tangential. \$\endgroup\$ – mxyzplk Nov 7 '15 at 3:26
  • \$\begingroup\$ I'm voting to close this question as off-topic because it does not leverage RPG expertise in any way. \$\endgroup\$ – mxyzplk Jan 24 '18 at 2:30
  • \$\begingroup\$ @mxyzplk Given that dice are a core tool in most RPG's, isn't this directly related to the hobby? I think I see what you are driving at, but I am not sure that this is too far removed from a lore type question. \$\endgroup\$ – KorvinStarmast Jan 24 '18 at 2:35
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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:

  1. Boil the water first. This gets rid of any dissolved air, and also helps with the next step.

  2. 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.

  3. 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.)

  4. 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.

  5. 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.)

  6. 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.

Observations

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.)

Results

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. :-(

Conclusions

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.

Further work

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.

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    \$\begingroup\$ This is an amazing answer, and it pretty much hit all the points that I was looking for. I'd really like to see the results of your chi-square test as well. \$\endgroup\$ – Sandwich Nov 7 '15 at 6:30
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    \$\begingroup\$ This test appears to have a similar quality to the chi-square test in that it reproduces more reliable results if you spend more time on it. I have to wonder how many times you could've rolled the dice (and write the result down) in the same amount of time that it required to prepare, execute and cleanup this experiment. Nonetheless, I am still greatly impressed and inspired by the effort and ingenuity displayed here, especially in improving the experimental method. For science! +1 \$\endgroup\$ – DaFluid Nov 7 '15 at 16:02
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    \$\begingroup\$ +5 for Science, and @DaFluid I was waiting to see if anyone would attempt a similar test, with varying results. Its always good to have a control you know. \$\endgroup\$ – Sandwich Nov 7 '15 at 21:57
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    \$\begingroup\$ This is a great post but... it doesn't actually answer the question. The question asks if it's a good way to test whether the dice are loaded (i.e., produce a non-uniform probability distribution). What you've shown is that it's a good way to determine whether dice have a mass imbalance. However, this test seems very sensitive: essentially, any unbalanced die where the buoyancy torque exceeds the viscous friction will rotate to be light-side up. One could imagine this test saying, "Whoa, this die is hella biased" when, actually, it's very close to uniform. \$\endgroup\$ – David Richerby Nov 13 '17 at 16:00
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    \$\begingroup\$ @DavidRicherby: That's a fair point, although also a question of semantics to some extent: the original meaning of "loaded", whether applied to dice or otherwise, is simply "weighted; bearing a load", but the term can indeed also be used to refer to other kinds of biased dice. Some of those biases may be detectable by the float test, if they make the shape of the die asymmetric, but not all of them. For example, "shaved dice" that are flattened along one axis, to make them more likely to land with that axis upright, may still pass the test. And yes, it can also be oversensitive to some biases. \$\endgroup\$ – Ilmari Karonen Nov 13 '17 at 18:39
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This experiment can show that a die has an extremely imbalanced center of mass if the same number always shows up no matter how you agitate the die, but it can fail to reveal a less extreme imbalance that still causes a bias. (Also note that the center of mass is not the only thing affecting the outcome of rolling a die on a surface -- asymmetry in the die's shape and unevenness in the friction of its surface and in the sharpness of its edges also affect the outcome.)

The only sure way to measure the fairness of a die is to roll it many times, tally the results, and do a chi-square test. The math of that is beyond the scope of this question. However, you can still use water to amplify the effect of an imbalanced center of mass in order to make a bias become apparent after fewer rolls, alleviating the tedium of rolling a die over and over again. The basic idea is simple: (1) prevent the die from bouncing, and (2) maximize the time it spends falling.

To do that, take the tallest, narrowest container you can find, like a vase or a graduated cylinder, and fill it with water. "Roll" the die several times by dropping it in the water and seeing which number faces up when it lands. When you let go of the die, each number should face up an equal amount of times. (i.e. hold it with the 1 facing up and drop the die n times, then hold it with the 2 facing up and drop it n times, etc.) If you want to slow the die's fall through the water, then add some salt, but not so much that the die floats.

Tally the results, sort them by frequency, and see if there is a set of numbers adjacent on the die that show up much more often than the other numbers.

If I wanted to find out if a die were loaded, then to save time, I would run tests on it in the following order until one of them revealed an obvious bias:

  • floating in saltwater
  • dropping several times through water
  • rolling many times on a surface
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  • \$\begingroup\$ Essentially, you're saying that the test isn't sensitive enough to mass imbalances. It looks like it should be very sensitive, to me. What evidence do you have that relates the sensitivity of this mass imbalance test to actual non-uniformity of the dice's behaviour? And how does testing a die by dropping it through a cylinder of water tell you anything about its behaviour when thrown through air onto a table. When you actually roll a die, it bounces a lot, yet you're claiming that the most accurate test is one that removes bouncing. Why? \$\endgroup\$ – David Richerby Nov 13 '17 at 16:04
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    \$\begingroup\$ Also, doesn't running the tests "until one of them reveal[s] an obvious bias" guarantee that you decide that every die is biased? \$\endgroup\$ – David Richerby Nov 13 '17 at 16:05
  • \$\begingroup\$ While "rolling" dice in a tall narrow container might require fewer dice rolls, does it actually safe time? (It takes time to get the dice out afterwards.) \$\endgroup\$ – Paŭlo Ebermann Nov 13 '17 at 21:48
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From the question, it seems that you're concerned with d6. A visual inspection of the dice should be applied before using the salt water method to float test them.

Look for: - Rounded or bloated side - A distended point from where the plastic was injected into the mold - Discolored pips - Air bubbles in Transparent dice - The condition of the edges: they should be uniformly shaped.

The general appearance of the dice will indicate if the dice were baked, drilled, or shaved. The distended point from the plastic injection will be on an edge, and should demonstrate a noticeable deflection. Pips that are mismatched, or too soft in comparison to other sides indicated that material was drilled out.

Float Test It will work for checking to see if a dice lacking other indications has been biased towards one side or the other. It is readily serviceable for d4, d6, d8, and d10, but as you shift towards dice with more faces approaching spherical, the bias will be harder to determine.

The problems with the float test procedure you've posted: you only want to put in enough salt to make the die that you are trying to test slightly better than neutrally buoyant. To little salt and it will drift to the bottom instead of free spinning. Too much and it will broach, changing the dynamics of how it sits.

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  • \$\begingroup\$ Curiously, based on my experiments, the float test seems to be easier for the rounder dice (d12, d20), since they'll freely rotate even on the surface. Dice with less sides tend to turn so that one face touches the surface, and then stick that way. To observe any imbalance in them, I had to carefully tweak the salt concentration so that the dice would float in the middle of the glass, away from the surface. \$\endgroup\$ – Ilmari Karonen Nov 7 '15 at 2:15
  • \$\begingroup\$ @IlmariKaronen - The precise adjustment of the salt concentration is the key, such that the die is as near to neutrally buoyant as possible, if slightly positively buoyant. The more faces the object is, the harder it is to ascertain which face is favored by the bias. \$\endgroup\$ – Drunk Cynic Nov 7 '15 at 3:01

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