Ridiculously Weighted Die (For Fun & Statistics Demonstration)

Everyone's had that time that they were playing a dice-based game and that guy just kept on getting sixes (or ones or doubles or whatever he needed). Some of us may have been quick to call out, "that die is weighted!" But in real life, weighted dice don't always roll you what you want, and sometimes even fair dice go on a winning streak. In this instructable, I'll show you how to make your own ridiculously weighted die, and then how to test that die, or any other die for fairness (or lack thereof) using statistics.

The only thing you'll actually need (aside for a 3D printer) is a 10mm x 10mm element cube. I get mine from luciteria.com (not sponsored in any way, I just love their products). A hammer or mallet isn't necessary, but can be useful in the assembly stage. Similarly, superglue can be used to seal the die if you don't mind sacrificing an element cube.

The principle behind this weighted die is that it has a center of gravity that is very biased towards one side. In this design, the metal cube very close to the '1' side on the die. Because of this, whenever you roll the die, that side tries to get as low as possible relative to the rest of the die. Since on the opposite side of the 1 is 6, this means in practice you should spin way more 6's then you otherwise would have!

Since there's only space inside this design for a 10mm cube, the denser that cube is the better. I'm using a tungsten cube which has an incredible density of ~20 grams per cm^3, compared to PLA at 1.24g. This shifts the weight balance overwhelmingly in the cube's favor. It's also not too expensive, I picked mine up at lucitera.com for about 12 bucks, and you can find them pretty cheap on amazon. Other metals, like copper or iron are potentially even cheaper, but will come at somewhat of a performance discount. I'd recommend tungsten, but you can use whatever you like. Just remember, the denser the better!

After you have obtained a cube, you must 3D print the two halves of the die (Die-Top and Die-Bottom, linked below). Use your favorite slicer to slice the stl files below. Good tolerances are helpful to make sure everything fits snugly together, so it's best to print slow. Infill on Die-Top should be kept at 10% or lower, because we want to have as little mass as possible on the unweighted side.

After a successful 3D print, you now have to put the cube into the bottom half of the die. It should be a pretty tight fit, so you might want to use a hammer or mallet to get the cube in. In lieu of that, you can just bang it against a table or the floor to get it in. If you don't mind keeping the cube there forever, you can add a drop of superglue to the bottom before you put it in for a more permanent fit.

Next, put on the Die-Top piece, similarly to how you installed the cube on the base. But before you press it down, make sure that the dots on the side of the die are aligned properly. Then, you can use whatever method you used last time to press the top down onto the base. Again, for a more permanent fit you can put a drop of superglue inside the top cover.

Perhaps not entirely necessary, but it definitely brings the quality up a notch to color the dots ("pips") on the sides of the die. Gently press a permanent marker into every pip on the die. If you're using a darker filament, like I did, then you should use a metallic sharpie. If the filament is lighter in color, then use whatever color you like!

You should now have a 100% working ridiculously weighted die. Unlike "regular" weighted dice which use comparatively small weights, the weight distribution here is over 10:1 in favor of the cube! And while it's actually rather difficult to tell the bias in most weighted dice (which is honestly on purpose), the skew of this die is so great it's readily apparent (which is fine, as this die was created as a demonstration of probability and not actually for cheating). Hence the name, ridiculously weighted die. Roll the die a few times to get a feel for how the weight makes it always roll toward six.

Now that you have your weighted die, how can we prove that it's actually weighted and not just random chance? This is exactly what statistics are used for in the real world.

To test, grab a scrap piece of paper and a pencil. A calculator doesn't hurt, either. Roll the die numerous times (I did 103), and tally up all the results. At the end, you should have written down the amount every number was rolled, like this in the picture.

To test the results, we'll use a standard Chi Square Test. To do the test, we simply calculate: (Observed Value - Expected Value)^2 ÷ Expected Value.

Since I rolled a suspicious number of sixes, I'll test for that. In my case, the expected value of sixes would have been one out of every six rolls, and 103/6 = 17.16 (it's allowed to be a decimal). However, I observed 45 (!) sixes, so plugging those numbers into the formula gives a Chi Squared Value of ~45. To see if this is significant, we simply need to check this value on a standard Chi Squared Table. But first we need to know one last bit, the degrees of freedom. The degrees of freedom is one less then the possible choices, so if the die is six-sided, the degrees of freedom would be 5. Comparing this on a Chi Squared Table, we see that the largest value they have for 5 degrees of freedom is 20.15, with a probability of 0.001. 45 is way beyond that, so the likelihood of this result being due to random chance is significantly less than 1 in 1000! In other words, this die is very loaded.

Now that you have a functional weighted die, and the knowledge to test it, you can go out there and show it off to others. It's a great demonstration of how probability and statistics can be used to calculate the fairness of dice. Or you can just use it to astonish people with the amount of consecutive sixes it rolls. Just don't cheat!