Seven 23mm d20s, one of each color as pictured, all nestled in an elegant case! These d20s feature my mathematically balanced numbering layout. The numbers are spread evenly across the die, avoiding any concentrations of high or low values. This reduces the impact any physical bias will have on the average roll (since physical biases generally favor a group faces in an area of the die). Also, I tested these dice for fairness against 4 other brands of resin d20s for a total of 10,000 rolls. Results are below. There are 40,548,366,802,944,000 possible d20 numbering layouts to analyze. It took me hundreds of hours to write and run the programs to design the balanced numbering layouts for these dice. Part of that was because I had to teach myself to code. There wasn't any vibe coding here, just good old fashioned googling syntax combined with months of trial and error. 🤣 Here are the details for the dice nerds out there like me! To mathematically balance a die you consider groups of faces and measure the sum of their numbers. The optimal result is that all the groups have the same sum, though in most cases no such layout exists and the optimal result is the closest possible layout that does exist. I balanced my d20 using half sums and vertex sums. I compare my d20 layout to the standard d20 layout and also the Magic-Numbered balanced d20 layout by The Dice Lab. It is an elegant layout and I include it only to show how my approach differs from theirs, since I've been asked that question. I’m a fan of The Dice Lab and if you are interested enough to read this I bet you will love their dice too so go check them out! Half sum: The sum of the faces on one half of the die. On the d20 that is a center face, three adjacent faces that share an edge with the center face and 6 faces that share an edge with those faces. This is a total of 10 faces. There are 20 half sums on a d20. If all the half sums were the same the sum of each would be 105. After weeks of analysis I conclusively determined that a layout where all the halves sum to 105 does not exist. The optimal possible layout has 12 half sums of 105, 4 of 104 and 4 of 106. Vertex sum: The sum of the faces that share a vertex. On a d20 there are 12 vertices and each has 5 faces that share that vertex. Thus there are 12 vertex sums on a d20. If all the vertex sums were the same the sum of each would be 52.5, which is not possible. The optimal possible layout has 6 vertex sums of 52 and 6 of 53. Total Deviation: For a given type of sum (such as half sum) the total deviation is the sum of the differences of each sum from the optimal sum for that type. The optimal sum for a sum group is the mean of the die multiplied by the number of faces in the group (for half sum that is 10.5 x 10 = 105). If the total deviation is zero it means that all the sums are the same and equal the optimal sum. Balancing the half sums was my first priority, followed by the vertex sums. I was able to achieve both the optimal possible half sum layout and a well balanced but not quite perfect vertex sum layout. As you can imagine I was extremely pleased with this result. To my knowledge this is the first half sum optimized d20 ever created and I was ecstatic to achieve a vertex sum total deviation of only 12! I also made a couple of other improvements over the standard layout. Down with the n+1 rule! It is a convention in dice making that the opposite faces of a die sum to the number of faces on the die plus 1. For example, on d20s if you add two opposite faces they will sum to 21. 1 will be opposite 20, 10 will be opposite 11. After a great deal of pondering I have come to the conclusion that this is a tradition we should abandon for the sake of dice fairness. It doesn't do anything to spread the numbers in a balanced way and it actually makes the impact of any weight bias worse. For example, if there is a weight imbalance on a die, it makes some faces more likely to come up. It will equally make the opposites of those faces less likely to come up. If the 1 is opposite the 20, a weight imbalance that favors the 1 will also disfavor the 20. You will not only roll more 1s but also you will roll fewer 20s. If instead the 1 was opposite the 2, you would roll more 1s but fewer 2s. You can see that the second option would be far less impactful to the average of a set of rolls on this hypothetical die, while the n+1 convention would actually maximize the impact of the bias on the mean. With my numbering layouts I now put similar values opposite each other. Who put the 6 next to the 9?! On the standard d20 layout the 6 is next to the 9. Not only that, but because one is written upside down in relation to the other, they are the exact same shape with nothing to differentiate them except for a period or line. It is a minor thing I suppose but it always drove me bananas. On these d20s you can rest assured that the 6 and the 9 are never next to each other and it will be easier for you to instantly tell which one you rolled. Fairness Testing It's important when testing dice to use enough rolls to ensure meaningful results. I rolled each of the five d20s 2,000 times, for a total of 10,000 d20 rolls. I tested my d20 against four other poured resin, sharp edge d20s. I chose a mix of expensive and less expensive dice to test. Interestingly, the expensive dice didn't fare particularly better than the inexpensive ones in these tests. I am using a conservative 99% confidence threshold to reject the hypothesis of fairness. That means any dice that failed had rolling data so biased that there is a less than 1% chance a fair die could have produced those rolls. The Honest Dice d20 performed the best with a test statistic of 15.28, well within the range of results you would expect from a fair die. I was pretty shocked to see how badly the others performed. D20 #3 was the only one that passed with a test statistic of 26.34. D20 # 1, #2, and #4 all failed with test statistics of 38.22, 71.20, and 56.20 respectively. We can reject the hypothesis that each of the other three dice are fair with the following (approximate) level of confidence: d20 #1 rejected with 99.443% confidence, d20 #2 rejected with 99.9999942% confidence, d20 #4 rejected with 99.99848% confidence. If those results are representative, it would imply that an appalling percentage of poured resin d20s on the market are badly biased. Four dice is a tiny sample size, and I hope that I was just unlucky to get so many biased dice in my testing, but the only way to find out would be to test more d20s and I was exhausted after 10,000 rolls! 🤣 Safety and other important notes: Dice number layouts are protected intellectual property, copyright 2026, Flying Horseduck, all rights reserved. Hex d4 design covered by patent # US D1,064,085 S. The solid hexagon on the high faces, "Hex d4", and "Honest Dice" are protected trademarks of Flying Horseduck aka Honest Dice. These dice contain small parts and could be a choking hazard for children. Keep away from children under age 5. These dice are intended for adult collectors, not for children.
