Resin and Gemstone Polyhedral Sets:
Imagine a d20 that has all blank faces except for one which you have written “20” on. How many ways could you finish numbering this die? You might be surprised to find out there are 121,645,100,408,832,000 possibilities.
Optimizing dice numbering layouts became a bit of a fixation for me. I spent hundreds of hours of programming and testing over the course of years. I started with 60-sided and 120-sided dice and now finally the standard 7 piece dice set. I am not a mathematician or a programmer but I love hard problems and this is the best I've found.
Most of you probably don't think much about the mathematical elegance of the numbers on your dice, but I feel compelled to tell you about it because it was super hard and I'm proud of myself :D
Comparing layouts
I am only going to go into detail about the d20 here (this is already going to be long), but I balanced all the dice in the dice set using similar methods. I will be comparing 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 here only to show how my approach differs from theirs. 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!
What does mathematically balanced mean?
The goal of mathematically balancing a die is to spread the numbers evenly across the die in order to avoid any areas with concentrations of high or low numbers. A die with a balanced layout will be more fair because any bias of the die to land on a face or group of faces more often will have a smaller impact on the mean roll than it would on a standard non-balanced layout.
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 die using the groups of faces shown in the image below.
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 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. Have a look at a d10 and you'll see the same thing! It is a minor thing I suppose but it always drove me bananas. On these dice sets 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.
120-sided dice:
There are over 10^196 ways to arrange 120 numbers on a 120-sided die. That is trillions of times more possibilities than there are atoms estimated to exist in the observable universe.
Solving the numbering layouts of these dice became a bit of a fixation for me. I spent hundreds of hours of programming and testing over the course of 3 years, each failed approach bringing me closer to a solution. I am not a mathematician or a programmer. I really had no idea if I would even find a solution, but I love a hard problem and this was the best I've found.
Most of you probably don't think much about the mathematical elegance of the numbers on your dice, but I feel compelled to tell you about it because it was super hard and I'm proud of myself :D
Halves sum to the same. As mentioned towards the very top of this long Kickstarter page, no matter how you turn it, the sum of the 60 faces on any one half of the die will always add up to 3,630 on the d120 and 630 on the d20. This is the main demonstration of how incredibly well mathematically balanced the numbers on this die are.
Pointy vertices sum to the same. There are 12 vertices that stick out the most... they are the pointiest ones for lack of a better description. Each of these vertices has 10 faces that surround it. If you sum them up they will always have the same value. On the d20 they sum to 105 and on the d120 they sum to 605.
Numbers are far away from themselves. On the d20 each number appears 6 times. I spaced them so they are as far as possible from each other. No matter how you turn the d20, it is impossible to see more than three of any particular number at one time (unless you cheat with a mirror). The d120 is similar but it is harder to tell without the repeated numbers. For example, the 120, 119, 118, 117, 116, and 115 are spread as far apart from each other as possible.
Neighbors are different. Any two faces that share an edge have a minimum difference so you don't end up with similar values right next to each other. On the d120 the minimum difference is 13, so the number 45 could be next to 32 but not next to 33. On the d20 the minimum is 3, so a 16 could be next to 13 but a 16 can't be next to 14.
Down with the n+1 rule! It is the convention in dice making that the opposite faces of a die sum to the number of results on the die plus 1. For example, on most d20s, if you add a face to the one on the opposite side 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 makes the impact of any weight bias worse. 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 the most similar values opposite each other. On the d120, the difference between two numbers on opposite faces will be 1. For example, 84 is opposite 85.
Opposites should be equal(ish). The neat thing about having a 120-sided d20 is that I have 6 instances of each number. That means I can put every value opposite itself! Adding on to the example above, if a weight imbalance favored a face with a 1 on it, it would disfavor the opposite face, which also is a 1. So you would roll more 1s on the first face, but fewer 1s on the opposite face. Since favoring one face equally disfavors the opposite face, the effect cancels out! That means the weight imbalance wouldn't affect the fairness of the rolling results. I did this with every face on the die, so theoretically, the 120-sided d20 I made should be fair even if the center of gravity was off. Now, I use extremely high quality metal that has a super consistent density, so there won't be a weight imbalance. So I decided to drill a few dozen holes in it to test the theory.
Fairly ugly. After removing about 6% of the weight of the die by drilling holes mostly on one side, I had created a die with a huge weight bias. I drilled the holes in the faces where they wouldn't mess up the edges. Since I am rolling on a flat surface, this shouldn't have any impact on how the die rolls in terms of the shape of the die. I also slightly countersunk the edges of the holes to make sure nothing protruded above the face of the die. Then I got to work rolling 2,000 rolls again to compare to the 2,000 times I rolled this die before drilling. Test Methodology
Still fair! The test showed that even after a very substantial weight bias the rolling results were still within what you would expect from a fair die. The Pearson's chi-squared test statistics were remarkably similar. So ha! I invented a die you can drill holes in and have it remain fair. Because... reasons.
60-sided dice:
Rules for Numbering Layouts
Rules for all 60-sided dice:
- Faces with the highest value on a die are each always exactly opposite a face with the lowest value
- The sum of the five faces of any pentagon on a die is within 1 of the sum of any other pentagon (all equal is not possible)
- Pentagons directly opposite each other on the die have equal sums
- Highest and lowest values are spread across the die as evenly as practical
Rules Specific to Each Die:
Notes and Definitions:
- The term "pentagon" is used to collectively refer to 5 faces that completely surround a vertex
- The term "opposite" means the other side of the die; if a die is resting on a face, the opposite face is up
- The term "adjacent" refers to two faces which share an edge
- On the D10 "0" is the value 10 and is considered the highest value on the die
- On the D00 "00" is the value 100 and is considered the highest value on the die
- On the D8 the re-roll faces are considered to have the value 4.5, which is the average roll of a D8
- * Re-roll faces are ignored in the "Difference Between Adjacent Faces" calculations