I discovered other interesting properties of these dice: As I was watching the video, I paused frequently to write down the numbers on the TOP HALF of the die (centered around the 20) and the numbers on the BOTTOM HALF of the die (centered around the 1). The dice in this video have the following properties: TOP HALF: Eight numbers (80%) range from 1-10; Two numbers (20%) range from 11-20 Seven numbers (70%) are even; Three numbers (30%) are odd 2, 3, 4, 6, 7, 8, 9, 10, 16, 20 BOTTOM HALF: Two numbers (20%) range from 1-10; Eight numbers (80%) range from 11-20 Three numbers (30%) are even; Seven numbers (70%) are odd 1, 5, 11, 12, 13, 14, 15, 17, 18, 19 But now compare this with a standard d20: TOP HALF: Five numbers (50%) range from 1-10; Five numbers (50%) range from 11-20 All Ten numbers (100%) are even 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 BOTTOM HALF: Five numbers (50%) range from 1-10; Five numbers (50%) range from 11-20 All Ten numbers (100%) are odd 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
I don't actually have any d20s and assumed this was exactly how they were set up... I imagined the best way to do it and came up with a remarkably similar settup trying to make sure its truly random or at least fair in its biases
I watched the video slowly and single frame for much of it. I did that to get a good view of the d20 sides and vertices and adjacent faces. To make a complete map of the d20. Then I placed that data into a spreadsheet to prove that the five faces around a vertex, and the three sides adjacent to each each face, are all numerically balanced. I was not surprised to find out they were. Then I went and bought a bunch of dice.
@@henryseg I have a blue d120. And a red d120 on order today. Thanks! I have a large spreadsheet proving the numerical balance of the d120. Crude but it works. Rows 1 thru 120. Columns for every one of the 62 vertices. Each column proves a sum of 605 or 363 or 242 depending on the number of faces.
This is helpful since some dice seem to have their numbers in different orders, except the opposing sides always sum up to the same - so opposing sides on my D20s always add up to 21, the D8s always have their opposing sides adding up to 9 etc. But the other thing with the 52 and 53 with the surrounding numberswasnew. Gonna check that out more often and keep it in mind when I consider getting new dice sets. Thank you for the information.
You should compare in a file Excell ex. 100 roll of your numerically balanced d20 with a standard d20 and the random function of the program. It would be interesting to see the distribution of values in the 3 cases and verify which is the better.
The problem is modelling the die roll. The naive thing to do is suppose that each face of a d20 comes up with equal probability, in which case there is no difference between any of the numberings. To see differences, you need a model of how a physical die with physical imperfections rolls. And you need a model of how those imperfections are likely to arise in manufacturing. It's all very complicated.
@@henryseg I understand. Thank you for the answer. In one of the videos of the opti dices I pointed out how you can also create the d20. You can do: 01, 02, 03, 04... 18, 19, 20 with the exagons font. You must only change the shape of the number 1 because now is readable upside down so 10 is equal to 01. Changing the shape of the disposition of the ten exagons of the number 1 you can do it!
Vertices numbered by lowest numbered faces. Vertex 1 has faces 1,5,11,17,19, sum 53. Vertex 2 has faces 1,5,14,15,18, sum 53. Vertex 3 has faces 1,11,12,13,15, sum 52. Vertex 4 has faces 2,4,10,16,20, sum 52. Vertex 5 has faces 2,4,13,15,18, sum 52. Vertex 6 has faces 2,9,10,14,18, sum 53. Vertex 7 has faces 3,6,7,16,20, sum 52. Vertex 8 has faces 3,6,8,17,19, sum 53. Vertex 9 has faces 3,7,11,12,19, sum 52. Vertex 10 has faces 4,7,12,13,16, sum 52. Vertex 11 has faces 5,8,9,14,17, sum 53. Vertex 12 has faces 6,8,9,10,20, sum 53.
It seems to me that it might be better to try to include the central face for the face-balancing: I can't think of a common situation that would raise/lower the probabilities of the neighboring faces, but not the central one. Was this a conscious decision? Is it just not possible if we include the central face, or is there something else that I'm missing?
@@AndrewBlechinger The numbers 1-4 have to be spaced out and then to make it balance the last numbers don't fit. You end up needing a 22 and a -1 or something somewhere.
is that mathematically the most you can do? Like, given those three constraints together, how many sides do you have to fix until you arrive at a fixed configuration? And is there enough leeway to add any other constraint that could improve the balance of these dice?
+Kram1032 Bob tells me that there are only a few solutions that have opposite numbers adding to 21 and balanced around vertices and balanced around faces. So likely any other constraint would not be possible to satisfy on the nose.
I have tried doing some further research on this. According to my findings there are 6 solutions which satisfy the constraints given, so I tried hard to find more properties which could be used to rank them. We could try to make the variance of edge sums (add two numbers adjacent to the given edge) as low as possible, but the variances are equal in all 6 solutions. Same for the variance of full face sums (add the number at the given face and numbers around it). Same for the third moments, in both cases. But each of the 6 solutions has a different fourth moment of edge sum. There are 4 values for fourth moments of full face sums, and solutions with low value for edge sums tend to have high values for full face sums.
So what you are saying is we could pick and choose solutions based on the kurtosis or pointiness of the solution's distribution of either the edge sums or the face sums, and the solutions which are more Gaussian (I assume that'd be the goal) for edge sums tend to be less Gaussian for face sums and vice versa? Could we express the solutions in some kind of linear combination which then could be minimized together? (I'm only sorta going out on a limb here. I don't really have much of a clue how these things are done but I've seen plenty of seemingly entirely unrelated things suddenly be interpreted in a very precise and natural manner as some sort of vector space with geodesics and all, so I can only assume this is true here too)
I have been analyzing the dice in my collection. I have been sitting here wondering why did other makers of dice not try to do a better job of distributing the (pips) or numbered faces. Almost makes me want to throw them all away in favor of balanced versions. Almost. They are still pretty. If not too smart. Insert evil grin here.
Hey I read your paper and it was insanely useful in my own paper on dice, thank you so much for making it.
I discovered other interesting properties of these dice:
As I was watching the video, I paused frequently to write down the numbers on the TOP HALF of the die (centered around the 20) and the numbers on the BOTTOM HALF of the die (centered around the 1).
The dice in this video have the following properties:
TOP HALF:
Eight numbers (80%) range from 1-10; Two numbers (20%) range from 11-20
Seven numbers (70%) are even; Three numbers (30%) are odd
2, 3, 4, 6, 7, 8, 9, 10, 16, 20
BOTTOM HALF:
Two numbers (20%) range from 1-10; Eight numbers (80%) range from 11-20
Three numbers (30%) are even; Seven numbers (70%) are odd
1, 5, 11, 12, 13, 14, 15, 17, 18, 19
But now compare this with a standard d20:
TOP HALF:
Five numbers (50%) range from 1-10; Five numbers (50%) range from 11-20
All Ten numbers (100%) are even
2, 4, 6, 8, 10, 12, 14, 16, 18, 20
BOTTOM HALF:
Five numbers (50%) range from 1-10; Five numbers (50%) range from 11-20
All Ten numbers (100%) are odd
1, 3, 5, 7, 9, 11, 13, 15, 17, 19
I don't actually have any d20s and assumed this was exactly how they were set up...
I imagined the best way to do it and came up with a remarkably similar settup trying to make sure its truly random or at least fair in its biases
I watched the video slowly and single frame for much of it. I did that to get a good view of the d20 sides and vertices and adjacent faces. To make a complete map of the d20. Then I placed that data into a spreadsheet to prove that the five faces around a vertex, and the three sides adjacent to each each face, are all numerically balanced. I was not surprised to find out they were. Then I went and bought a bunch of dice.
You might also want to check our work on the d120!
@@henryseg I have a blue d120. And a red d120 on order today. Thanks! I have a large spreadsheet proving the numerical balance of the d120. Crude but it works. Rows 1 thru 120. Columns for every one of the 62 vertices. Each column proves a sum of 605 or 363 or 242 depending on the number of faces.
This is helpful since some dice seem to have their numbers in different orders, except the opposing sides always sum up to the same - so opposing sides on my D20s always add up to 21, the D8s always have their opposing sides adding up to 9 etc. But the other thing with the 52 and 53 with the surrounding numberswasnew. Gonna check that out more often and keep it in mind when I consider getting new dice sets. Thank you for the information.
You should compare in a file Excell ex. 100 roll of your numerically balanced d20 with a standard d20 and the random function of the program.
It would be interesting to see the distribution of values in the 3 cases and verify which is the better.
The problem is modelling the die roll. The naive thing to do is suppose that each face of a d20 comes up with equal probability, in which case there is no difference between any of the numberings. To see differences, you need a model of how a physical die with physical imperfections rolls. And you need a model of how those imperfections are likely to arise in manufacturing. It's all very complicated.
@@henryseg
I understand. Thank you for the answer.
In one of the videos of the opti dices I pointed out how you can also create the d20.
You can do:
01, 02, 03, 04... 18, 19, 20 with the exagons font.
You must only change the shape of the number 1 because now is readable upside down so 10 is equal to 01. Changing the shape of the disposition of the ten exagons of the number 1 you can do it!
@@GiuliSnow i believe the issue was also more that you cannot fit two numerals in that font onto a single face
Can you do this with other dice, like the d12 or d8?
I imagine it's possible with any die, but the calculated numbers would be different.
Vertices numbered by lowest numbered faces.
Vertex 1 has faces 1,5,11,17,19, sum 53.
Vertex 2 has faces 1,5,14,15,18, sum 53.
Vertex 3 has faces 1,11,12,13,15, sum 52.
Vertex 4 has faces 2,4,10,16,20, sum 52.
Vertex 5 has faces 2,4,13,15,18, sum 52.
Vertex 6 has faces 2,9,10,14,18, sum 53.
Vertex 7 has faces 3,6,7,16,20, sum 52.
Vertex 8 has faces 3,6,8,17,19, sum 53.
Vertex 9 has faces 3,7,11,12,19, sum 52.
Vertex 10 has faces 4,7,12,13,16, sum 52.
Vertex 11 has faces 5,8,9,14,17, sum 53.
Vertex 12 has faces 6,8,9,10,20, sum 53.
It seems to me that it might be better to try to include the central face for the face-balancing: I can't think of a common situation that would raise/lower the probabilities of the neighboring faces, but not the central one. Was this a conscious decision? Is it just not possible if we include the central face, or is there something else that I'm missing?
+Abca Def It turns out that it isn't possible to balance the numbering at all well with the central face included in the face balancing.
So you can't make it such that a face and the three adjacent ones always add up to 42? Huh. Interesting.
@@AndrewBlechinger not while keeping the other aspects balanced as well
@@AndrewBlechinger
The numbers 1-4 have to be spaced out and then to make it balance the last numbers don't fit. You end up needing a 22 and a -1 or something somewhere.
Is it just me or does this gentleman sound like voice actor Liam O'Brian of Critical Role?
Nicki Upson I
My thoughts exactly, haha!
Is Henry German? That would explain why his accent sounds like Caleb's.
too bad it costs almost 7 times the actual dice just for international shipping
Try wish i would say
Great video. I look forward to buying some of these. My current d20 doesn't even have opposing sides that add up to 21. It's really weird.
Might be an MTG d20, those have a spiralling pattern of 20 all the way down to 1
I Bought 5 D20s To Use On Board Games I Own. Also I’m Making My Own Board Games Using Paper.
Your first link in your description gives an error.
Are opposite vertex-sums complementary?
Is this a unique solution? If not, how many solutions are there?
Not a unique solution. I do not know how many there are. I may write some python to find out.
At least 2 because it can be opposite chirality
is that mathematically the most you can do?
Like, given those three constraints together, how many sides do you have to fix until you arrive at a fixed configuration?
And is there enough leeway to add any other constraint that could improve the balance of these dice?
+Kram1032 Bob tells me that there are only a few solutions that have opposite numbers adding to 21 and balanced around vertices and balanced around faces. So likely any other constraint would not be possible to satisfy on the nose.
I have tried doing some further research on this. According to my findings there are 6 solutions which satisfy the constraints given, so I tried hard to find more properties which could be used to rank them. We could try to make the variance of edge sums (add two numbers adjacent to the given edge) as low as possible, but the variances are equal in all 6 solutions. Same for the variance of full face sums (add the number at the given face and numbers around it). Same for the third moments, in both cases. But each of the 6 solutions has a different fourth moment of edge sum. There are 4 values for fourth moments of full face sums, and solutions with low value for edge sums tend to have high values for full face sums.
So what you are saying is we could pick and choose solutions based on the kurtosis or pointiness of the solution's distribution of either the edge sums or the face sums, and the solutions which are more Gaussian (I assume that'd be the goal) for edge sums tend to be less Gaussian for face sums and vice versa? Could we express the solutions in some kind of linear combination which then could be minimized together?
(I'm only sorta going out on a limb here. I don't really have much of a clue how these things are done but I've seen plenty of seemingly entirely unrelated things suddenly be interpreted in a very precise and natural manner as some sort of vector space with geodesics and all, so I can only assume this is true here too)
I have been analyzing the dice in my collection. I have been sitting here wondering why did other makers of dice not try to do a better job of distributing the (pips) or numbered faces. Almost makes me want to throw them all away in favor of balanced versions. Almost. They are still pretty. If not too smart. Insert evil grin here.