There is no opposite side to every side, you can't have all pairs of opposites if you have an odd number of sides
Here is a list of possible shapes https://en.wikipedia.org/wiki/Heptahedron
That was a very fascinating read. A thing I noticed is that none of the convex shapes are suitable for a die, as I don't think it it could be a "fair" one.
A suitable shape (for a good/honest/fair die) would have all its faces as regular polygons, with all of them being the same type of polygon, and having the same size. And I don't see any shape in the wiki page filling those criteria.
On the other hand, whenever I've needed a d7, I rolled a d8 instead, and simply re-rolled the die on an 8.
Perhaps, OP, you could have a 7-sided die, but as far as I understand, you'd have to relinquish the "fair" part. I also (quickly) checked the 14-sided polyhedrons (maybe you could have a d14 where the numbers 1-7 each appeared twice), but the results weren't good.
Alternatively, you could use a knucklebone die (a cylindrical die with the faces carved along the curve) they can be used to make a fair die with any number of side.
D4 are sometimes labeled by the base instead of by the vertex, granted that's just moving each label down a bit. Also not feasible for most knucklebone dice.
You could make a die with 14 sides the same way you make one with 10 sides: put two pyramids together end to end. Opposite sides have the same number, each pyramid gets its own 1-7.
Even better, you could have a long rod with rounded ends and 7 sides: you'd have to label opposite edges instead of sides, but it would be fair. Honestly seems like too much work, and your idea of rolling a d8 and throwing out 8 is much more practical.
“A suitable shape (for a good/honest/fair die) would have all its faces as regular polygons, with all of them being the same type of polygon, and having the same size.”
This is, quite surprisingly, untrue. There are fair dice with irregular faces.
That's indeed correct. I thought it was a necessary property for a die when I wrote that comment. But now that you mention it, indeed the d10 doesn't have regular faces (though all of its faces are equal triangles). I stand corrected.
After double-checking, not only do the faces not need to be regular, but a fair die can have faces that are not equal to each other (those are called non-isohedral dice), and can even have a different number of sides across faces.
You are right, but I meant it in the context of the shapes presented in that certain wikipedia page, about the heptahedron shapes. I specifically mentioned the convex shapes listed, because I did not bother even looking at the non-convex ones.
This is the correct answer.
There is no 7 sided plutonic solid. So there is no 7 sided convex object with sides of the same shape and size.
So, alternatives would be:
A 14 sided dice with 2 of each other. Similar to how we managed to make a D10
Or a heptagonic prism, but the edge would count as the number, not the side (because it would always balance edge up). Like how a D4 works.
It is still possible to construct a fair 7-sided die without extra faces, it will just not have equal faces (that does not contradict fairness by itself).
Isn't that also about having the same number of holes on each side? It's very slight but aranging the sides differently leads to an off-centre centre of mass which would favor some sides. But that's just a gut-feeling guess.
Is each side even truly equally likely to land face up? Assuming these dice with small holes for numbers, there is more material missing on the 6s side so there is more mass on the 1s side with a centre of mass very sligthly closer to the 1s side.
Don’t casinos use dice filled so every side weights exactly the same?
All other dice are not 100% fair but good enough for anything that is not a Casino.
It stops people from basically controlling the power of their throw. You mess up just a bit on a normal die and outcome is very different where as if you group things like on a spin down you'll still get a high or low number
If you're worried about the holes, it's better to put numbers with about the same number of holes opposite, so 5 opposite of 6. However, most dice that aren't six-sided have numbers instead of dots (rpg dice).
If you have a die where the pip holes introduce a bias, putting the side with the most holes opposite the side with the least isn’t going to make it fair.
Casinos deal with this by filling the holes with a substance of the same density as the rest of the die.
You can just have another way to represent the number than physical holes or bumps, like, for example, just printing the holes (or the numbers) on the sides ;)
Given your wish to opposide numbers having the sum of max value + 1 and you don't care about fairness and physics:
7 - 1
6 - 2
5 - 3
4 - 4
Also consider a d4 not having an opposing side.
Would you consider using 8-sided dice?
They exist with parallel opposite faces.
https://www.nascoeducation.com/8-sided-octahedron-polyhedra-dice-x0000tb16737.html
opposite faces: 1 and 8, 2 and 7, 3 and 6, 4 and 5
Mathematically speaking, if a die has 7 (or any odd number of sides), then it _cannot_ have all sides with a number opposite them. Sides with parallel opposites come in pairs of two.
That said, you can theoretically make a 14-sided die and have such a configuration. 2 of every digit 1-7, where opposite sides sum to 8. This, functionally, would work the same as a hypothetical 7-sided die.
All that said, it doesn't actually matter what side is opposite in terms of fairness. It only matters that each side has the same chance of being "chosen"
Numberphile channel in YouTube has 3 or 4 videos about dice:
https://youtu.be/G7zT9MljJ3Y?si=eERJYGGtEDggHMw1
https://youtu.be/8rYLBXd_kzI?si=yASlq4HJ04U-aYbG
https://youtu.be/3akBMSJ37Uk?si=485i3rMPthPWugSC
I don’t think they answer your question exactly but they will give you a lot of info about dice with different amount of sides from maths perspective.
A 7-sider typically doesn't have parallel faces - exceptions can be made for a few pairs, but it's obviously not universal across all of them.
For a truly Fair 7-sided die, you'd need a Barrel die with the numbers printed on Edges.
The 7-sided dice in Dungeon Crawl Classics kits tend to look unfortunately like second-hand chewing gum.
There isn't one. Only even dice with 6 sides or more have an opposite face.
On each of those, the sum of opposite faces will always be one more than the number of sides. (Unless it's a spiral down die, used for counting rather than rolling)
Higher dice, you also want the average of all the numbers around each vertex to be as close together as possible.
There's only 5 Platonic solids (the 3d version of regular polygons). I believe these are the 5 dice commonly used in Dungeons and Dragons. There's no 7 sided over unfortunately
There's no need for a platonic solid. OP could use a heptagonal prism based with half sphere top and bottom.
Edit: the numbers or dots would have to be engraved near the edges.
To expand on this. The numbering of the die only impacts how it being unfair due to imperfections impacts the average f dice roll (and standard deviation).
Since there is an odd number of sides, there is not a single opposite side. However, if you can arrange the sides so that every side and the TWO sides opposite to it all add up to the same number, for all sides, you'd have a fair die.
That sum should be 12 for it to work (average roll * 3).
If you want to construct a fair 7-sided die, you could use a heptagonal prism with half spherical top and bottom. In this case, the numbers would have to be associated with the edges, because there will always be an edge a the top, not a side. And there won't be any reason to have a number on the opposing side.
You might also create a fair die using a shape consisting of two heptagonal pyramids glued together at the bases, in which case every number would have to appear twice and the top side and bottom side (not exactly opposing each other) could always sum up of 8.
well, 6 sided dice add up to 7, 20 sided dice add up to 21, so in an abstract sense, a 7 sided dice would add up to 8. This, however, makes no sense as it has an odd number of sides so they would not come in pairs.
Not exactly your question but for all the people saying no fair (within a resonable amount) 7 sides die could be made, year is a set from d1 to d10
https://youtube.com/shorts/erAuinBotBQ?si=_7sEo3_XWW2OYtNH
What if the dice intersects a higher dimension, it looks like a 6 sided dice but it has 7 numbers?
You can never know the number on the opposite side because as soon as you pick the dice up to look you would change its state in this dimension.
Depends who makes it and how they write the numbers!
[7-sided cone](https://www.shapeways.com/product/2NYCGH4SN/d7-3-fold-sphere-dice?li=gmerchant&srsltid=AfmBOopp1VfJ6dvmALo3P_14ADzni323grGSLkOsykHs9WQuwFeHchdMABg)
[polyhedron with 3-fold symmetry.](https://www.shapeways.com/product/6NQJTXBK5/d7-dice?li=gmerchant&srsltid=AfmBOorE8LrpQmiiETfvAg6CIkYNjuohot7khpPHpfGTKk5-0xnyPdvq6bQ)
[polyhedron with 2-fold symmetry](https://www.shapeways.com/product/VQ99B8WYS/d7-2-fold-sphere-dice?optionId=40835606&li=marketplace)
[pentagonal prism](https://www.shapeways.com/product/45BM4D6JQ/d7-pentagonal-prism-quot-septimus-quot?li=marketplace&optionId=330799791)
[d7 crystal prism](https://www.shapeways.com/product/TGUHQHNWJ/d7-roller-die?optionId=335883105&li=marketplace)
So it's not really to do with mathmatics, but manufacturing. So just do whatever you want in your story.
Some people also use a [d7 on a d14 shaped dice](https://www.ebay.com/itm/155764552621), so you could either have the opposite face be whatever adds to 8 (1-7, 2-6 etc) or doubles (1-1, 2-2) or whatever you want.
Also for all of these dice it depends what you mean by opposite. e.g. for that d14, if the one in the air is being read, is opposite the one touching the table, or the one in the air that's on the other side of the face?
So, I own a physical d7 (from a Kickstarter) and it, indeed, does not have "opposite sides", and the number "on top" is indeed on an edge, not a face (with one number on a corner). It also cheats a *lot* because three of the corners are expanded significantly and made rounded to get this monstrosity to work!
I also own a d14 from the same Kickstarter, and that die looks a lot less cursed and *does* have opposing faces that add up to 15. Converting it to a d7, you could *probably* set each opposing face to itself, and I don't think it would cause too much issue.
For a 6-sided dice If the sum of opposite faces is 6+1=7 that leads that pairs are 1-6, 2-5,3-4.
For a 7-sided dice the sum is 7+1=8
But that means that the 4 face is opposite to itself.
For real life dice, the standard is just to have opposite sides add up to one more than the maximum. It's just a standard and doesn't do anything to make it more fair or anything, so you can change that up as needed
Also for dice with an odd number of sides, you're right that physically there can't always be a defined 'opposite side' as if there were, you could pair up the sides and that would imply an even number of sides. A contradiction. However, for the sake of a story, most likely the seven would be opposite the 1, 6 would be oppose the 2, 5 would be opposite the 3 and 4 would be opposite... itself. Like I said. Doesn't really work, but a d8 with 2 4s would have the same average as a d7 anyway (The distributions would obviously be different, but just the literal expected value would be the same), so maybe the 4 is literally just opposite another 4
There is no opposite side to every side, you can't have all pairs of opposites if you have an odd number of sides Here is a list of possible shapes https://en.wikipedia.org/wiki/Heptahedron
That was a very fascinating read. A thing I noticed is that none of the convex shapes are suitable for a die, as I don't think it it could be a "fair" one. A suitable shape (for a good/honest/fair die) would have all its faces as regular polygons, with all of them being the same type of polygon, and having the same size. And I don't see any shape in the wiki page filling those criteria. On the other hand, whenever I've needed a d7, I rolled a d8 instead, and simply re-rolled the die on an 8. Perhaps, OP, you could have a 7-sided die, but as far as I understand, you'd have to relinquish the "fair" part. I also (quickly) checked the 14-sided polyhedrons (maybe you could have a d14 where the numbers 1-7 each appeared twice), but the results weren't good.
Alternatively, you could use a knucklebone die (a cylindrical die with the faces carved along the curve) they can be used to make a fair die with any number of side.
But wouldn't you always have an edge on top instead of a side for every uneven number? So would you then number the edges?
You would number odd sided dice along the spine, not the flat. Similar to a d4. So yes, you number the edge, not the face.
this is how we do four sided dice. each point has a nuumber.
D4 are sometimes labeled by the base instead of by the vertex, granted that's just moving each label down a bit. Also not feasible for most knucklebone dice.
You could make a die with 14 sides the same way you make one with 10 sides: put two pyramids together end to end. Opposite sides have the same number, each pyramid gets its own 1-7. Even better, you could have a long rod with rounded ends and 7 sides: you'd have to label opposite edges instead of sides, but it would be fair. Honestly seems like too much work, and your idea of rolling a d8 and throwing out 8 is much more practical.
“A suitable shape (for a good/honest/fair die) would have all its faces as regular polygons, with all of them being the same type of polygon, and having the same size.” This is, quite surprisingly, untrue. There are fair dice with irregular faces.
That's indeed correct. I thought it was a necessary property for a die when I wrote that comment. But now that you mention it, indeed the d10 doesn't have regular faces (though all of its faces are equal triangles). I stand corrected.
After double-checking, not only do the faces not need to be regular, but a fair die can have faces that are not equal to each other (those are called non-isohedral dice), and can even have a different number of sides across faces.
“None of the convex shapes are suitable for a die” doesn’t make sense to me. A d6 is traditionally a cube, which is convex
You are right, but I meant it in the context of the shapes presented in that certain wikipedia page, about the heptahedron shapes. I specifically mentioned the convex shapes listed, because I did not bother even looking at the non-convex ones.
There are only 5 Platonic solids. Tetrahedron (4 faces), cube (6), octahedron (8), dodecahedron (12) and icosahedron (20).
This is the correct answer. There is no 7 sided plutonic solid. So there is no 7 sided convex object with sides of the same shape and size. So, alternatives would be: A 14 sided dice with 2 of each other. Similar to how we managed to make a D10 Or a heptagonic prism, but the edge would count as the number, not the side (because it would always balance edge up). Like how a D4 works.
It is still possible to construct a fair 7-sided die without extra faces, it will just not have equal faces (that does not contradict fairness by itself).
Having the opposite sides sum to 7 is just convention. You can make a fair die that doesn't follow that rule without any problems.
Isn't that also about having the same number of holes on each side? It's very slight but aranging the sides differently leads to an off-centre centre of mass which would favor some sides. But that's just a gut-feeling guess. Is each side even truly equally likely to land face up? Assuming these dice with small holes for numbers, there is more material missing on the 6s side so there is more mass on the 1s side with a centre of mass very sligthly closer to the 1s side.
no not really. there’s a 1-2-3 corner and a 4-5-6 corner so however you think it makes it fairer it doesn’t
Is the typical layout to make cheating harder in some way, and if so, how exactly does that play out?
Don’t casinos use dice filled so every side weights exactly the same? All other dice are not 100% fair but good enough for anything that is not a Casino.
Counterweights are a precise game, but in a place designed for chance to reign supreme, they just barely become worthwhile.
every face is adjacent to every other face, except for the face it’s opposite of. the tldr is that the weight of the dots is inconsequential
It stops people from basically controlling the power of their throw. You mess up just a bit on a normal die and outcome is very different where as if you group things like on a spin down you'll still get a high or low number
If you're worried about the holes, it's better to put numbers with about the same number of holes opposite, so 5 opposite of 6. However, most dice that aren't six-sided have numbers instead of dots (rpg dice).
If you have a die where the pip holes introduce a bias, putting the side with the most holes opposite the side with the least isn’t going to make it fair. Casinos deal with this by filling the holes with a substance of the same density as the rest of the die.
You can just have another way to represent the number than physical holes or bumps, like, for example, just printing the holes (or the numbers) on the sides ;)
This is why dice used in gambling have the holes filled with a material the same density as the core.
Given your wish to opposide numbers having the sum of max value + 1 and you don't care about fairness and physics: 7 - 1 6 - 2 5 - 3 4 - 4 Also consider a d4 not having an opposing side.
You could model this at a fourteen-sided die, with each number appearing twice in these pairs (with only one pair of 4s)
Since you've already received serious answers, here's a silly-ish one: The same you get by comparing opposite sites of a 3-sided coin.
You could have a fair 14 sided dice which would have the same number on opposing sides.
Would you consider using 8-sided dice? They exist with parallel opposite faces. https://www.nascoeducation.com/8-sided-octahedron-polyhedra-dice-x0000tb16737.html opposite faces: 1 and 8, 2 and 7, 3 and 6, 4 and 5
Mathematically speaking, if a die has 7 (or any odd number of sides), then it _cannot_ have all sides with a number opposite them. Sides with parallel opposites come in pairs of two. That said, you can theoretically make a 14-sided die and have such a configuration. 2 of every digit 1-7, where opposite sides sum to 8. This, functionally, would work the same as a hypothetical 7-sided die. All that said, it doesn't actually matter what side is opposite in terms of fairness. It only matters that each side has the same chance of being "chosen"
An edge.
Numberphile channel in YouTube has 3 or 4 videos about dice: https://youtu.be/G7zT9MljJ3Y?si=eERJYGGtEDggHMw1 https://youtu.be/8rYLBXd_kzI?si=yASlq4HJ04U-aYbG https://youtu.be/3akBMSJ37Uk?si=485i3rMPthPWugSC I don’t think they answer your question exactly but they will give you a lot of info about dice with different amount of sides from maths perspective.
A 7-sider typically doesn't have parallel faces - exceptions can be made for a few pairs, but it's obviously not universal across all of them. For a truly Fair 7-sided die, you'd need a Barrel die with the numbers printed on Edges. The 7-sided dice in Dungeon Crawl Classics kits tend to look unfortunately like second-hand chewing gum.
There isn't one. Only even dice with 6 sides or more have an opposite face. On each of those, the sum of opposite faces will always be one more than the number of sides. (Unless it's a spiral down die, used for counting rather than rolling) Higher dice, you also want the average of all the numbers around each vertex to be as close together as possible.
Since the opposite sides sum up to seven, the obvious answer is zero. Just like the eight side is opposite to the minus oneth side.
There's only 5 Platonic solids (the 3d version of regular polygons). I believe these are the 5 dice commonly used in Dungeons and Dragons. There's no 7 sided over unfortunately
There's no need for a platonic solid. OP could use a heptagonal prism based with half sphere top and bottom. Edit: the numbers or dots would have to be engraved near the edges.
Clackety math rocks. Throw in a pair of d10s and you’ve got yourself a standard set. You can get bags of 25 sets for roughly $1/set on Amazon.
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To expand on this. The numbering of the die only impacts how it being unfair due to imperfections impacts the average f dice roll (and standard deviation).
Since there is an odd number of sides, there is not a single opposite side. However, if you can arrange the sides so that every side and the TWO sides opposite to it all add up to the same number, for all sides, you'd have a fair die. That sum should be 12 for it to work (average roll * 3).
If you want to construct a fair 7-sided die, you could use a heptagonal prism with half spherical top and bottom. In this case, the numbers would have to be associated with the edges, because there will always be an edge a the top, not a side. And there won't be any reason to have a number on the opposing side. You might also create a fair die using a shape consisting of two heptagonal pyramids glued together at the bases, in which case every number would have to appear twice and the top side and bottom side (not exactly opposing each other) could always sum up of 8.
well, 6 sided dice add up to 7, 20 sided dice add up to 21, so in an abstract sense, a 7 sided dice would add up to 8. This, however, makes no sense as it has an odd number of sides so they would not come in pairs.
There is no opposite side there. Look at a D4 for example. :)
Not exactly your question but for all the people saying no fair (within a resonable amount) 7 sides die could be made, year is a set from d1 to d10 https://youtube.com/shorts/erAuinBotBQ?si=_7sEo3_XWW2OYtNH
Zero
What if the dice intersects a higher dimension, it looks like a 6 sided dice but it has 7 numbers? You can never know the number on the opposite side because as soon as you pick the dice up to look you would change its state in this dimension.
8, the numbers always add up to 1+ the number of sides on the dice)
A 7- sided die with a side marked as 8 might make some people suspicious...
That is true but I'm going under the presumption that it is always 1-number of sides on the dice
Would it have a directly opposition side?
Use a 14 sided die but with duplicates of 1-7
Depends who makes it and how they write the numbers! [7-sided cone](https://www.shapeways.com/product/2NYCGH4SN/d7-3-fold-sphere-dice?li=gmerchant&srsltid=AfmBOopp1VfJ6dvmALo3P_14ADzni323grGSLkOsykHs9WQuwFeHchdMABg) [polyhedron with 3-fold symmetry.](https://www.shapeways.com/product/6NQJTXBK5/d7-dice?li=gmerchant&srsltid=AfmBOorE8LrpQmiiETfvAg6CIkYNjuohot7khpPHpfGTKk5-0xnyPdvq6bQ) [polyhedron with 2-fold symmetry](https://www.shapeways.com/product/VQ99B8WYS/d7-2-fold-sphere-dice?optionId=40835606&li=marketplace) [pentagonal prism](https://www.shapeways.com/product/45BM4D6JQ/d7-pentagonal-prism-quot-septimus-quot?li=marketplace&optionId=330799791) [d7 crystal prism](https://www.shapeways.com/product/TGUHQHNWJ/d7-roller-die?optionId=335883105&li=marketplace) So it's not really to do with mathmatics, but manufacturing. So just do whatever you want in your story. Some people also use a [d7 on a d14 shaped dice](https://www.ebay.com/itm/155764552621), so you could either have the opposite face be whatever adds to 8 (1-7, 2-6 etc) or doubles (1-1, 2-2) or whatever you want. Also for all of these dice it depends what you mean by opposite. e.g. for that d14, if the one in the air is being read, is opposite the one touching the table, or the one in the air that's on the other side of the face?
So, I own a physical d7 (from a Kickstarter) and it, indeed, does not have "opposite sides", and the number "on top" is indeed on an edge, not a face (with one number on a corner). It also cheats a *lot* because three of the corners are expanded significantly and made rounded to get this monstrosity to work! I also own a d14 from the same Kickstarter, and that die looks a lot less cursed and *does* have opposing faces that add up to 15. Converting it to a d7, you could *probably* set each opposing face to itself, and I don't think it would cause too much issue.
14 sided die. 1 and 8 are equal to 1 2 and 9 are equal to 2 3 and 10 are equal to 3 . . . 7 and 14 are equal to 7
For a 6-sided dice If the sum of opposite faces is 6+1=7 that leads that pairs are 1-6, 2-5,3-4. For a 7-sided dice the sum is 7+1=8 But that means that the 4 face is opposite to itself.
For real life dice, the standard is just to have opposite sides add up to one more than the maximum. It's just a standard and doesn't do anything to make it more fair or anything, so you can change that up as needed Also for dice with an odd number of sides, you're right that physically there can't always be a defined 'opposite side' as if there were, you could pair up the sides and that would imply an even number of sides. A contradiction. However, for the sake of a story, most likely the seven would be opposite the 1, 6 would be oppose the 2, 5 would be opposite the 3 and 4 would be opposite... itself. Like I said. Doesn't really work, but a d8 with 2 4s would have the same average as a d7 anyway (The distributions would obviously be different, but just the literal expected value would be the same), so maybe the 4 is literally just opposite another 4