Two more cheats for a d7

[jack]

1. No-one says rolling dice has to result in a *face*. It's obviously more convenient, but if you roll your dice in a V-shaped valley, you'll end up with them landing on an edge. And you could get a corner by using a conical well. (Though you need to shake it to make sure it doesn't get stuck.) Then any polyhedron with 7n equivalent faces, or edges, or vertices would do the trick. Unfortunately, I don't think it helps, I think you need 7N faces or regular septagons to get 7N edges or vertices, which you can't have.

2. You can have it in six dimensions, though. A 6-dimensional tetrahedron has seven 5-d-tetrahedral faces.

Comments

[bugshaw.livejournal.com]

lucian_albrecht points me at another variant on d7s.

[pseudomonas.livejournal.com]

I would need quite some convincing that those were equiprobable.

[cartesiandaemon.livejournal.com]

I don't think they can be. Or, at least, according to the guy I linked to who did a lot of thinking about it, when well-made were good enough for all practical purposes, and as good as many d6s.

But I don't think they can be actually be theoretically fair. I don't have a rigorous proof, but the argument in previous posts, based on the other guy I linked to, went something like this. Have a matrix of the probabilities of going from one side to another. The values depend on the friction and restitution of the surface. So changing the surface will change the probabilities of different transitions by different amounts, so an asymmetric die fair for one surface can't be fair for another.

That's not rigorous, we haven't proved you can't cancel out the changes, but it doesn't seem likely.

[alextfish.livejournal.com]

Surely the thing to do is just make a d14 by analogy with a d10, and just have two copies of each digit in [1,7]?

[cartesiandaemon.livejournal.com]

Doh. For some reason, I'd actually assumed d10s were cheats, no, I see they're perfectly fair (they're not platonic solids, but the shape is one of the valid face-symmetric results for a fair die).

However, I'm sure anything where the symmetry involves just one axis of rotation is a cheat, though inconveniently can't formalise that constraint. It works for seven, but it bugs me because it works for a 100 too, but isn't very elegant for that...