The set with 5 dice is also more insane (you can buy all this stuff from http://www.mathsgear.co.uk/), you can create a circle of dice that beat each other going clockwise. But if you then double up the dice (two red, two blue etc), the order switches to anti-clockwise. There's also a second (star-shaped) loop in case someone works out how the first loop works.
Otherwise -- cool subject. I know about non-transitive voting (it was fairly relevant to what I did in academia for a while). But I hadn't heard much before about the dice. ;)
The prescriptivists lost a long, long time ago: language is defined by usage.
You can be "correct" according to some prescriptivist definition of the english language, or you can be understood by the people who actually use the language to communicate. Since nearly everyone uses "dice" for both the singular and plural form, and most people will never even have heard "die" being used in this context, dice is the correct form.
A friend of mine and I sat down and defined what we believe to be the best set of 4 non-transitive dice. Our set has some great properties, like the numbers 1-24 each occur one time, each die has the same sum on its faces as the others, each die has the number 1, 2, 3, or 4 on it (so they're easy to identify.)
We manufactured about 10 sets of them, if I remember correctly.
I've wondered if we should Kickstart it. I can't imagine there'd be much interest.
EDIT: I found my old rendering of what they'd look like:
Try playing existing adversarial games with them. Instead of a single dice, each player gets a set of them and picks one of them before rolling, without showing it.
Risk gets much more fun that way, for example - because suddenly, even the individual battles are about imagining you can read the other players intentions.
> You've probably come here via the dice we handed out at one of our recruiting
> events. By now you've hopefully noticed what's special about the 4 dice you're
> looking at: "red beats green beats blue beats white beats red", each with 66.7%
> probability. They are known as Efron's dice, and you can find some more info on
> this Wikipedia page.
> Your mission, should you choose to accept it, is to generalise the situation.
> For example, can you improve on the 66.7%? What if the dice are replaced by
> arbitrary random variables? What is the best "mutual beating probability" if
>you have only three dice? Or more than four?
It's a fun challenge and I'd recommend giving it a go if you have an interest in maths.