A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See <a href="http://en.wikipedia.org/wiki/Nontransitive_dice"> wikipedia</a>)

For some sets, the deviation from transitivity is small in the sense that A beats B beats C beats A with probabilities $p_{ij}$ only slightly greater than $0.5$ . <a href="http://en.wikipedia.org/wiki/Nontransitive_dice#Efron.27s_dice"> Efron's dice</a> (there are 4 of them) beat each other nontransitively with probability $2/3$. 

Can we make a strictly better set of $4$ six-sided dice? That is, a set of 4 six-sided dice such that they beat each other nontransitively with all probabilities $> 2/3$ ?

Can we make a strictly better set of $4$ $n$-sided dice for some small $n$ which one can conveniently make a die out of, e.g. $n = 4, 8, 12, 20 $ ?

Can we make a strictly better set of <b>$5$</b> $n$-sided dice for some small $n$ which one can conveniently make a die out of, e.g. $n = 4, 6, 8, 12, 20 $ ?

Can we make a strictly better set of $3$, $4$ or $5$ dice, each having a potentially  different number of sides ($4, 6, 8, 12$ or $20$) ?

Ideally I would like to find a fairly small set of fairly easy-to-make, preferably platonic-solid dice which beat each other nontransitively with probabilities > 80%. They would make an excellent teaching aid and magic trick. There is an <a href="http://math.stackexchange.com/questions/260072/what-is-the-most-unfair-set-of-three-nontransitive-dice">answer</a> on math.stackexchange which claims that the best you can do with 3 dice is $p = 0.58$, which is disappointingly close to $0.5$; for a teaching aid you need to be able to beat students almost every time for them to spot the pattern quickly. Efron's dice are substantially better at $2/3$, but is that really the best we can do? ([Crossposted][1] from math.stackexchange)

EDIT: I missed <a href="http://math.stackexchange.com/questions/57338/how-far-can-probability-intransitivity-be-stretched?rq=1">this answer</a> which argues that the probability cannot be > than 0.75 irrespective of the details of the dice. Still, it would be nice to know what the "simplest" set of "simple" dice is that gets you above, say, 70%, 72%, etc. 

  [1]: http://math.stackexchange.com/questions/286193/what-is-the-most-extreme-set-4-or-5-nontransitive-n-sided-dice