These are nontransitive coins such that each coin has the same pair of probabilities $p, 1-p$ for its two sides, which are labelled from the set $\{1,2,3,4,5,6\}$:
Side with probability Coin p 1-p ---- --- --- X 2 6 Y 3 4 Z 5 1
Because $P(X < Y) = P(Y < Z) = p$ and $P(Z < X) = 1-p^2$, it follows from $p = 1-p^2$ that $p$ must be the "little" golden ratio $\frac{\sqrt{5} - 1}{2} \approx 0.6180$.
NB: This can be seen as an embellishment of the answer by Brian Scott, modified to use a set of consecutive positive integers, and to make the coins equally "biased". (It was merely coincidental that I happened to assign prime numbers to the sides with probability $p$ and non-primes to the opposite sides, and also that the golden ratio happens to be involved.)