  The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either
the $n$th card or the $(n-k)$th card to the top of the deck, with probability
half each. We determine the spectral gap for the location of a single card,
which, as a function of $k$ and $n$, has surprising behavior. For example,
suppose $k$ is the closest integer to $\alpha n$ for a fixed real
$\alpha\in(0,1)$. Then for rational $\alpha$ the spectral gap is
$\Theta(n^{-2})$, while for poorly approximable irrational numbers $\alpha$,
such as the reciprocal of the golden ratio, the spectral gap is
$\Theta(n^{-3/2})$.

Probability