  What is the smallest number of random transpositions (meaning that we swap
given pairs of elements with given probabilities) that we can make on an
$n$-point set to ensure that each element is uniformly distributed -- in the
sense that the probability that $i$ is mapped to $j$ is $1/n$ for all $i$ and
$j$? And what if we insist that each pair is uniformly distributed?
  In this paper we show that the minimum for the first problem is about
$\frac{1}{2} n \log_2 n$, with this being exact when $n$ is a power of $2$. For
the second problem, we show that, rather surprisingly, the answer is not
quadratic: $O(n \log^2 n)$ random transpositions suffice. We also show that if
we ask only that the pair $1,2$ is uniformly distributed then the answer is
$2n-3$. This proves a conjecture of Groenland, Johnston, Radcliffe and Scott.
