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We study an example of a {\em hit-and-run} random walk on the symmetric group$\mathbf S_n$. Our starting point is the well understood {\em top-to-random}shuffle. In the hit-and-run version, at each {\em single step}, after pickingthe point of insertion, $j$, uniformly at random in $\{1,\dots,n\}$, the topcard is inserted in the $j$-th position $k$ times in a row where $k$ is uniformin $\{0,1,\dots,j-1\}$. The question is, does this accelerate mixingsignificantly or not? We show that, in $L^2$ and sup-norm, this acceleratesmixing at most by a constant factor (independent of $n$). Analyzing thisproblem in total variation is an interesting open question.

The hit-and-run version of top-to-random