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It is well-known that the convergence time of a Markov chain can be decreased by adding a few additional (randomly chosen) transitions. In a related phenomenon, Diaconis and Chatterjee (2021) showed that, for many chains, alternating stochastic moves with a bijection satisfying suitable conditions can lead to fast mixing. These bijections are known to exist in many cases, yet finding explicit examples of these bijections, even for simple chains, can be challenging. We discuss these results and consider a simple explicit bijection for the $d$-hypercube, showing the resulting chain mixes in $d$ steps with a cut-off. Joint work with Chandan Tankala.