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Recent demonstrations of unexcised black holes traversing acrosscomputational grids represent a significant advance in numerical relativity.Stable and accurate simulations of multiple orbits, and their radiated waves,result. This capability is critically undergirded by a careful choice of gauge.Here we present analytic considerations which suggest certain gauge choices,and numerically demonstrate their efficacy in evolving a single moving punctureblack hole.Comment: 8 pages, 13 figures. Edited slightly for clarity as requested by the  referee and editor. Also corrected errors due to a misplaced factor of alpha.  Main results unchange

How to move a black hole without excision: Gauge conditions for the numerical evolution of a moving puncture