The microtubule cytoskeleton is a dynamic structure in which the lengths of the microtubules are tightly regulated. One regulatory mechanism is the depolymerization of microtubules by motor proteins in the kinesin-13 family ¹. These proteins are crucial for the control of microtubule length in cell division^2-4, neuronal development⁵ and interphase microtubule dynamics^6,7. The mechanism by which kinesin-13 proteins depolymerize microtubules is poorly understood. A central question is how these proteins target to microtubule ends at rates exceeding those of standard enzyme-substrate kinetics⁸. To address this question we developed a single-molecule microscopy assay for MCAK, the founding member of the kinesin-13 family⁹. Here we show that MCAK moves along the microtubule lattice in a one-dimensional (1D) random walk. MCAK-microtubule interactions were transient: the average MCAK molecule diffused for 0.83 s with a diffusion coefficient of 0.38 μm² s^-1. Although the catalytic depolymerization by MCAK requires the hydrolysis of ATP, we found that the diffusion did not. The transient transition from three-dimensional diffusion to 1D diffusion corresponds to a "reduction in dimensionality"10 that has been proposed as the search strategy by which DNA enzymes find specific binding sites¹¹. We show that MCAK uses this strategy to target to both microtubule ends more rapidly than direct binding from solution.