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We identify a tradeoff curve between the number of wheels on a train car, andthe amount of track that must be installed in order to ensure that the traincar is supported by the track at all times. The goal is to build an elevatedtrack that covers some large distance $\ell$, but that consists primarily ofgaps, so that the total amount of feet of train track that is actuallyinstalled is only a small fraction of $\ell$. In order so that the train trackcan support the train at all points, the requirement is that as the traindrives across the track, at least one set of wheels from the rear quarter andat least one set of wheels from the front quarter of the train must be touchingthe track at all times.  We show that, if a train car has $n$ sets of wheels evenly spaced apart inits rear and $n$ sets of wheels evenly spaced apart in its front, then it ispossible to build a train track that supports the train car but uses only$\Theta( \ell / n )$ feet of track. We then consider what happens if the wheelson the train car are not evenly spaced (and may even be configuredadversarially). We show that for any configuration of the train car, with $n$wheels in each of the front and rear quarters of the car, it is possible tobuild a track that supports the car for distance $\ell$ and uses only$O\left(\frac{\ell \log n}{n}\right)$ feet of track. Additionally, we show thatthere exist configurations of the train car for which this tradeoff curve isasymptotically optimal. Both the upper and lower bounds are achieved viaapplications of the probabilistic method.

Train Tracks with Gaps: Applying the Probabilistic Method to Trains