The Lonely Runner Conjecture
<p>Imagine 8 runners on a circular track of 1 unit length all running at different constant speeds and all starting from the same place. Will each runner be at least 1/8 units of the track away from every other runner at some point in time?</p>
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<p>Will each runner eventually become lonely?</p>
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<p>Believe it or not but this is an unsolved problem in mathematics! In fact, it is not just unsolved for the number 8 but for all numbers k ≥ 8 where k is the number of runners.</p>
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<p>The lonely runner conjecture is one of the famous mysteries of mathematics. So easy to understand yet so hard to grasp in its entirety.</p>
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<p>In this article, we will explore this problem and provide some animation code in Python so that the reader can play around with it using custom values.</p>
<h1>Introduction</h1>
<p>When there are k runners on the track, we define a runner as lonely if he or she is at least 1/k distance (of the circular track) away from any other runner. The speeds of the runners are considered unknown variables and the conjecture says that no matter which speeds the runners have, all of the k runners will eventually be lonely.</p>
<p>The speeds are allowed to be any real numbers including irrational (even negative velocities are allowed), however, as will soon be clear, one can make several simplifications to this broad setup.</p>
<p>One of the reasons this problem is so interesting is that one can view it through several different mathematical lenses. We can see it as a problem in geometry, number theory, graph theory, and a field called Diophantine approximation theory.</p>
<p>It has been proved to be true in cases 2 ≤ k ≤ 7.</p>
<p>The following GIF shows a possible scenario with 6 runners in which the dots are colored yellow when they become lonely.</p>
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