## The Lonely Runner ConjectureAugust 11, 2011 When I was a college freshman, my dorm mates talked me into being the captain of our intramural track team. It wasn't because I was fleet of foot. In fact, they said I wouldn't need to run at all. I would just act as a records keeper. Perhaps they admired my skill at holding a clipboard. What they didn't tell me was that when there was an absent runner, the captain needed to run the race. That happened once, I finished dead last, but it did teach me a valuable lesson. When you're volunteering for something, make certain you know what you're volunteering for. There's an interesting mathematics problem called the lonely runner conjecture. It involves runners who start a race together on a circular track, they all run at different rates, and they're indefatigable; that is, they don't just run till they drop, they run at their constant speed and never drop. The conjecture is that there will be times at which any given runner will be at a distance of at least a fraction (1/n) of the track length from any other runner. There is one caveat, however. The runners' rates must be pairwise distinct; that is, all rates are different. As typical, the easiest stated conjectures turn out to be hard to prove. Proofs are known up to seven runners, and the seven runner proof was published in 2008.[1-3] This problem is quite easy to simulate, so I wrote a simulation program, which can be found here. For six runners, the following graph shows the percentage separation from the closest runner for one particular runner over 10,000 steps. The program assigns random running rates to the runners. The fastest runner in this case made a complete circuit in 109 steps, and the slowest runner completed the circuit in 304 steps. A peak separation of 37.9% occurs at 9,000 steps.Separation from the closest runner for a particular runner over 10,000 steps in a simulation of the lonely runner conjecture. There's a peak separation of 37.9% at 9,000 steps. (Graph via Gnumeric).
The schematic of the runners' track in the figure below shows the position of the runners at 9,000 steps. There are two "cozy" runners who are separated by just 0.37% of the track length, and the one lonely runner who's 37.9% distant from the nearest runner.
## References:- Lonely runner conjecture, Open Problem Garden Web Site.
- J. Barajas and O. Serra, "The lonely runner with seven runners," arXiv Preprint Server, October 24, 2007.
- J. Barajas and O. Serra, "The lonely runner problem with seven runners," The Electronic Journal of Combinatorics, vol. 15, no. R48, pp. 111-116.
- Florian Cajori, "A History of Mathematics," MacMillan Company, (New York. 1919), Second Edition, p. 59f. (via Scribd),
- Florian Cajori, A History of Mathematics, MacMillan Company, (New York, 1894) , p. 73f. (via Google Books).
- Archimedes Cattle Problem, Greek text, New York University.
- Sebastian Czerwinski, "Random runners are very lonely," arXiv Preprint Server, February 22, 2011.
- C. Harold Horvat and Matthew Stoffregen, "A Solution to the Lonely Runner Conjecture for Almost All Points," arXiv Preprint Server, March 8, 2011.
Linked Keywords: College freshman; dorm; intramural; track team; clipboard; mathematics; lonely runner conjecture; conjecture; mathematical proof; computer simulation; Diophantine approximation; Diophantine equations; piebald; Archimedes cattle problem; Archimedes; Eratosthenes; Greek language; cattle; Helios; equation; Florian Cajori; arXiv preprint server; Fourier analysis; Googler; A History of Mathematics. |
RSS Feed
## Google Search
Latest Books by Dev Gualtieri
- Voice Synthesis - June 26, 2017
- Refining Germanium - June 22, 2017
- Granular Capillarity - June 19, 2017
- Kirchhoff–Plateau Problem - June 15, 2017
- Self-Assembly - June 12, 2017
- Physics, Math, and Sociology - June 8, 2017
- Graphene from Ethylene - June 5, 2017
- Crystal Alignment Forces - June 1, 2017
- Martian Brickwork - May 29, 2017
- Carbon Nanotube Textile - May 25, 2017
- The Scent of Books - May 22, 2017
- Patterns from Randomness - May 18, 2017
- Terpene - May 15, 2017
- The Physics of Inequality - May 11, 2017
- Asteroid 2015 BZ509 - May 8, 2017
- Fuzzy Fibers - May 4, 2017
- The Sofa Problem - May 1, 2017
- The Wisdom of Composite Crowds - April 27, 2017
- J. Robert Oppenheimer and Black Holes - April 24, 2017
- Modeling Leaf Mass - April 20, 2017
- Easter, Chicks and Eggs - April 13, 2017
- You, Robot - April 10, 2017
- Collisions - April 6, 2017
- Eugene Garfield (1925-2017) - April 3, 2017
- Old Fossils - March 30, 2017
- Levitation - March 27, 2017
- Soybean Graphene - March 23, 2017
- Income Inequality and Geometrical Frustration - March 20, 2017
- Wireless Power - March 16, 2017
- Trilobite Sex - March 13, 2017
- Freezing, Outside-In - March 9, 2017
- Ammonia Synthesis - March 6, 2017
- High Altitude Radiation - March 2, 2017
- C.N. Yang - February 27, 2017
- VOC Detection with Nanocrystals - February 23, 2017
- Molecular Fountains - February 20, 2017
- Jet Lag - February 16, 2017
- Highly Flexible Conductors - February 13, 2017
- Graphene Friction - February 9, 2017
- Dynamic Range - February 6, 2017
- Robert Boyle's To-Do List for Science - February 2, 2017
- Nanowire Ink - January 30, 2017
- Random Triangles - January 26, 2017
- Torricelli's law - January 23, 2017
- Magnetic Memory - January 19, 2017
- Graphene Putty - January 16, 2017
- Seahorse Genome - January 12, 2017
- Infinite c - January 9, 2017
- 150 Years of Transatlantic Telegraphy - January 5, 2017
- Cold Work on the Nanoscale - January 2, 2017
- Holidays 2016 - December 22, 2016
- Ballistics - December 19, 2016
- Salted Frogs - December 15, 2016
- Negative Thermal Expansion - December 12, 2016
- Verbal Cues and Stereotypes - December 8, 2016
- Capacitance Sensing - December 5, 2016
- Gallium Nitride Tribology - December 1, 2016
- Lunar Origin - November 27, 2016
- Pumpkin Propagation - November 24, 2016
- Math Anxiety - November 21, 2016
- Borophene - November 17, 2016
- Forced Innovation - November 14, 2016
- Combating Glare - November 10, 2016
- Solar Tilt and Planet Nine - November 7, 2016
- The Proton Size Problem - November 3, 2016
### Deep ArchiveDeep Archive 2006-2008
Blog Article Directory on a Single Page |

Copyright © 2017 Tikalon LLC, All Rights Reserved.

Last Update: 06-26-2017