Lonely Runner Conjecture

Justin Leung

As the world begins to open up following the pandemic, you decide it’s time for a new start. One of your resolutions is to pick up running in order to keep fit, but as you don’t want to be lonely you ask your friend Bob to run with you around the local running track. However, Bob says it doesn’t matter whether you run together or not because you both run at different speeds, and so both of you will feel lonely at least once at a certain point during the run. And furthermore, he says this can’t be fixed by inviting more runners. So, is Bob correct? Let’s do some maths and try to work this out…

Puzzle 1

(a). Let’s say you and Bob each run at a constant speed. It takes you 70 seconds and Bob 90 seconds to complete a lap. Given that one lap is 400 meters long, how fast are you and Bob running? [Remember that (distance) = (speed) x (time)]

(b). How long would it take for you to feel lonely? Where we define “lonely” to be when you and Bob are at exactly opposite sides of the track (see diagram below).

(c). Bob is having a good day and he’s running slightly quicker. Instead of 90 seconds, he can complete a lap in 85 seconds. How long will it take for the both of you to feel lonely now?

(d). It’s a rainy day, and you forgot your spikes. As a result, you can only complete a lap in 80 seconds while Bob can maintain his usual speed. How long does it take for the both of you to feel lonely?  

(e). It’s a cold day, and both of you are running 1 m/s slower than your usual speed. How long does it take for the both of you to feel lonely? How are the results compared to part a?

(f). Plot your results from parts (a)-(d) on a graph using the axes shown below. The x-axis is the reciprocal of the difference in speed (1/(v1-v2)) and the y-axis is the first time of loneliness. What pattern(s) do you see?

(g). How long would it take for both of you to feel lonely if you and Bob were running at general speeds v1 and v2 respectively (assuming v1 > v2)? 

*answers at the bottom of the page


Puzzle 2

You decide that you and Bob might not feel lonely if you invite a third friend (John) to run with you. John is super fit and can run a lap in 55 seconds. Now, in order to understand the problem, we use one of the runners as a reference point. This means setting one of the runner’s speeds to zero, and then expressing the speeds of the other runners relative to the one at zero speed. For example, if we set runner B’s speed to 0, (vB = 0), then runner A and runner C’s speed will be (vA-vB) and (vC-vB) respectively. This will help us to see more clearly when a runner is “lonely”.

Using the track as the reference pointUsing a runner as the reference point

(a). Using Bob as the reference point, when will Bob start being lonely from John? When will he stop being lonely from John? In this case, “lonely” is defined when the nearest runner is at least ⅓ of the track away from you (since we now have 3 runners).

(b). When will Bob be lonely again from John? How often do these episodes of “loneliness” occur?

(c). When will Bob be lonely from you? How often do these episodes of “loneliness” occur?

(d). Bob will only experience “loneliness” when he is “lonely” from the both of you at the same time. When will he be lonely? How periodic are his episodes of “loneliness” and how long do they last?

(e). When will John and yourself be “lonely”? Use a similar approach to parts (a)-(d).

*answers at the bottom of the page


Puzzle 3

Do you think Bob is right? In other words, do you think that there are a number of runners k or above where runners will not feel “lonely” anymore? 

Working through the puzzles above, you will have no doubt realised that the situation gets more complicated with more runners. When there are 2 runners, it’s quite simple: we know both runners will feel lonely at the same time, when they are at opposite sides of the track. With 3 runners, it is slightly more difficult: we need to figure out whether each and every runner feels lonely.

You also might be wondering why we described the runners as being “lonely” in the first place. And that is because the problems that you have worked through above are part of a puzzle called the Lonely Runner Conjecture:

Suppose k runners are running around a circular track that’s 1 unit long. All the runners start at the same point, but they run at different speeds. A runner is said to be “lonely” if they are at least a distance 1/k along the track from every other runner. The lonely runner conjecture states that each of the runners will be lonely at some point.



yellow dots = runners that have experienced loneliness

So far, the conjecture has been proven to be true up to k = 7 runners, but it remains unsolved for k ≥ 7. So, there’s not really a right answer for puzzle 3! If you think Bob is wrong, he could be, we simply don’t know at the moment. You could even conduct your own experiment by inviting 7 friends to run with you and see if each of you feel “lonely” at some point.

So, what may have seemed like a simple problem at first, quickly turned into an unsolved mathematical conjecture puzzling the greatest minds of our time. And who knows? You might be the person to find the answer and receive eternal mathematical glory!


Continue scrolling for answers to puzzle 1

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Puzzle 1 answers

(a). You = 5.71 m/s, Bob = 4.44 m/s
(b). 157.5 seconds
(c). 198.33 seconds
(d). 360 seconds
(e). 157.5 seconds
(f). A straight line going through the origin with gradient = 200
(g). t=200v1-v2, where v1 > v2

Continue scrolling for answers to puzzle 2

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Puzzle 2 answers

(a). Start = 47.14s, stop = 94.29s
(b). Bob will be lonely again from John at 188.57 seconds. These episodes of loneliness happen every 141.43 seconds.
(c). From 85.56 to 171.11 seconds, and it happens every 256.77 seconds.
(d). 85.56 seconds
(e). John = 141.43 seconds, You = 105 seconds.


References

https://demonstrations.wolfram.com/TheLonelyRunnerConjecture/
https://www.futilitycloset.com/2017/03/31/the-lonely-runner/

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