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Reaching the Unreachable #### December 15, 2022

Have you ever felt like giving up? Wanting to quit something because it seems impossible to reach your goal (for example passing Difference- and Differential Equations)? Of course you have, but don’t give up too fast! Sometimes, it is possible to succeed in what seemed like a hopeless mission (like passing Difference- and Differential Equations). Today we will discuss the “Ant on the rubber band” paradox which perfectly displays such a situation.

The ant on the rubber band

The backstory of the “Ant on the rubber band” paradox is as follows. Imagine a rubber band which is 1 km long, and an ant on this rubber band at one of the end points. The ant starts crawling along the rubber band with a speed of 1 cm per second and wants to reach the other end point of the rubber band. At the same time, the rubber band begins streching at a constant rate of 1 km per second, so that after 1 second the band is 2 km long, after 2 seconds it is 3 km long, etcetera. Then, the question is: will the ant ever reach the other side of the rubber band?

At first you might think that the answer is obvious. After all, the rubber band is stretching way faster than the ant is crawling . But the key to understanding this paradox is noticing that when the band stretches, it will also move the ant. We can see this by looking at the proportion of the band that the ant has covered after n seconds. For example, after 1 second the ant has travelled 1 cm which is equal to of the length of the rubber band. But don’t forget that the band also stretches at the same time. After 1 second, the rubber band stretches from 1 km to 2 km. The crucial part is that the band stretches evenly across its full length, thus the 0,99 km in front of the ant becomes 1,98 km and the 0,01 km behind the ant doubles to 0,02 km.  Then, the ant has covered of the length of the rubber band. Thus despite the band stretches, the proportion of the length of the band covered by the ant has not changed. Let us formalize this mathematically.

Mathematical proof

Consider a rubber band on the x-axis, with starting point and end point where are constants and denotes the time. The ant starts moving along the rope at with a constant velocity relative to the rope at its current position. Assume the rope stretches suddenly after each second, so that the end point moves from to at , and from to at , etc.

Let be the proportion of the distance from the starting point to the end point which the ant has covered at time . After 1 second, the ant has travelled distance but the rope has also streched and because of that the end point moved to .  Then, In the next second, the ant has again travelled distance but the length of the rope also stretched to , thus the ant travelled of the distance from the starting point to the end point.  Then, In general, for any , we have Note that for any , we have Thus We know is a harmonic series which diverges. Consequently we can find an such that which implies that Recall that is the proportion of the length of the rope travelled in total by the ant at time . We have just shown that there exists an such that and therefore the proportion of the length covered by the ant will eventually be 1. Hence, given enougn time, the ant will always reach the other side of the rubber band!

To return to the question discussed in the introdution of this article, yes, it is possible to succeed in what seemed like an impossible mission. However, that does not mean this is always easy. Let us take a look at our ant crawling on the rubber band. It is to be expected that, with the given speed of the ant of 1 cm per second, the speed of 1 km per second by which the rubber band is stretching and the starting length of the rubber band of 1 km, the ant will need a really long time to reach the other side of the rubber band. To be precise, in this case it will take approximately seconds until the ant has reached the other side. Nevertheless, if the ant really has the ambition of reaching the other side, it is possible. The moral of this story is that you can succeed at things which appeared to be undoable. Just like the ant, do not give up too fast because you are capable of a lot more than you think!