The ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the principles of the puzzle remain the same.
The details of the puzzle can vary, but a typical form is as follows:
At first consideration it seems that the ant will never reach the end of the rope, but whatever the length of the rope and the speeds, provided that the length and speeds remain steady, the ant will always be able to reach the end given sufficient time — in the form stated above, it would take years. There are two key principles: first, since the rubber rope is stretching both in front of and behind the ant, the proportion of the rope the ant has already walked is conserved, and, second, the proportional speed of the ant is inversely proportional to the length of the rubber rope, so the distance the ant can travel is unbounded like the harmonic series.
A formal statement of the problem
For sake of analysis, the following is a formalized version of the puzzle.
The statement of the puzzle from the introduction corresponds to when is 1 km, is 1 km/s, and is 1 cm/s.
Solutions of the problem
A discrete mathematics solution
Although solving the problem appears to require analytical techniques, it can actually be answered by a combinatorial argument by considering a variation in which the rope stretches suddenly and instantaneously each second rather than stretching continuously. Indeed, the problem is sometimes stated in these terms, and the following argument is a generalisation of one set out by Martin Gardner, originally in Scientific American and later reprinted.
Consider a variation in which the rope stretches suddenly and instantaneously before each second, so that the target-point moves from to at time , and from to at time , etc. Many versions of the problem have the rope stretch at the end of each second, but by having the rope stretch before each second we have disadvantaged the ant in its goal, so we can be sure that if the ant can reach the target point in this variation then it certainly can in the original problem, or indeed in variants where the rope stretches at the end of each second.
Let be the proportion of the distance from the starting-point to the target point which the ant has covered at time t. So . In the first second the ant travels distance , which is of the distance from the starting-point to the target-point (which is throughout the first second). When the rope stretches suddenly and instantaneously, remains unchanged, because the ant moves along with the rubber where it is at that moment. So . In the next second the ant travels distance again, which is of the distance from the starting-point to the target-point (which is throughout that second). So . Similarly, for any , .
Notice that for any , , so we can write
.
The term is a partial harmonic series, which diverges, so we can find such that , which means that .
Therefore, given sufficient time, the ant will complete the journey to the target point. This solution could be used to obtain an upper bound for the time required, but does not give an exact answer for the time it will take.
An analytical solution
A key observation is that the speed of the ant at a given time is its speed relative to the rope, i.e. , plus the speed of the rope at the point where the ant is. The target-point moves with speed , so at time it is at . Other points along the rope move with proportional speed, so at time the point on the rope at is moving with speed . So if we write the position of the ant at time as , and the speed of the ant at time as , we can write:
This is a first order linear differential equation, and it can be solved with standard methods. However, to do so requires some moderately advanced calculus. A much simpler approach considers the ant's position as a proportion of the distance from the starting-point to the target-point.
Consider coordinates measured along the rope with the starting-point at and the target-point at . In these coordinates, all points on the rope remain at a fixed position (in terms of ) as the rope stretches. At time , a point at is at , and a speed of relative to the rope in terms of , is equivalent to a speed in terms of . So if we write the position of the ant in terms of at time as , and the speed of the ant in terms of at time as , we can write:
where is a constant of integration.
Now, which gives , so .
If the ant reaches the target-point (which is at ) at time , we must have which gives us: