Samuli kemppi ant on a rubber rope
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.
Contents
- Samuli kemppi ant on a rubber rope
- Ant on a rubber rope top 6 facts
- A formal statement of the problem
- An informal reasoned solution
- A discrete mathematics solution
- An analytical solution
- Intuition
- Metric expansion of space
- References
The details of the puzzle can vary, but a typical form is as follows:
An ant starts to crawl along a taut rubber rope 1 km long at a speed of 1 cm per second (relative to the rubber it is crawling on). At the same time, the rope starts to stretch uniformly by 1 km per second, so that after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc. Will the ant ever reach the end of the rope?At first consideration it seems that the ant will never reach the end of the rope, but in fact it does (although in the form stated above the time taken is colossal). Whatever the length of the rope and the relative speeds of the ant and the stretching, providing the ant's speed and the stretching remain steady the ant will always be able to reach the end given sufficient time. Once the ant has begun moving, the rubber rope is stretching both in front of and behind the ant, conserving the proportion of the rope already walked by the ant and enabling the ant to make continual progress. This is similar to the paradox of Achilles and the tortoise.
Ant on a rubber rope top 6 facts
A formal statement of the problem
The problem as stated above requires some assumptions to be made. The following fuller statement of the problem attempts to make most of those assumptions explicit.
Consider a thin and infinitely stretchable rubber rope held taut along anAn informal reasoned solution
If the speed at which the target-point is receding from the starting-point is less than the speed of the ant on the rope, then it seems clear that the ant will reach the target-point (because it would eventually reach the target-point by walking along the axis, and walking along the rope can only carry it further forward).
However, though it doesn't seem clear at first, the ant will always reach the end of the rope, no matter the ant's speed or the speed of the rope's expansion. This can be reasoned by the following. Assuming the typical form of the puzzle mentioned above, the ant moves 1 cm/s. As a figurative example, let's say the ant has covered 1/1000 of the rope after one second. In the second second, the ant moves the same distance, but it is smaller compared to the size of the rope. Let's say this is 1/2000. This will continue for a long time, with the ant's distance covered in a second decreasing relative to the length of the rope. That means our fraction will continue getting smaller. However, if we add all these fractions up, we will get a part of the harmonic series, which diverges. This means in the end, the ant will get to the end of the rope, even though it will take an exceedingly long time.
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
Let
Notice that for any
The term
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
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
Now,
If the ant reaches the target-point (which is at
(For the simple case of v=0, we can consider the limit
For the problem as originally stated,
Intuition
The key fact is that the ant moves together with the points of the rope when the rope is being stretched. At any given point of time we can find the proportion of the distance from the starting-point to the target-point which the ant has covered. Even if the ant stops and the rope continues to be stretched, this proportion will not decrease and will in fact remain constant as the ant travels together with the point on the rope where the ant stopped (because the rope is stretched uniformly). Therefore, if the ant moves forward this proportion is only going to increase.
If the rope is stretched with constant speed, these increments in proportion get smaller over time, but form a diverging arithmetic series. If the rope is stretched with increasing speed the series is not guaranteed to be diverging.
Metric expansion of space
This puzzle has a bearing on the question of whether light from distant galaxies can ever reach us given the metric expansion of space. The universe is expanding, which leads to increasing distances to other galaxies, and galaxies that are far enough away from us will have an apparent relative motion greater than the speed of light. It might seem that light leaving such a distant galaxy could never reach us.
By thinking of photons of light as ants crawling along the rubber rope of space between the galaxy and us, we can see that just as the ant can eventually reach the end of the rope, so light from distant galaxies, even some that appear to be receding at a speed greater than the speed of light, can eventually reach Earth, given sufficient time.
However, the metric expansion of space is accelerating. An ant on a rubber rope whose expansion increases with time is not guaranteed to reach the endpoint. The light from sufficiently distant galaxies may still therefore never reach Earth.