A rubber band is 1 meter long. An ant starts at one end, crawling at 1 millimeter per second. At the end of each second, the rubber band is instantaneously stretched by an additional meter. (So, at the end of the nth second, the rubber band becomes n+1 meters long.)

Does the ant ever reach the far end of the band? If so, when?

I take it we should assume that the stretching is both instantaneous and uniform, so that the ratio of the distance that the ant is from his starting point to the total length is the same after the stretch as it was before.

Because if the stretching all takes plae behind the ant, he will reach the starting point in 1000 seconds = 16 minutes, 4 seconds. If all the stretching is in front of the ant, he'll never complete the circuit.

If the stretching is instantaneous and uniform, then the first stretch doubles both distances, making the distance behind the ant 2mm. the ant crawls another mm before the second stretch (3 mm), which then expands all distances by half (3 * 1.5 = 4.5 mm). the next stretch moves the ant from 5.5mm (4.5 + 1) to 7.333mm (5.5 * 4/3). On the fourth stretch, the distance goes from 8.333 to 10.41666.

So, after one second, the ant is 2/2000 = 0.1% of the distance he must travel

After 2 seconds, he is 4.5/3000 = 0.15%

After 3 seconds, he is 7.333/4000 = 0.183333%

After 4 seconds, he is 10.4166/5000 = 0.2080813333%

So the ant is making some progress, but by a smaller increase in percentage each time. The question becomes does this sequence converge to 100% or less (in which case the ant will never reach his starting point) or does it converge to a number higher than 100% or maybe even diverge (in which case, he reaches the end just before the first stretch that would result in a distance of 100% or more.)?

Although I intuitively suspect the former (that the ant will never reach the end), the very fact that this puzzle exists suggests that there is a time when the ant returns to his starting position. The only way to be sure is to do the math.

Posted by TomM on 2002-10-03 09:54:11