I have tried Wikipedia, I have tried youtube but now you are my only hope reddit. I have been assigned to present the ant on a rubber rope paradox to class and no matter how hard I try it still baffles me.
In: Mathematics
Basically the ant is always making progress towards the end of the rope. and the fraction of what is left to cover is always being reduced.
The assumption is that when the rope is stretched equal distances are added in front and behind the ant. Hence the relative position of the ant to the end of the rope doesn’t change.
Basically if the ant is halfway to the end of the rope and then you stretch it, then the ant is still half way to the end of the rope regardles of how long the new length of the rope. Any progress the ant makes still counts even if you keep stretching the rope.
The paradox come from the fact that yes the ant is making progress and should theoretically reach the end of the rope. However as long as you keep stretching the rope faster than the ant can move them overall length become greater, the rate of progress for the ant becomes smaller and smaller and the ant would never move fast enough to reach the end.
imagine adding 0.09 to 0.9 this second then 0.009 next second then 0.0009 the next second and repeating this infinitely trying to get to 1. the numbers have positive value and are increasing the whole sum but the rate of increase gradually gets smaller and smaller and seems like it should never be able to get to 1 however in mathematical theory this series is equal to 1
hence mathematically 0.99999… (repeating indefinitely) = 1.
So, the idea with the puzzle (going by the version on Wiki) is that the ant is on a 1km rope walking at 1cm/s, but the rope gets stretched at 1km/s. It seems like the rope is always getting more longer than the ant has to walk, so that ant should never reach the end.
The problem with that thinking is that the ant gets dragged along when the rope stretches. How far the ant gets dragged depends on how far along the rope it is.
Imagine the ant finally gets to be 1cm from the end of the rope. But now the rope stretches by another 1km. Does that mean the ant in now 1km away? No, because the rubber that is under the ant gets pulled too. This would mean the ant is still about 1cm away from the end.
Hopefully that’ll get you started on the idea. And if not there’s a [Vsauce2](https://youtu.be/OM9KepKsg6U) video on this exact riddle!
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