There is a famous arithmetic question :
Two trains $150$ miles apart are traveling toward each other along the same track. The first train goes $60$ miles per hour; the second train rushes along at 90 miles per hour. A fly is hovering just above the nose of the first train. It buzzes from the first train to the second train, turns around immediately, flies back to the first train, and turns around again. It goes on flying back and forth between the two trains until they collide. If the fly's speed is $120$ miles per hour, how far will it travel?
It is easy to determine the distance travelled by the bee.
But how to determine how many times it touches first/second train?
or
Which train it touches last?
Best Answer
Here is some graphical intuition for understanding why the bee touches trains infinitely often.
The bee's trajectory in space-time is a zig-zag path which is self-similar; if you zoom in onto successive pairs of turnarounds, you get a copy of the original:
So the bee touches both trains infinitely often, in a finite amount of time. The duration of time between successive touches gets shorter and shorter.
To make it obvious without needing to zoom, take a faster bee (or slower trains). Here is what happens for a bee 10 times as fast:
EDIT. Given the discussion about bees of nonzero length, here's what happens to a 20 mile long bee going at 1200 mph: