Q: If two trains move towards each other at certain velocities, and a fly flies between them at a certain constant speed, how much distance will the fly cover before they crash?

The brain teaser comes in a many variations. For example:

Trains A and B, 700 miles apart, are heading toward each other on a straight piece of track. Train A is going 85 mph while train B is going 55 mph. At the same moment, a bee that flies 110 mph is sitting on the nose of train A and begins flying toward train B. When it reaches train B it makes an instantaneous reversal of direction and flies back toward train A. It continues to change direction every time it runs into a train until both trains and the bee meet in a spectacular crash. What total distance did the bee fly before the big collision?

Mathematician: The difficult way to solve this problem is to figure out how much distance the bee (or fly) traveled before turning around each time it approached a train, and then sum these distances together. The easy way to solve it is simply to figure out how long it took the trains to crash, and then calculate how far the bee, which travels at a constant speed, must have gone during this amount of time.

More specifically: The bee always travels at the same speed V. If we can figure out how much time, T, the bee flew before the trains collided with each other, then the total distance D it flew will just be V T, the product of the velocity and time. We know V, so all that remains is to figure out T. To do this, we just need to calculate how long it takes for the trains to crash. If the first train has velocity v1 and the second v2, and the distance between them initially is d, then the time T before the crash will just be d/(v1+v2), which is equivalent to the amount of time that it takes a train going velocity v1+v2 to travel the distance d. The total distance traveled by the bee is given by:

D = V d / (v1+v2)

= (700 miles)  * (110 mph)/((55 mph)+(85 mph))

= 550 miles