At school, a number of years ago now, I remember a question being set to the class. It was designed to make us think, and intentionally impossible – although I still tried to answer it logically, missing the point that it must be answered as it was set – that was the difficulty.

Well, I spent a number of hours driving this weekend which reminded me of the question – and gave me time to think about the solution.

The question was this:

Cars A and B are driving at different but constant speeds. B is going faster than A and behind it, approaching it. You choose a moment and recall where A and B are. You note that when B gets to where A was, A has moved forward to a new position. For each successive observation you note that when B gets to where A was, A has moved on. So how can B overtake A?

It was perhaps something of a pointless question, as the true answer is pretty obvious, but the idea was to get us thinking outside the box. If I recall correctly, our teacher gave up on this happening pretty quickly.

My first line of thought was definitely outside the box, but I think would have been hard to defend at ultimately classified as a ‘nice try’. The idea is as follows:

Everything in the universe boils down to a collection of energy and interaction. As such, the movement and energy that can be acheived must adhere to strict, discrete levels according to the harmony of subatomic particles, energies and interactions. Thus, as you take the scenario in question further into its iterations, you will eventually come to a point where the distance that car A should be moving does not account for a whole discrete movement, while car B does and so passes it.

Clearly it’s a work of fiction, but seems a nice idea. It took a lot of time to think about, which is more than I can say for the answer that I believe is the true solution:

As you follow the scenario in question through long enough, the time between each iteration will become shorter and shorter. The reason for this is that car B is known to travel faster than A, therefore car A will not travel as far from its old spot as B does to reach it in any given time. If you follow these iterations to their conclusion, the time between iterations will shorten to zero. At zero, car B takes no time to reach car A’s old position, and so car A can go nowhere, to which car B takes no time to reach and so on.

We are stuck at the point in time where B is level with A. The increments have reached zero change in time and distance. The model is no longer progressing. Logically, time always progresses, so we have to investigate what happens at the next point in time. It doesn’t matter what amount of time we choose, B will now pass A, as B is travelling faster than A.

It’s pretty obvious now, but I am a decade older and have some education and experience behind me. If you have any other solutions, feel free to post them in the comments.