The MAN IN BLACK has a cunning plan. He intends to beat the famous athlete Achilles in a race. Okay, it may be a little late - there was the small matter of an unfortunate foot injury - but the strategy still has some merit.
The ancient Greeks had a famous conundrum involving a race between Achilles and a tortoise. If you haven't met it, it goes like this. Achilles runs ten times as fast as the tortoise, so the tortoise is conceded a 100 metre start. In the time it takes Achilles to run that 100m, the tortoise runs 10m. When Achilles runs that 10m the tortoise runs 1m, and so on for ever. Greek mathematicians wondered if this meant that Achilles would never overtake the tortoise.
Modern mathematicians recognise this as an infinite series - 100, 110, 111, 111.1 .... We say it converges - it gets closer and closer to a final value, despite never actually arriving at it. This series converges to 111.11111... metres, or 1/9 km. Achilles takes the lead after running that distance.
The MAN IN BLACK has a better plan. He may not run as fast as Achilles (when he was alive at any rate), but has a decent amount of stamina and can keep running for ages. So, the plan is to ask for a 1km start. In the time it takes Achilles to run 1km, the MIB aims to run half a kilometre. While Achilles runs that 1/2 km, the MIB will run 1/3 km. While Achilles runs 1/3km, he will run 1/4 km, and so on.
The difference here is that this series doesn't converge - 1 + 1/2 + 1/3 + 1/4 + 1/5 ... is infinite, as any maths student should know. Achilles will never overtake the MAN IN BLACK.
This gave our hero an immense sense of smugness, which lasted about ten seconds. After a few minutes more, he deduced exactly where the flaw in his argument lies. Try to work it out yourself before clicking on the button below.