# Hare and the Tortoise Puzzle

This isn’t really a puzzle… But it makes for an interesting thought exercise for those who look at the trees rather than the forest.

A turtle and a hare decide to have a race. The turtle has finally figured out a way, he thinks, that will guarantee him a win, or at least not a loss. And, if he has figured it correctly, it doesn’t matter how fast the hare can run! The only stipulation is that he has to have a head start… Here’s his thinking:

Say the turtle gets an X minute head start, and travels at Y miles/hour. Then the hare takes off at Z miles/hour (Z >> Y). Here’s the idea… After a few moments, the turtle gets to point “a”. The hare, of course, takes some amount of time to catch up to point “a”. But by then, the turtle has gone a bit further, to point “b”. The hare will take some finite amount of additional time to get to point “b”. By then, the turtle will have moved on to point “c”. Etc, etc, etc…. It appears, to the turtle, that the hare will never catch up!

Clearly, the turtle is a little “slow”. We know the hare will win give reasonable values for speed, race distance, and the head start. Using simple math, we can figure out the point that the hare overtakes the turtle. So, where did the turtle go wrong in his thinking?

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as the hare is faster than the turtle , the time needed for the hare to catch up to the turtles new position decreases because the distance between them decreases.

the turtle is looking at a sequence that is converging at a particular distance from the starting point. At that point , the turtle and hare are neck to neck.