# The Famous Runner and the Tortoise Puzzle

• ArielGenesis
In summary, the runner can't catch up to the tortoise because the time between each instance decreases with each cycle, and the rate of these events increases until the runner catches up with the tortoise.

#### ArielGenesis

i forget the name of the puzzle but it is somewhere in the brain teaser.

there is a famus runner in a race with a tortoise.
and to win he had to go after the tortoise
but before that he must travel that half of the distance
but before that he must travel that half of half of the distance
but before that he must travel that half of half of half the distance
and so on
so he go nearrer each time but he cannot go after the tortoise.

isn't this is a bit simmilar with constantly accelerting and trying to catch up the light speed.

The reason you can't catch light is you can't travel faster than light. In your example given a constant rate of travel for the rabbit and the turtle if the rabbit's velocity is faster than the turtle then he will eventually overtake him with time. Time is the key here sure we could keep going 1/2 way into infinity mathematically. But you can also make a formula that will show when in time the rabbit will overtake the turtle. The only way to stop him from catching the turtle is to stop time and that's not possible.

It's just a case of how you've describe the progress. The tortoise has a head start, by the time the runner gets to where the tortoise was, the tortoise is a bit further adhead. Seems like the runner can't catch up, until you take into account that the time between each instance of the runner catching up to where the tortoise was gets smaller with each cycle, and the rate of these events (versus time) increases until the runner catches up with the turtle where the time between events is zero, and the rate is infinite.

A similar process is more common place, a bouncing ball. A very hard ball, like a pool (billard) ball bouncing on a very hard surface (tile) is a good example. The ball loses energy on each bounce. If the ball bounces, say 80%, of it's previous height each time, how long before the ball stops bouncing? Turns out it does stop bouncing in a fixed amount of time, but the frequency approaches inifinity as the time progresses to the point the ball stops bouncing. You can hear this frequency increase up to a point. In a real life situation, the bouncing stops sooner, when the height becomes less than the deformation of the ball and surface.

The runner is achilles
It was a paradox that baffled Archimedeans(sp?) for centuries. They couldn't solve the problem because they didnt know limits. :tongue:

Last edited:
Ryoukomaru, yeah that's what i meant.

so how about the light speed?

Ryoukomaru said:
The runner is achilles
It was a paradox that baffled Archimedeans(sp?) for centuries. They couldn't solve the problem because they didnt know limits. :tongue:
Don't bring Archimedes into this.
True, Greek PHILOSOPHERS puzzled over this; I've yet to see any evidence that Archimedes was puzzled by this.
And, you don't need the concept of limits to understand why the "paradox" is flawed:
Pick any stick of finite length.
IDEALLY, you could chop up this stick into infinitely many pieces, and then hand over the bundle of pieces to some poor jerk who are obliged to measure the total length of the pieces.
What he's faced with, is to sum up infinitely many pieces, but still that sum would be equal to the stick's original length.
Thus, the "paradox" is based on a false premise.

This is an argument which would have been easily understood by any intelligent Greek, and I am quite certain that we have only been handed down the puzzlements of mathematical incompetents, not the intelligent Greeks' resolution of the "paradox".

ArielGenesis said:
Ryoukomaru, yeah that's what i meant.

so how about the light speed?
It doesn't really have anything to do with moving at the speed of light. As others explained, the situations are not analagous and converting the word-problem to an equation will demonstrate that.

## 1. How does the puzzle go?

The puzzle is a classic tale about a race between a fast runner and a slow-moving tortoise. The runner is confident in his speed and challenges the tortoise to a race. The tortoise accepts the challenge and agrees that the runner can start ahead while the tortoise starts from a designated point.

## 2. Why is this puzzle famous?

The puzzle is famous because it presents a paradox that has puzzled thinkers for centuries. The question is, can the faster runner ever catch up to the tortoise?

## 3. How do people typically try to solve the puzzle?

Many people try to solve the puzzle using mathematical equations and calculations. They often focus on the speed of the runner and the tortoise and try to determine if the runner can catch up to the tortoise based on different scenarios.

## 4. What is the solution to the puzzle?

The solution to the puzzle is that the runner will never be able to catch up to the tortoise. This is because the distance between the two will continuously be divided and the runner will only ever be able to close the gap by a fraction, but will never be able to fully catch up.

## 5. Are there any real-life applications of this puzzle?

Yes, this puzzle has been used to explain various concepts in mathematics, such as the concept of infinity and limits. It has also been used to illustrate the concept of Zeno's paradox, which questions the possibility of motion and change. Additionally, the puzzle has been adapted into various forms, such as in computer science and game theory, to explore different strategies and outcomes.

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