Math problem: Zeno's paradox

[PeteH 0]

Zeno was a famous mathematician from Elea, a Greek city on the Italian coast. Zeno was well known for posing puzzling paradoxes that seemed impossible to resolve. One of his his most well known paradoxes was that of Achilles and the tortoise.

Suppose you have a race between Achilles and a tortoise. Now suppose that Achilles runs 10 times as fast as the tortoise and that the tortoise has a 10 meter head start at the beginning of the race. Zeno argued that in such a situation, it would take Achilles an infinite amount of time to catch the tortoise. His argument went as follows:

By the time Achilles runs the 10 meters to the point where the tortoise began, the tortoise will have traveled one meter and will therefore still be one meter ahead of Achilles. Then, by the time Achilles covers a distance ofone meter, the tortoise will have traveled one tenth of a meter and is

still ahead of Achilles. After Achilles travels one tenth of a meter, the tortoise will have traveled 1/100th of a meter. Each time Achilles reaches the previous position of the tortoise, the tortoise has reached another position ahead of Achilles. As long as it takes Achilles some amount of time to traverse the distance between the point where he is and the point where the tortoise is, the tortoise will have time to move slightly beyond that point. No matter how long the race goes on, Achilles will have to move through every point where the tortoise has been before he can pass him. Each time Achilles reaches such a point, the tortoise is at another point. Therefore, Achilles will have to pass through an infinite number of points in order to catch up with the tortoise. If it takes him some time to pass through each one of these points, it will take hiim forever to catch up. Can you find the faulty logic in the above argument? Can you do it without google?

My current inspiration to solve mathematical problems came from that three door problem, so if you got more, keep 'em coming.

[quade 4]

Of course, the "modern" answer might have to include a discussion of Planck Length. Obviously if you keep halving distances and you get to a point at which you can't do that any further because at that point time and space become digital, Achilles would over take the tortoise by ten Planck Lenghts over the course of one Planck Time. He is, after all traveling ten times as fast. (This does imply that he also becomes faster than light, but I digress.)

But, of course, even the ancients solved the problem without resorting to that.

quade -
The World's Most Boring Skydiver

[bor 0]

Imagine that you've found ten bags of gold coins. And you know that there is one bag with false coins. The weight of real gold coin is 1 gr, the weight of false coin is 1,1 gr. How can you determine which bag has false coins by measuring the weight, if you can to measure coins only one time?

[falxori 0]

personally, i'd take all bags.
even fake coins are good for something...
i'd take 1 out of the 1st bag.
2 out of the 2nd bag
3 out of the 3rd bag and so on.
so i'll have (if they were all real):
1*1+1*2+1*3+1*4+1*5+1*6+1*7+1*8+1*9+1*10=55.
in fact, we'll get 55.x where x is caused by the 0.1gr *#coins. hence its also the bag number...

now where is my gold ?

O

"Carpe diem, quam minimum credula postero."

[samhussey 0]

Err just a guess but.....
the problem doesnt have exact numbers for the speed. If you put numbers in place of the relative fractions for each ones speed, then you can calculate that achilles will overtake the tortoise and win. The point is that there is no reference speed supplied.

Is that right?

[falxori 0]

you can look at the whole thing in another way.
u r trying to get from point a to point b,
you walk half the way. then walk half the way of whats left, and so on and so on...

u'll never get there.
the problem here is that you keep changing the resolution.

"Carpe diem, quam minimum credula postero."

[samhussey 0]

I think that was quades point about Plank length. I stilll think its because there is no actual speed given. Put some numbers in and it doesn't work anymore.

[PeteH 0]

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The point is that there is no reference speed supplied.

Nope, there's no need for speed in solving this problem.

[samhussey 0]

Really? Damn...

[falxori 0]

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Put some numbers in and it doesn't work anymore

it will
because you keep changing speed, time and distance resolutions.
he closes the first 10 meters in 1 second.
then another 1 meter in 0.1 sec.
then 0.1 meter in 0.01 second and so on.
your sampling rate is not constant. so its not that he wont pass him, you simply never get to smaple it.
and yes, its similar to quades point

"Carpe diem, quam minimum credula postero."

[samhussey 0]

So the question is, do infinitely declining fractions of a total add up to the total, or not?

[mr2mk1g 10]

A will pass T. He still travels 10 x as far as A each time. Each time you give the example the distance traveled decreases but the factor between the two stays the same. Eventually the two lines will cross - they are not paralell.

[samhussey 0]

Ah, so the time frame isn't constant, it slows down as well until the point where hercules is infinitely close to the tortoise?

[falxori 0]

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Ah, so the time frame isn't constant, it slows down as well until the point where hercules is infinitely close to the tortoise?

exactly
if u keep sampling at the same intervals, sure he'll pass him.

looking at the problem like it was presented is like saying that A wont pass T because he hasnt passed him yet.

ok , any other "fun" problems? and where the hell is my gold ?

"Carpe diem, quam minimum credula postero."

[samhussey 0]

Woohoo! I understand! Where's the aspirin.....

[SkydiveMonkey 0]

How far away is the finish line from Achilles? If it's only 10.5 metres ....

____________________
Say no to subliminal messages

[falxori 0]

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How far away is the finish line from Achilles? If it's only 10.5 metres

this question defines the difference between math and engineering..

another one of the same kind (but not the same):
The Arrow: Time is made up of instants, which are the smallest measure and indivisible. An arrow is either in motion or at rest. An arrow cannot move, because for motion to occur, the arrow would have to be in one position at the start of an instant and at another at the end of the instant. However, this means that the instant is divisible which is impossible because by definition, instants are indivisible. Hence, the arrow is always at rest.

"Carpe diem, quam minimum credula postero."

[BETO74 0]

How about the two trains going on the same track opposite directions. the distance between the trains in 100 miles, and they travel at 100 miles per hour. A super bird goes from train A to train B touches it and comes back to train A touches and goes back to train B, the super bird travels at 150 miles per hour.
How many trips the bird have made before the trains collide?

100MPH 100MPH
--------------> <--------------
A_______________________________________________B
100 MILES

http://web.mac.com/ac057a/iWeb/AC057A/H0M3.html

[mr2mk1g 10]

infinate - the trains are going in opposite directions...

<---A ___________________ B--->

Or I suppose the answer could be related to the cercumferance of the earth.

[BETO74 0]

Sorry same direction!! you know the train is going to crash

http://web.mac.com/ac057a/iWeb/AC057A/H0M3.html

[falxori 0]

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How many trips the bird have made before the trains collide?

werent you paying attention?
the trains will never collide, they will only get closer and closer...

O

"Carpe diem, quam minimum credula postero."

[mr2mk1g 10]

damnit - thought that was too easy... that means I'd actually have to think to work this one out... too bad - back to work in 3 minutes.

[BETO74 0]

Also reminds me the question.

Can you have infinite spaces on a finite space?
Why?

If you have a letter size paper, can you divided oon half? can you divided that half on half, can you keep going forever?

http://web.mac.com/ac057a/iWeb/AC057A/H0M3.html

[happythoughts 0]

Makes you wonder why anyone even remembers this Zeno guy, if he can't figure out the obvious stuff.

Of course, speed is relative. Imagine a snail sitting on the back of the tortoise yelling "Wheeeee!".

[PeteH 0]

Achilles problem is about geometric series.

There's infinite distance intervals, but 'cos these intervals decrease geometrically the total distance Achilles needs is finite
11.1111111m (10 m + 10/9 m)