The Tortoise and the Hare
Little Johnny was not convinced when his teacher said that the hare would finally overtake the tortoise. He remembers the fable from the bed time stories and was sure that the tortoise always wins the race.
"You said that the tortoise was a thousand meters ahead," said Johnny. "Yes," replied the teacher, "but the hare runs ten times as fast as the tortoise."
"That makes little difference, Sir; because by the time the hare covers those thousand meters, the tortoise would have gone ahead by hundred." "That's right."
"While the hare is over with his hundred, the tortoise would have covered ten meters more. When the hare makes up these ten, the tortoise would have managed another meter. By the time the hare covers this one whole meter, the tortoise would be leading by one tenth of a meter. And while the hare is still at it, the tortoise covers another one-hundredth, and so on. I agree the lead is diminishing and if you carry on it may reduce to one-millionth of a meter or even less but the hare will always be lagging behind."
"No, Johnny, that can't be right as by the time the hare covers two thousand meters, the tortoise covers just two hundred meters and would be eight hundred meters behind."
"Well, Sir, let's not jump into conclusions. I have just explained to you the way I look at the problem and it seems logical.
What do you think ? Don't let the fable affect your decision.
"You said that the tortoise was a thousand meters ahead," said Johnny. "Yes," replied the teacher, "but the hare runs ten times as fast as the tortoise."
"That makes little difference, Sir; because by the time the hare covers those thousand meters, the tortoise would have gone ahead by hundred." "That's right."
"While the hare is over with his hundred, the tortoise would have covered ten meters more. When the hare makes up these ten, the tortoise would have managed another meter. By the time the hare covers this one whole meter, the tortoise would be leading by one tenth of a meter. And while the hare is still at it, the tortoise covers another one-hundredth, and so on. I agree the lead is diminishing and if you carry on it may reduce to one-millionth of a meter or even less but the hare will always be lagging behind."
"No, Johnny, that can't be right as by the time the hare covers two thousand meters, the tortoise covers just two hundred meters and would be eight hundred meters behind."
"Well, Sir, let's not jump into conclusions. I have just explained to you the way I look at the problem and it seems logical.
What do you think ? Don't let the fable affect your decision.
Labels: friday special, puzzle, thinktank
posted by Rajesh Lal at
8:36 AM
30 Comments:
Where's the finish line?
If the finish line is 111 meters or less away from the turtle, he wins.
If the finish line is 112 meters or more away, the hare wins.
Math people could probably figure the exact distance the hare will pass the tortoise, but that doesn't seem the point.
The error in the boy's thinking is dividing time/distance into infinitely smaller units up to the point that the hare makes the pass.
I think the Kid is right.
The turtle should run with speed <= 100 meter/Hour and drop his speed by 1/10 every Hour (due to fatique, I guess!)
This is Zeno's famous paradox involving Achilles and a tortise. 24 centuries of philosophers and mathematicians have argued over this paradox, with no conclusive answer either way. The essence of the paradox is that the "hare" must perform an infinite amount of tasks in order to overtake the tortoise, which is theoretically impossible. Of course, if a person were to watch such a race, they would see this seemingly impossible feat performed. Such is the nature of infinity.
The chicken came first. Immaculate conception.
The hare would pass the turtle. The turtle covers 1/10 of every 1000meters the hare runs, not 1/10 of what distence hair is at at every moment of the race.
If it were to be that case, it would be a rediculously long and boring race that would virtualy never end.
All this infinitessimal unit of space/time contemplation is making my head hurt. To determine the exact smallest unit of space at which the hare will have no option but to simply be tied with the tortoise before overtaking it, I will have to consult my genie - who will, in response, ten times as quickly pose the same question to his meta-genie - who will, in response, ten times as quickly pose the same question to his meta-meta-genie - who will, in response, ten times as quickly pose the same question to his meta-meta-meta-genie - who will, in response, ten times as quickly pose the same question to his meta-meta-meta-meta-genie...
i believe the hare would take over the hare. if Turtle speed is 1mph (just an example) then Hare speed is 10mph
Just narrow down to make it easier to explain, we'll say the distance between the 2 is 100
Hare speed
10.20.30.40.50.60.70.80.90.100.110,
120.130.140.150.160.170.180.190.
1h.2h.3h.4h.5h.6h.7h.8h.9h.10h.11h,
12h.13h.14h.15h.16h.17h.18h.19h.
Turtle speed
101.102.103.104.105.106.107.108,
109.110.111.112.113.114.115.116
1h .2h .3h .4h .5h .6h .7h .8h ,
9h .10h.11h.12h.13h.14h.15h.16h.
Complicated i know, but the point is, eventually the hare passes the turtle, in my example between 11h and 12h. You can probably substitute different speeds, but eventually you get the same outcome.
That's what i think, and that's how I choose to prove it.
srry, the first line should say, i believe the hare would take over the turtle, and yes i know, its a tortoise not a turtle but i dont care
well, this puzzle only works id the tortoise moves then the hare moves but you have to take into consideration that they will both be constantly moving at the same time. therefore when the tortoise moves 10 metres the hare will have moved 100 and then the tortoise will move a further ten metres and the hare will have moved a total of 200 therfore the hare must have taken over the tortoise.
Johnny is correct.
His teacher should quit his job.
Obviously johnny is correct because the hare would take a nap and therefore never be able to pass him
According to the paradox, that hare will only overtake the tortoise when the distance between them is the smallest unit of space. After that point, when the hare moves, it will pass the tortoise. So, the trick to this is to know that Time and space are not infinetly divisible.
I don't know anymore, it hurts my brain. both the way i explained it and the way the kid explains it works and it hurts my head
hey the tortoise won cuz in the version i heard the hare fell asleep witch put the totrtoise in front so he won da race
The teacher is right and johnny is wrong. by the looks of it johnny is thinking of the distancce as exponential when it shouldn't be.
obv johnny is wrong because he is proposing that something in 2nd place will never pass another thing. so y do millions of ppl flock to race tracks each year to see the 2nd place driver pass the first?
H= distance moved by hare
T= distance moved by tortoise
H T
1000 100
100 10
10 1
1 1/10
1/10 1/100
and so on..
The values will approaches 1/infinite = 0
Hence, the hare will overtake the tortoise at one point of time
in my veiw there is no way to know because we dont know how long it will take for the race to be over.
The key to this puzzle is the variable the teacher gives. The Hare can ONLY run as fast as 10 times the Tortoise,thus if the Tortoise slows down, the Hare must slow down as well, proving Johnny's theory correct. That's my view on it.
lol just to say I'm 12 so don't flame me but there could be a point were the hare either tripped up or the tortoise cheated, but also when you think wen will you ever b able to make a hare and a tortoise to race.
It totally depends on the length of the finishing line and the speed of the tortoise and the hare. If the hare is 100 miles per hour and the tortoise is only 1/10th of the speed which is 10 miles per hour. Therefore the hare gains 90 miles every 10 miles that the tortoise gain.So he is only 910 miles away from the tortoise. If this goes on.sooner or later the hare would catch up. Therefore the distance of the tortoise to the finish line has to take into account. then it goes on when the tortoise reaches 20 miles the distance between the tortoise and the hare is only 820 miles. By the 110 mile that the tortoise travel the hare is only 10 miles away. So it has to be more then 116miles betwen the tortoise and the end to make the hare win. Thus this shows that the distance and the speed totally depends and it is possible that the hare would win
I agree with the previous comment. I dont know if you guys wanna hear how I think it goes. But here's my solution: lets say the race is Xmiles long. the tortoise has to run (X-1000) while the hare runs Xm. (Assuming that we start looking at the situation when the race begins). So the speed of the hare is V_r lets say. Therefore making the speed of the tortoise (V_r/10). Using this info, you end up the with time the hare takes to complete the race being X/(V_r) and the time for the tortoise being (X-1000)/(V-r/10). (Using the formula time=distance/speed). Equate the two together to get the distance the hare must run before the catches up to the hare.
Long story short: the hare only beats the tortoise after 111.1miles.
I think this is right..
This is one of Zeno's Paradox's called Achilles and the tortoise. It is similar to the dichotomy paradox, stating you can never go anywhere because before you get somewhere you have to go half way first, then when you get there you still have to go half way again, and then half way again, and so on and so forth. He has an infinite halves to cross and thus will never reach his goal. I love paradoxs.
From Wikipedia...
"Proposed solutions both to Achilles and the tortoise, and to the dichotomy
Both the paradoxes of Achilles and the tortoise and that of the dichotomy depend on dividing distances into a sequence of distances that become progressively smaller, and so are subject to the same counter-arguments.
Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances.
Before 212 BCE, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method. These methods allow construction of solutions stating that (under suitable conditions) if the distances are always decreasing, the time is finite."
For anyone who has taken calculus you would recognize this problem as being one of infinite sums.
Thank you
The question is impossible because the world is round... at anyone point the hare is in front of the turtle and the turtle is also ahead of the hare
The kid is assuming the finish line is the limit of when the hare approaches the tortose. As the hare approaches the tortuse, the limit is infinately small.
Stupid Kid.. haha
In reality the hare should win...
As a few have already mentioned, this is one of Zeno's paradoxes. It was created to defend the view of another Philosopher, Parmenides, who offered that all is one, and as such movement is impossible, and anything that appears to move is an illusion.
It can of course be solved by modern day Calculus.
The main point of this false paradox is to show that an infinite number of processes, distances, or points can exhist inside of a finite distance or time.
Mathematically, the hare will finally reach the destination. This is the presently accepted way of disproving this false paradox.
This was the first paradox we studied in my Philosophy class. It is a false paradox because it can be proven wrong. Not only by experience, but also logic.
~Cenyu
say: tortoise's speed = 1m/unit time;
hare's speed would be = 10m/unit time (hare 10 times faster)
then:
tortoise traveled distance:
1000 + 1*(time)
hare traveled distance:
0 + 10*(time)
by the time = 1000/9 unit time
both is at the distance 1111.111.. meters
any time more than 1000/9 unit time,
or at any mark pass 1111.11.. meters the hare will be at lead.
Post a Comment
Links to this post:
Create a Link
<< Home