Archive for the ‘mathematics’ Category

The dichotomy paradox

Posted: April 28, 2009 in logic, paradox
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Its amazing to see how the logic fails to explain common things we see around leading to brain teasing paradoxes.

“That which is in locomotion must arrive at the half-way stage before it arrives at the goal”   – Aristotle.

Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on.This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.

Hmmm…weird…so if u have read my previous post, then neither that tortoise can ever finish the race,altough he might be declared as the winner for always maintaining a lead.

PS: Contains adopted contents

Achilles and the tortoise

Posted: April 28, 2009 in logic, paradox
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“In a race, the quickest runner can never overtake the slowest unless they start from the same point, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead”.

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise’s starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.

PS: Adopted contents

The Grandfather Paradox

Posted: April 19, 2009 in logic, paradox
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Suddenly i got obsessed with paradoxes. Infact I have always been in love with it. So here is another one and this time its The Grandfather Paradox . This one is for those who believe in the fictions we read especially the time travel stuff. Ooops..

Ok, the paradox is simple,”suppose a man traveled back in time and killed his biological grandfather before the latter met the traveller’s grandmother”. Well then atleast one of his parents will not be born and as a result the traveller itself is not born. Now if he is not born, then there is no way he could have gone back in time and killed his grandfather.If he doesnt go back and kill his granpa then granpa meets grandma and as result he is born.Now if he is born he can then go back and kill his granpa….huh, i’m tired of typing. Isn’t that enough for a paradox.

This is a recursive logical paradox as each case implies its own negation. ie, ~p => p. Well anyway the grandfather is safe for now since there is no way you can travel back in time. But that doesnt make him safe forever,we might invent it any time..Oh are u a cynic and telling me thats not possible at all..Ok then i think u shud watch this ad from RedHat:

See it can happen anytime. So are you a grandfather already or else try not to be. Anyway find your soulmate before some pranky grandchild of yours gets the idea of trying this out..

The Birthday Paradox

Posted: April 18, 2009 in logic, mathematics, paradox
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This is one theory that really interested me. Can you guess what is the probability that any two people in a football court ,out of a total of 23 including the referee,share the same birthday. It is more than 50 %. Atleast we have mathematical proof for that.

The actual statement is “In a group of at least 23 randomly chosen people, there is more than 50% probability that some pair of them will both have been born on the same day. For 57 or more people, the probability is more than 99%, and it reaches 100% when the number of people reaches 367 (there are a maximum of 366 possible birthdays)”.This happens because we have to consider the total number of pairs in the field rather than the number of players on the field.

Suppose when there are n players there can be n.(n-1)/2 pairs. So with 23 players there can be 23×11,ie 253 pairs possible. Now we can see how does it calculate the probability. Let the probability that atleast one pair of them share  a birthday be p(n). Then the probability that all the birthdays are on different dates is 1-p(n) which we can call as ~p(n).

Now if n>365, Pigeonhole principle shows that ~p(n) is 0. If n<365, then:

~p(n)= 365/365 x 364/365 x 363/365 x …… x (365-n-1)/365,

since the second person cant have the birthday of the first and third person cant have the birthday same as 1st or second.

=> ~p(n)= 365! / ((365^n).(365-n)!)

Keeping in mind that p(n)=1-~p(n),if we substitute values for n, then we get :




Now that was a simple proof right. And this theory is the basis of the famous birthday attack( Given a function f, the goal of the attack is to find two inputsx1,x2 such that f(x1) = f(x2).) in cryptography.

PS: In reallife there are a lot more parameters that affect the result like  leap years, twins, seasonal or weekday variations.