Problem solving at Sangam

Here is an interesting problem:

There is a wierd man who goes to a noodle shop and orders noodle soup. He asks for exactly 100 noodles in his soup. Once his noodle soup arrives, he gets to work. He extracts two noodle ends from under the soup, ties a knot and lets it slip into the soup again. He does the above 100 times so that now all the noodle ends are tied.

Now he reaches his hand into the soup. What is the probability that he would extract a garland containing all the 100 noodles.

Well let me tell you why this problem is so interesting.

On the first glance if you were asked to guess what the probability would be what would you say? I would say that it is close to zero. Or it would be one in a million. If you thought like I did you couldn't be more wrong. (But if you didn't then hats off!). I left the problem for a long time and didn't attempt to solve it untill I showed it to Mr. DVS Ravikant over dinner at Sangam and within a minute he said: " Oh I see.. ". (Well you cant be JEE 13 for nothing)

So I decided to give it another try and I got the way to do it in about a day. I was surprised at the result: it was 8.87% !!

The point of telling the story is that if you have the right intuition of what is going on in the problem then you already solved the problem. The rest is routine algebra.

My intuition was that there are so many outcomes to the problem, i.e the end result of the knot tying can be one of the plethora of possibilities that can be realized. And I thought all are equally probable. But later I realized I was wrong. The fact is it is more probable that the man ends up with a 100 noodle garland than any of the other possible combinations(Can you guess why?).

Let me tell you another example(Its a cheeky problem. Don't curse me after you read the solution. Ravi thought it couldn't be solved, But I persisted on it for a while and then it struck me..)

A Pack of cards has 52 cards. You are in a dark room with this pack of cards. You have been told that inside the pack there are 42 cards facing down, 10 cards facing up. You have been asked to reorganize this pack of cards into two decks so that eack deck contains an equal number of cards that face up.

Remember you are in darkness and can't see. How will you do it?

Hint:(It was given so in the problem although I wouldn't call it a hint)
The two decks need not contain an equal number of cards.

Well Ravi and me were going over this again in sangam and it was quite obvious to Ravi right in the first sight that it cannot be split since no matter how you slpit the pack you can never ensure that there are equal number of cards facing up.

But then that is the problem isn't it?

I pondered on it for a while and then it struck me. You can flip the cards can you not? There was the solution. Just pick 10 cards at random and flip them. The remaining 42 cards and this pack of 10 would contain an equal number of cards that face up.(can you guess why?)

more of these can be found at
http://www.cisl.columbia.edu/grads/tuku/brainteasers.html