How many conversations?

We have a network of 6 computers that have some pre-loaded software on them that help in conversing with other computers. A conversation means exchange of information between the 2 computers and thus a single conversation helps in both computers knowing what information is present on each of their disks. If at some point of time all the computers have something new to share with the others, then at the most how many conversations should occur between the computers until all of them have all the information? By conjecture how many such conversations should occur at the most for a 'n' connected computer network? Do not assume any relation to an actual computer network problem, hence don't apply your network protocol skills here. Its a pure combinatorics problem!

Difficulty: Medium

Partitioning

You blindfolded and let into a room. The room has an infinitely many quarters scattered around on the floor. your friend tells you that that 20 of these quarters are tails and the rest are heads. He also says that if you can split the quarters into 2 piles where the number of tails is the same in both piles, then you win all of the quarters. You are allowed to move the quarters and to flip them over, but you can never tell what state a quarter is currently in (the blindfold prevents you from seeing, and you cannot tell by feeling it). How do you go about partitioning the quarters so that you can win all of them?

Source: William Wu's puzzle forum

Male-Female balance

In a certain land to increase the number of females so as the females can outnumber the males, a ruler, ordered the following: "As soon as a mother gave birth to her first son, she would be forbidden to have any more chilren." the ruler argued that some families can have more girls but no family would have more than one boy thereby creating a higher ratio of girls to boys. Now do you really think the ruler's strategy would work? Why and why not?

Profit and Loss

If I sold a bike to my friend for $100 and my friend sells it back to me for $80 and I sell it back to him for $90, how much total profit or loss have I made or incurred?

The Evil Shylock

There lived a farmer, his daughter, and a broker. The farmer for some reason borrowed a lot of money from the broker by pawning the only farm, the farmer had. The time given to the farmer to repay the money and get his farm released was over and the broker, the evil shylock, proposed the following: he would place a black stone and a white stone in a bag, and the farmer's daughter would need to pick out one of the stones from the bag in front of the entire town. If the daughter drew the white stone, the broker would return the farmer's farm and forgive him of the money. If she drew the black stone, the broker would marry the farmer's daughter and take the farm as well. The farmer had no choice but to agree. The farmer's daughter does not trust the broker and she knows that the broker might as well place 2 black stones in the bag. How can she get out of marrying the broker and save the farm for her father?

What's the deal with 0.999999999999....

Compare the numbers 0.99999... (infinitely many 9s) and 1. Which of the following statements is true? Why?

* 0.99999 ... <> 1

Birthday matching

At a movie theater, the manager announces that he would give a free ticket to the first person in line whose birthday is the same as someone who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday and that birthdays are distributed randomly throughout the year what position in line gives you the greatest chance of being the first duplicate birthday so you can get a free ticket?