Interesting/tricky problems: (Back to home)

1. What is the probability of  two natural numbers (positive integers) be co-prime?  

2. Egg problem:

* You are given 2 eggs.
* You have access to a 100-storey building.
* Eggs can be very hard or very fragile means it may break if dropped from the first floor or may not even break if dropped from 100 th floor.Both eggs are identical.
* You need to figure out the highest floor of a 100-storey building an egg can be dropped without breaking.
* Now the question is how many drops you need to make. You are allowed to break 2 eggs in the process.

3. Door Toggling Puzzle Or 100 Doors Puzzle

There are N doors in a row numbered from 1 to N. Initially all are closed.
Then you make N passes by the N doors. In pass 1 you toggle the all the doors (1,2,3,4....)starting from the first door. In the second pass you toggle every second door(2,4,6,8,...). In the third pass you toggle all third doors(3,6,9...).Similarly you make N passes. Question is what is the state of door k after N passes.

4. (Ball Weighting Problem)You are given 13 balls.The odd ball may be either heavier or lighter.Find out the odd ball in 3 weightings.
More generally, you are given N balls, at least how many weightings are required to get the odd ball?

5. (Two Coin Flipping Problem, by Matt McCutchen) Someone flips a coin repeatedly. She stops when she gets either 3 heads in a row or 8 tails in a
row. What is the probablity that she gets 3 heads in a row?

6. (A Puzzle for Pirates, by Ian Stewart) Ten pirates have gotten their hands on a hoard of 100 gold pieces and wish to divide the loot. They are democratic pirates, in their own way, and it is their custom to make such divisions in the following manner: The fiercest pirate makes a proposal about the division, and everybody votes on it, including the proposer. If 50 percent or more are in favor, the proposal passes and is implemented forthwith. Otherwise the proposer is thrown overboard, and the procedure is repeated with the next fiercest pirate. All the pirates enjoy throwing one of their fellows overboard, but if given a choice they prefer cold, hard cash. They dislike being thrown overboard themselves. All pirates are rational and know that the other pirates are also rational. Moreover, no two pirates are equally fierce, so there is a precise pecking order— and it is known to them all. The gold pieces are indivisible, and arrangements to share pieces are not permitted, because no pirate trusts his fellows to stick to such an arrangement. It’s every man for himself. What proposal should the fiercest pirate make to get the most gold? For convenience, number the pirates in order of meekness, so that the least fierce is number 1, the next least fierce number 2 and so on. The fiercest pirate thus gets the biggest number, and proposals proceed in reverse order from the top down.