Question: You have a bag with 'N' strings in it. You randomly grab two ends and tie them together until there are no more loose ends.
In the end, what is the expected number of loops (strings tied to their own end)?
Answer: 1 + 1/3 + 1/5 ... + 1/(2N-1).
Each time you tie two together the number of string ends available decreases by 2. The chance of grabbing a string and its end is 1/(2N-1).
Question: There are 1600 people sitting around a circular table. The first person (person 1) has a sword and kills the second person then hands it to the next alive person (in this case person 3). Person 3 stabs person 4 and gives the sword to person 5. This goes on until person 1499 kills person 1500. Then person 1 kills person 3 and so on. This is repeated until there is only a single person remaining.
Who remains in the end?
Answer: Person 1153.
If you have any number of people equal to a power of 2 (2, 4, 8, etc.) then the first person will be the last remaining. The closest power of 2 to 1600 is 1024 (210). So the first person to go of the 1600 when there is 1024 people left will be the last person remaining. 1600 - 1024 = 576. 576 * 2 = 1152. Person 1152 will be the 576th person killed and person 1153 will be the first person to go of the remaining 1024 people.
Question: There are 100 coins scattered in a dark room. 90 have heads facing up and 10 are facing tails up. You cannot tell which coins are which. How do you sort the coins into two piles that contain the same number of tails up coins?
Answer: The piles don't need to be the same size, so make a pile of 10 coins and a pile of 90 coins, flip all of the ten coins and it is guaranteed that the piles have the same number of tails.
Question: Three guests check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 for himself. Each guest got $1 back: so now each guest only paid $9; bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?
Hint: Make a list of all of the people involved and how much money they ended up with/spent.
Answer: The $9 paid by each guest accounts for the $2 that went to the bellhop. So rather than adding $27 to the $2 kept by the bellhop, the $27 accounts for the bellhops money. The $27 plus the $3 kept by the guests does add up to $30.
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