## Thursday, March 3, 2011

### How many has Same Birthday Hardest Puzzle

How many people must be gathered together in a room, before you can be certain that there is a greater than 50/50 chance that at least two of them have the same birthday?

### A Man In The Island Covered In Forest

A man is stranded on an island covered in forest.

One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole

island, killing the man in the process.

There are cliffs around the island, so he cannot jump off.

How can the man survive the fire? (There are no buckets or any other means to put out the fire)

One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole

island, killing the man in the process.

There are cliffs around the island, so he cannot jump off.

How can the man survive the fire? (There are no buckets or any other means to put out the fire)

## Tuesday, March 1, 2011

### Bad King & Prisoners

A bad king has a cellar of 1000 bottles of delightful and very expensive

wine. A neighboring queen plots to kill the bad king and sends a servant to

poison the wine. (un)fortunately the bad king’s guards catch the servant

after he has only poisoned one bottle. Alas, the guards don’t know which

bottle but know that the poison is so strong that even if diluted 1,000,000

times it would still kill the king. furthermore, it takes one month to have

an effect. The bad king decides he will get some of the prisoners in his

vast dungeons to drink the wine. Being a clever bad king he knows he needs

to murder no more than 10 prisoners - believing he can fob off such a low

death rate - and will still be able to drink the rest of the wine at his

anniversary party in 5 weeks time.

explain how….

wine. A neighboring queen plots to kill the bad king and sends a servant to

poison the wine. (un)fortunately the bad king’s guards catch the servant

after he has only poisoned one bottle. Alas, the guards don’t know which

bottle but know that the poison is so strong that even if diluted 1,000,000

times it would still kill the king. furthermore, it takes one month to have

an effect. The bad king decides he will get some of the prisoners in his

vast dungeons to drink the wine. Being a clever bad king he knows he needs

to murder no more than 10 prisoners - believing he can fob off such a low

death rate - and will still be able to drink the rest of the wine at his

anniversary party in 5 weeks time.

explain how….

### Light Bulbs & Switch Puzzles

You are standing in a hallway next to three light switches, all of which are

off. Each

switch operates a different incandescent light bulb in the room at the end

of the hall.

You cannot see the lights from where the switches are. Determine which light

corresponds to each switch. You may go into the room with the lights only

once.

off. Each

switch operates a different incandescent light bulb in the room at the end

of the hall.

You cannot see the lights from where the switches are. Determine which light

corresponds to each switch. You may go into the room with the lights only

once.

## Monday, February 28, 2011

### Infinite Balls Paradox

You have a basket of infinite size (meaning it can hold an infinite number of objects). You also have an infinite number of balls, each with a different number on it, starting at 1 and going up (1, 2, 3, etc...).

A genie suddenly appears and proposes a game that will take exactly one minute. The game is as follows: The genie will start timing 1 minute on his stopwatch. Where there is 1/2 a minute remaining in the game, he'll put balls 1, 2, and 3 into the basket. At the exact same moment, you will grab a ball out of the basket (which could be one of the balls he just put in, or any ball that is already in the basket) and throw it away.

Then when 3/4 of the minute has passed, he'll put in balls 4, 5, and 6, and again, you'll take a ball out and throw it away.

Similarly, at 7/8 of a minute, he'll put in balls 7, 8, and 9, and you'll take out and throw away one ball.

Similarly, at 15/16 of a minute, he'll put in balls 10, 11, and 12, and you'll take out and throw away one ball.

And so on....After the minute is up, the genie will have put in an infinite number of balls, and you'll have thrown away an infinite number of balls.

Assume that you pull out a ball at the exact same time the genie puts in 3 balls, and that the amount of time this takes is infinitesimally small.

You are allowed to choose each ball that you pull out as the game progresses (for example, you could choose to always pull out the ball that is divisible by 3, which would be 3, then 6, then 9, and so on...).

You play the game, and after the minute is up, you note that there are an infinite number of balls in the basket.

The next day you tell your friend about the game you played with the genie. "That's weird," your friend says. "I played the exact same game with the genie yesterday, except that at the end of my game there were 0 balls left in the basket."

How is it possible that you could end up with these two different results?

A genie suddenly appears and proposes a game that will take exactly one minute. The game is as follows: The genie will start timing 1 minute on his stopwatch. Where there is 1/2 a minute remaining in the game, he'll put balls 1, 2, and 3 into the basket. At the exact same moment, you will grab a ball out of the basket (which could be one of the balls he just put in, or any ball that is already in the basket) and throw it away.

Then when 3/4 of the minute has passed, he'll put in balls 4, 5, and 6, and again, you'll take a ball out and throw it away.

Similarly, at 7/8 of a minute, he'll put in balls 7, 8, and 9, and you'll take out and throw away one ball.

Similarly, at 15/16 of a minute, he'll put in balls 10, 11, and 12, and you'll take out and throw away one ball.

And so on....After the minute is up, the genie will have put in an infinite number of balls, and you'll have thrown away an infinite number of balls.

Assume that you pull out a ball at the exact same time the genie puts in 3 balls, and that the amount of time this takes is infinitesimally small.

You are allowed to choose each ball that you pull out as the game progresses (for example, you could choose to always pull out the ball that is divisible by 3, which would be 3, then 6, then 9, and so on...).

You play the game, and after the minute is up, you note that there are an infinite number of balls in the basket.

The next day you tell your friend about the game you played with the genie. "That's weird," your friend says. "I played the exact same game with the genie yesterday, except that at the end of my game there were 0 balls left in the basket."

How is it possible that you could end up with these two different results?

### Gameshow:Three Doors, 1 Prize

You are on a gameshow and the host shows you three doors. Behind one door is a suitcase with $1 million in it, and behind the other two doors are sacks of coal. The host tells you to choose a door, and that the prize behind that door will be yours to keep.

You point to one of the three doors. The host says, "Before we open the door you pointed to, I am going to open one of the other doors." He points to one of the other doors, and it swings open, revealing a sack of coal behind it.

"Now I will give you a choice," the host tells you. "You can either stick with the door you originally chose, or you can choose to switch to the other unopened door."

Should you switch doors, stick with your original choice, or does it not matter?

Answer

You should switch doors.

There are 3 possibilities for the first door you picked:

1. You picked the first wrong door - so if you switch, you win

2. You picked the other wrong door - again, if you switch, you win

3. You picked the correct door - if you switch, you lose

Each of these cases are equally likely. So if you switch, there is a 2/3 chance that you will win (because there is a 2/3 chance that you are in one of the first two cases listed above), and a 1/3 chance you'll lose. So switching is a good idea.

Another way to look at this is to imagine that you're on a similar game show, except with 100 doors. 99 of those doors have coal behind them, 1 has the money. The host tells you to pick a door, and you point to one, knowing almost certainly that you did not pick the correct one (there's only a 1 in 100 chance). Then the host opens 98 other doors, leave only the door you picked and one other door closed. We know that the host was forced to leave the door with money behind it closed, so it is almost definitely the door we did not pick initially, and we would be wise to switch.

You point to one of the three doors. The host says, "Before we open the door you pointed to, I am going to open one of the other doors." He points to one of the other doors, and it swings open, revealing a sack of coal behind it.

"Now I will give you a choice," the host tells you. "You can either stick with the door you originally chose, or you can choose to switch to the other unopened door."

Should you switch doors, stick with your original choice, or does it not matter?

Answer

You should switch doors.

There are 3 possibilities for the first door you picked:

1. You picked the first wrong door - so if you switch, you win

2. You picked the other wrong door - again, if you switch, you win

3. You picked the correct door - if you switch, you lose

Each of these cases are equally likely. So if you switch, there is a 2/3 chance that you will win (because there is a 2/3 chance that you are in one of the first two cases listed above), and a 1/3 chance you'll lose. So switching is a good idea.

Another way to look at this is to imagine that you're on a similar game show, except with 100 doors. 99 of those doors have coal behind them, 1 has the money. The host tells you to pick a door, and you point to one, knowing almost certainly that you did not pick the correct one (there's only a 1 in 100 chance). Then the host opens 98 other doors, leave only the door you picked and one other door closed. We know that the host was forced to leave the door with money behind it closed, so it is almost definitely the door we did not pick initially, and we would be wise to switch.

### Gameshow:Three Doors, 1 Prize

You are on a gameshow and the host shows you three doors. Behind one door is a suitcase with $1 million in it, and behind the other two doors are sacks of coal. The host tells you to choose a door, and that the prize behind that door will be yours to keep.

You point to one of the three doors. The host says, "Before we open the door you pointed to, I am going to open one of the other doors." He points to one of the other doors, and it swings open, revealing a sack of coal behind it.

"Now I will give you a choice," the host tells you. "You can either stick with the door you originally chose, or you can choose to switch to the other unopened door."

Should you switch doors, stick with your original choice, or does it not matter?

You point to one of the three doors. The host says, "Before we open the door you pointed to, I am going to open one of the other doors." He points to one of the other doors, and it swings open, revealing a sack of coal behind it.

"Now I will give you a choice," the host tells you. "You can either stick with the door you originally chose, or you can choose to switch to the other unopened door."

Should you switch doors, stick with your original choice, or does it not matter?

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