## MARBLE JARS I

You are a prisoner in a foreign land, and your fate will be determined by a little game. There are two jars, one with 50 white marbles and one with 50 black marbles. You are allowed to redistribute the marbles however you wish; the only requirement is that after you are done with the redistribution, every marble must be in one of the two jars. Afterwards, both jars will be shaken up, and you will be blindfolded and presented with one of the jars at random. Then you pick one marble out of the jar given to you. If the marble you pull out is white, you live; if black, you die. How should you redistribute the marbles to maximize the probability that you live, and what is this maximum probability (roughly)?

## CORK, BOTTLE, COIN

If you were to put a coin into a bottle of wine and then insert a cork in the bottle's opening, how could you remove the coin without touching the cork or breaking the bottle?

## ALL HORSES ARE THE SAME COLOR

Theorem: All horses are the same color.

Base Case: 1 horse. Clearly with just 1 horse, all horses have the same color.

Inductive Step: If it is true for any group of N horses that all have the same color, then it is true for any group of N+1 horses.

Proof: Given any set of N+1 horses, if you exclude a random horse, you get a set of N horses. By the inductive step these N horses all have the same color. But by excluding any other horse in the pack of N+1 horses, you can conclude that the remaining N horses also have the same color. Therefore all N+1 horses have the same color. QED

What is wrong with this theorem?

## PUNCTUATION I

Add punctuation to the following phrase to make something gramatically and logically coherent:

is is not not not is not is is is is not is not is it not

## BROWN EYES AND RED EYES

There is an island of monks where everyone has either brown eyes or red eyes. Monks who have red eyes are cursed, and are supposed to commit suicide at midnight. However, no one ever talks about what color eyes they have, because the monks have a vow of silence. Also, there are no reflective surfaces on the whole island. Thus, no one knows their own eye color; they can only see the eye colors of other people, and can not say anything about them. Life goes on, with brown-eyed monks and red-eyed monks living happily together in peace, and no one ever committing suicide. Then one day a tourist visits the island monastery, and, not knowing that he's not supposed to talk about eyes, he states the observation "At least one of you has red eyes." Having acquired this new information, something dramatic happens among the monks. What happens?

## WATER BUCKETS

Using only a 5-gallon bucket and a 3-gallon bucket, put exactly four gallons of water in the 5-gallon bucket. (Assume you have an infinite supply of water. No measurement markings on the buckets.)

## FORCEFIELD DETAINMENT

A group of prisoners are trapped in a forcefield. These prisoners are perfectly brave, meaning that each of them would attempt an escape on any positive probability of personal success. The prisoners are monitored by a guard who has only one bullet in his gun, but who is also renowned for his perfect marksmanship skills (he never misses). A maintenance technician needs to tune up the forcefield generator, and so for one second, the forcefield is released. How can the guard still keep all the prisoners detained?

## PENNY TRACK GAME

There are three one-dimensional tracks, of length 12, 7, and 5 spaces respectively. You start with pennies in the first space of each track; your opponent starts with pennies in the last space of each track. On your turn, you may move any one of your pennies any number of spaces in either direction along its track, however you are not permitted to bypass the other player's penny or occupy its space. If a player has no legal move, he loses.

What should your first move be?

## ACROBAT THIEF AND GOLD ROPES

An acrobat thief enters an ancient temple, and finds the following scenario:

- The roof of the temple is 100 meters high.
- In the roof there are two 1" holes, separated by 1 meter.
- Through each hole passes one gold rope, each rope going all the way to the floor.
- There is nothing else in the room.

The thief would like to cut and steal as much of the ropes as he can. However, he knows that if he falls from height that is greater than 10 meters, he will die. The only thing in his possession is a knife.

How much length of rope can the acrobat thief get? And how?

## 5 CARD MAGIC TRICK

This is a magic trick performed by two magicians, A and B, with one regular, shuffled deck of 52 cards. A asks a member of the audience to randomly select 5 cards out of a deck. the audience member -- who we will refer to as C from here on -- then hands the 5 cards back to magician A. After looking at the 5 cards, A selects one of the 5 cards and gives it back to C. A then arranges the other four cards in some way, and gives those 4 cards face down, in a neat pile, to B. B looks at these 4 cards and then determines what card is in C's hand (the missing 5th card). How is this trick done?

## TRUTHS, FALSEHOOD, RANDOMNESS

Of three men, one man always tells the truth, one always tells lies, and one answers yes or no randomly. Each man knows which man is who. You may ask three yes/no question to determine who is who. If you ask the same question to more than one person you must count it as question used for each person whom you ask. What three questions should you ask in order to determine which man is which?

## DUCK IN THE POND

A duck is in a circular pond with a menacing cat outside. The cat runs four times as fast as the duck can swim, but cannot enter the water. The cat starts at the closest possible point to the duck that it can get. Can the duck get to the perimeter of the pond without the cat catching him?

## UNIVERSAL TRUTH MACHINE

Someone claims to have invented a Universal Truth Machine (UTM), a machine that takes a proposition as input, and returns "true", "false", or "undecidable" as output. Example:

input output
----------- -------------
"2 + 2 = 4" "true"
"0 + 2 = 4" "false"
"this proposition is false" "undecidable"

Devise a true proposition that the UTM will claim to be false or a
false proposition that the UTM will claim to be true, thereby
disproving the inventor's claim.

Perhaps not too surprisingly (since I held almost 1/3 of the 22 tickets) I won the lottery and am thus the Meta Devil and will design next week's Meta Game!