RAY: You're sitting at a table with a bunch of pennies on it. Some are facing heads up and some are facing tails up. You're wearing a blindfold, and you're wearing mittens so that you can't actually feel the coins and tell which side is facing up. If we could trust you, we'd let you do it without gloves.

TOM: Well, you know me. I always sit around with mittens and a mask on anyway.

RAY: I will tell you that a certain number of the pennies are facing heads up. Let's say 10 are facing heads up.

TOM: You're telling me there are a bunch of pennies in front of me and 10 of them are heads up?

RAY: Right. With your mittens on, you can move the pennies around, you can pick them up, you can put them down again, you can shake them, you can do whatever you want. Here's the question: Is it possible to separate those pennies into two groups, so that each group has the same number of pennies facing heads up? How do you do it?

Five married couples attended a party. One of the ten people was a mathematician. During the party, many people shook hands, but nobody shook hands with his/her spouse, and nobody shook hands with him/herself.

The mathematician asked each of the other people, "With how many people did you shake hands?" Each person gave a different answer.

With how many people did the mathematician's spouse shake hands?

An oracle chooses two whole numbers from the range [2,100]. (That means the numbers can include 2 and 100, and the two numbers can be the same) There are two perfect logicians, Sum and Product. The oracle gives Sum the sum of the two numbers and Product the product of the two numbers. They have a small conversation:

**Product:**I don't know what the numbers are.**Sum:**I knew you didn't know what the numbers were.**Product:**I know what the numbers are.**Sum:**I know what the numbers are.

Welcome to perfect physics world. Imagine a circular frictionless bracelet with numbered balls randomly placed on it as in the example to the right. Each ball is traveling either clockwise or counter-clockwise at the same constant angular velocity. All collisions are elastic, i.e., no energy is lost and immediately afterwards the balls bounce back from whence they came. The balls do not compress during collision, and have non-zero diameter.

The initial condition is simply the position and direction of travel of the balls. Assume it would take one unit of time for one ball to travel a full revolution without collisions.

**Questions**

- Are you
**guaranteeed**that the initial condition will eventually repeat? I.e., ball number 1 is at the same position going the same direction, similarly for all the other balls. - Prove your answer.
- If you
*can*guarantee that the initial position will repeat, can you predict when it will? I.e., describe, as a closed function of the input position, how long it will take to repeat.

Assume you have n students. You would like to have lab sections in which every student chooses a partner to work together for that particular lab section. Over the course of the semester, you'd like to have every student work with every other student exactly once. As there are n students, this can be done in n-1 days optimally. How is it done? (e.g. show how to pair 20 students up)

- Pick a number from 1 through 9.
- Subtract 5 from the number (negative results are ok)
- Multiply the result by 3.
- Square the result.
- Add the digits of the result until there's only one digit. (Example: 64. 6+4=10. 1+0=1.)
- If the result is less than 5, add 5; if not, subtract 4.
- Multiply the result by 2.
- Subtract 6 from the result.
- Locate the corresponding letter in the alphabet. (Examples: 1=A, 2=B.)
- Pick a country that begins with that letter.
- Note the second letter in that country's name. Think of a mammal that begins with that letter.
- Now think of the color of that mammal. What do you have? (And why did I know it was a
*grey elephant from Denmark*?)

An explorer is in a strange land where the men always tell the truth and the women always lie. He meets three natives but cannot tell their sex by their appearance or voice, so he asks them.

The first one replies, but the explorer doesn't hear the answer. The second one makes the following three statements:

- The first person said, "I am a man"
- The first person is a woman
- I am a man

The third one makes these two statements:

- The second person is a woman
- I am a man

**Which are men, and which are women?**

Divide a sheet of paper into eight parts. Number them on one side as in the diagram. The problem is to fold the paper (along the lines) to form a packet (like a folded map) with No. 1 face up on top, followed by the other numbers in order.

7 | 4 | 3 | 2 |

6 | 5 | 8 | 1 |

A man is walking across a railroad bridge spanning a river. When he is 7/10 of the way across, he hears a train coming. Fortunately for him, no matter which way he runs at top speed, he can just make it to the end and jump clear as the train misses him. The train is traveling 60mph. How fast can the man run?

You meet two natives from different villages at a crossroads. One always lies and one always tells the truth, but you doesn't know which is which. You need to find out whether the left or right path leads to safety, but can only ask one of them a single question. What should you ask (and what should you do depending on what they answer)?

You have 100 lightbulbs, numbered 1-100, and 100 frogs, also numbered 1-100. Whenever a frog jumps on a lightbulb, it toggles between on and off. All lightbulbs are initially off.

- frog #1 jumps on every lightbulb (ie turning them all on).
- frog #2 jumps on every 2nd lightbulb, toggling some of them back off.
- ...
- frog #k jumps on every kth lightbulb.

Only one of John, Alice or Frank has sweets. John says that Alice does not have sweets; Alice says that neither John nor herself has sweets; Frank says that either John or Alice has sweets. If only one of them is telling the truth, who has the sweets?

Given:

ABCD - BDCA ------ 119?

Where `A`,`B`,`C`,`D` are digits, what is
the missing digit, the `?`

Name all the ways to extend a baseball game indefinitely. The methods have to be ones in which the umpires have no say. So a solution in which the pitcher just held the ball and didn't pitch wouldn't work, because the umpires could order the pitcher to pitch the ball and they would have a say.

Recombinant numbers are numbers whose digits can be recombined by simple
mathematical expressions to form the number themselves. e.g. 25 = 5^{2},
125 = 5^{(1+2)}, 625=5^{(6-2)}, 3215=5^{(3+2)*1},
15625=5^{(1+5+6)/2}, 78125=5^{(7*2-8-1)}, 390625=5^{(7*((3*9*0)+6+2)}

Question: Is the set of recombinant numbers finite or infinite?

by David Blackston (davidb@cs.berkeley.edu)

Imagine a chessboard that is 2^{n} by 2^{n} on a side.
Remove one square anywhere on the chessboard. Prove by induction that you
can tile the remaining chessboard with triominoes. (A triomino is an L-shaped
piece containing 3 squares, which could be formed by gluing the chess squares
a1, a2 and b1 together, which looks like:
)

Find a simple counting number (1,2,3,...) denominator, n, which has a repeating decimal equivalent 1/n (e.g. 3 has 1/3 = .333...) but whose multiples (2/n, 3/n, 4/n, etc) are themselves the EXACT same repeating decimal, just starting at a different place in the pattern.

Man #1 says he has two children. One of them is a boy. What's the chance
his other one is a girl? (supplementary question: Man #2 says he has two
children. **His first one** is a boy. What's the chance the other one
is a girl?)

Fold a dollar bill into a triangle using the least number of folds.

Three you're-only-allowed-to-do-these-in-your-head quickies:

- Approx. what is 15/16 plus 7/8? (options: 1, 2, 19, 20)
- What is the sum of all of the integers from 1 through 100 inclusive?
- What is the one's digit of the number 2
^{312}?

You are given a clock with the numbers removed and all but one of the number markers also removed, leaving the minute and hour hands and one of the hour markers (but you don't know which one). The clock is circular, so you can't tell the original orientation of it. With just a protractor (Protractor?! But I just met her!) and your logic, how can you instantaneously determine both the exact time and the original positions and numbering of the other 12 dots?

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