# Uptown Girl

A man has two girlfriends, one who lives uptown and the other downtown. He likes them equally, so he lets the trains decide which he will visit: He arrives at the train station at random times and takes whichever train arrives first.

Over time, he finds that he’s visiting the uptown girlfriend much more often than the downtown girlfriend, even though uptown and downtown trains arrive at the station equally often. Why?

# Water Battle

An odd number of people armed with water guns are standing in a field so that all the pairwise distances are distinct. At a signal, each shoots at his nearest neighbor and hits him. Prove that one person doesn’t get wet.

# Six Choices

On a multiple-choice test, one of the questions is illegible, but the choice of answers is listed clearly below. What’s the right answer?

(a) All of the below.
(b) None of the below.
(c) All of the above.
(d) One of the above.
(e) None of the above.
(f) None of the above.

# Black and White

From Stratagems of Chess, 1817. White to mate in two moves.

# Paint Scheme

How many colors are necessary to paint the squares of a chessboard so that no bishop can move between two squares of the same color?

# Black and White

By J. Berger. White to mate in two moves.

# Fowl Play

From a 1947 competitive examination for high school seniors conducted by Stanford’s math department:

My grandfather’s papers included an old invoice:

72 turkeys \$-67.9-

The first and last digits are illegible. What are the missing digits, and what was the price of one turkey?

# Rolling

I drove this car 20,000 miles and used five tires equally in accumulating the mileage. How many miles’ wear did each tire sustain?

# Black and White

By J.H. Graham. White to mate in two moves.

# Open and Shut

A warden oversees an empty prison with 100 cells, all closed. Bored one day, he walks through the prison and opens every cell. Then he walks through it again and closes the even-numbered cells. On the third trip he stops at every third cell and closes the door if it’s open or opens it if it’s closed. And so on: On the nth trip he stops at every nth cell, closing an open door or opening a closed one. At the end of the 100th trip, which doors are open?