You’re given a regular deck of 52 playing cards.  In the pile you’re given, 13 cards face up and the rest face down.  You are to separate the given cards into two piles, such that the number of face-up cards in each pile is the same.  In separating the cards, you’re allowed to flip cards over.  The catch:  you have to do this in a dark room where you cannot determine whether a card is face up or face down.

Warm-up:  You are given a box of matches and a piece of rope.  The rope burns at the rate of one rope per hour, but it may not burn uniformly.  For example, if you light the rope at one end, it will take exactly 60 minutes before the entire rope has burnt up, but it may be that the first 1/10 of the rope takes 50 minutes to burn and that the remaining 9/10 of the rope takes only 10 minutes to burn.  How can you measure a period of exactly 30 minutes?  You can choose the starting time.  More precisely, given the matches and the rope, you are to say the words “start” and “done” exactly 30 minutes apart.

The actual problem:  Given a box of matches and two such ropes, not necessarily identical, measure a period of 15 minutes.

Boris and Natasha live in different cities in a country with a corrupt postal service.  Every box sent by mail is opened by the postal service, the contents stolen, and the box never delivered. Except: if the box is locked, then the postal service won’t bother trying to open it (since there are so many other boxes whose contents are so much easier to steal) and the box is delivered unharmed.

Boris and Natasha each has a large supply of boxes of different sizes, each capable of being locked by padlocks.  Also, Boris and Natasha each has a large supply of padlocks with matching keys.  The padlocks have unique keys.  Finally, Boris has a ring that he would like to send to Natasha.  How can Boris send the ring to Natasha so that she can wear it (without either of them destroying any locks or boxes)?

Think of a positive integer, call it X.  Shuffle the decimal digits of X, call the resulting number Y.  Subtract the smaller of X,Y from the larger, call the difference D.  D has the following property:  Any non-zero decimal digit of D can be determined from the remaining digits.  That is, if you ask someone to hide any one of the non-zero digits in the decimal representation of D, then you can try to impress the other person by figuring out the hidden digit from the remaining digits.  How is this done?  Why does it work?

Two players are playing a game.  The game board is a circular table.  The players have access to an ample supply of equal-sized circular coins.  The players alternate turns, with each turn adding a single coin to the table.  The coins are not allowed to overlap.  Once a coin is placed on the table, it is not allowed to be moved. The player who has no place to put his next coin loses.  Develop a winning strategy for the player who starts.  (The table is large enough to accommodate at least one coin.)

Find two positive integers that together with 23 are the lengths of a right triangle.

Hint: There’s a simple technique that, given any odd positive integer, allows you to figure out the other two integer sides of a right triangle in your head (or with pen and paper if the numbers get too large).  Find this technique.

For any even number N, partition the integers from 1 to N into pairs such that the sum of the two numbers in each pair is a prime number.

Hint:  Chebyshev proved that the following property (Bertrand’s Postulate) holds:  for any k > 1, there exists a prime number p in the range k < p < 2*k.

At some point during a baseball season, a player has a batting average of less than 80%.  Later during the season, his average exceeds 80%.  Prove that at some point, his batting average was exactly 80%.

Also, for which numbers other than 80% does this property hold?