Favorite Logic Problems

1.    You have 12 balls which look exactly the same.   Eleven of them have the same weight, but one of them is either lighter or heavier than the others (you don't know which).  Using a balance scale, you can determine which ball is the "odd ball" (and whether it is too light or too heavy) in three weighings.  How do you do this?

2.*    Indiana Jones needs to cross a flimsy rope bridge over a mile-long gorge.  It is so dark that it is impossible to cross the bridge without a flashlight.  Furthermore, the bridge is so weak that it can only support the weight of two people.  The party has just one flashlight, which has a weak beam, so whenever two people cross, they are constrained to walk together, at the speed of the slower person.  Indiana Jones can cross the bridge in 5 minutes.  His girlfriend can cross in 10 minutes.  His father needs 20 minutes, and his father's sidekick needs 25 minutes.  They need to get everyone across safely in one hour to escape the bad guys.  Can they do it?

3.*    A group of jealous professors are locked up in a room.  There is nothing else in the room but pencils and one tiny scrap of paper per person.  The professors want to determine their average (mean, not median) salary so that each one can gloat or grieve over their personal situation compared to their peers.   However, they are secretive people and do not want to give away any personal salary information to anyone else.  Can they determine the average salary in such a way that no professor can discover any fact about the salary of anyone but himself or herself?  (For example, even facts such as "3 people earn more than $40,000" or "no one earns more than $90,000" are not allowed.)

*  From The Art and Craft of Problem Solving by Paul Zeitz, John Wiley and Sons, Inc., 1999.