Hats, Parties, and Driving Cars (Mathematical Puzzles/Riddles, Part III)

Hi, and happy (late) new year! The holidays are drawing to a close (or have already closed) and classes are starting (or have already started). In case you don’t want to think about all of that, here are some more neat puzzles/riddles on the usual subjects (hats, etc.) Enjoy!

ANOTHER HAT PROBLEM

A hundred mathematicians have traveled to a hattery for the annual conference on colored hat problems. The following game is presented to them. Every mathematician receives a card with a distinct random real number. These cards are taped to the mathematicians’ foreheads, so that each mathematician can see the numbers of all ninety-nine of his or her colleagues, but cannot see his or her own assigned number. The hundred mathematicians then each individually select either a white hat or a black hat from the haberdashery’s infinite collection of white and black hats. The mathematicians are lined up by the proprietor of the hattery in increasing order of the numbers on their cards. If the hats on their heads are alternating in color (i.e., BWBWB … or WBWBW …), then they all win a lifetime collection of hats. Can the mathematicians agree on a strategy beforehand to guarantee a win?

Note that the mathematicians cannot talk to each other while selecting their hats, nor can they see what hat colors the other mathematicians are selecting. Their strategy, of course, can involve each mathematician choosing a color based on the numbers they see on their colleagues’ foreheads.

For example, if there were only two mathematicians, say Alice and Bob, then the game would be trivial – Alice would agree beforehand to choose a white hat and Bob a black hat (or vice-versa). What about for three mathematicians?

A HAND-SHAKING PROBLEM

You and your spouse are at a party of five couples (including you). As the party goes on, various introductions and hand-shakings occur. You observe that nobody shakes hands with their spouse, as presumably they already know them. At some point, you stop the party and ask each of the nine other people how many hands they have shaken. You are surprised to obtain nine distinct answers (i.e., one person shook no hands, one person shook hands with exactly one other person, one person shook hands with exactly two other people, and so on). How many people did your spouse shake hands with?

GASOLINE ON A RACETRACK

You are driving a car at constant speed around a circular racetrack. Distributed around the racetrack are cans of gasoline. The net amount of gasoline on the racetrack is precisely enough to drive the car once around the track, but the gasoline has been divided into many small containers and scattered along the course of the ride. You start with no gasoline, but as you pass each can of gasoline you (instantaneously) pick it up and put it in the gas tank. You are allowed to start your ride at any point along the track. Is it always possible to complete a full lap?

For example, if the gasoline were divided into two containers, then you could start at whichever container had more gas and drive in the direction for which the distance to the other was shortest. (Check that this always makes it possible to complete a full lap!) In general, the gasoline may be divided into many containers of unequal size.

Special thanks this month to M. Miller and L. Alpoge for discussion and communication of the above riddles! If you have a riddle or puzzle that you think is cool, send it along (with a solution, if you like) and we may decide to post it.