Mathematical Brain Teasers and Logic Puzzles

Why do we create puzzles simply for the pleasure of solving them? After proposing a few theories, Professor Rosenhouse notes that mathematicians love puzzles, especially those that lead to deep mathematical insights. Get warmed up for the series with six brain teasers involving hourglasses, a restaurant order, a biased coin, the numbers on a clock face, and two chessboard scenarios.

Test your wits against the puzzle that likely inspired the famous expression “thinking outside the box.” Then apply this strategy to a variety of brain teasers, involving matchsticks, cards, light switches, and other objects in interesting and puzzling situations. Also ponder the legendary physics exam question: How can you find the height of a building by using a barometer?

First, find a shortcut solution to a classic word problem in algebra. This introduces the episoide’s theme: forget your algebra and use cleverness to solve problems without x’s and y’s. Along the way, you’ll learn that sometimes having too much information can make a problem harder. Also find out why transcontinental flights take longer in one direction than the other (not counting wind effects).

Now turn to logic puzzles, trying to distinguish between knights who only make true statements, and knaves who only tell falsehoods. Start with simple cases. Then introduce tricky “if–then” statements. Next, what if the knight or knave is insane and thus has false beliefs? This makes things trickier! Finally, add a third category: normal people who are sometimes truthful, sometimes not.

Lewis Carroll, author of Alice’s Adventures in Wonderland, wrote a book of logic puzzles for children. Take a crack at some of these fun exercises, which Carroll designed to illustrate the principles of Aristotelian logic. See what you can conclude from such categorical statements as “all wasps are unfriendly, and all puppies are friendly.” Carroll’s syllogisms get progressively more elaborate.

Return to the Island of Knights and Knaves to consider puzzles where asking the right questions is the point of the problem. Work your way up to the famous “heaven or hell” puzzle. Then close with an exercise in coercive logic, devised by noted mathematician and puzzle master Raymond Smullyan. Easy riches hinge on a very simple bargain that sounds too good to be true. Do you accept?

Learn about biconditional statements of the form, “p if and only if q.” Then tackle the “Hardest Logic Puzzle Ever,” devised by philosopher George Boolos. You have three yes/no questions to identify three gods: the god who always answers truthfully, the god who always lies, and the god who randomly mixes true and false answers. One big problem: They answer in a language you don’t speak.

Finish your study of logic with puzzles where you must draw conclusions based on what other people can infer from information they are given. Your first example is the “muddy children” puzzle, in which children with muddy faces must conclude with logical certainty (without looking in a mirror, feeling their faces, or being told) that they have muddy faces. Such puzzles are unusually subtle.

Ponder probability, starting with the chances of getting an ace of spades when you turn over the top cards on two well-shuffled decks. In probability, it’s a safe bet that your first instinct is wrong! Investigate other phenomena, including the chances that your suitcase is lost when 98 percent of the luggage has arrived at baggage claim, but yours has not.

Study the famous Monty Hall problem from the game show Let’s Make a Deal. Your quandary: A new car is hidden behind one of three doors; after making your choice, your door is left shut and one of the doors without the car is opened. Do you care to switch to the other closed door? Find out why one expert says, “No other statistical puzzle comes so close to fooling all the people all the time.”

Discover the fun of arithmetic and other simple mathematics. Begin with the game Krypto. Then try out the “four fours” puzzle. Next, see how perfect squares and perfect triangles reveal algebra and geometry working together. Finally, reason out why a negative number multiplied by itself a is a positive number.

Take up algorithmic puzzles, which require a carefully thought-out procedure (or algorithm) to solve. Algorithms have notable applications in computer science, but they also come in handy for dividing pirate gold, transporting hungry animals to an island, and solving life-or-death riddles posed by movie villains. At least, that’s the entertaining approach you take in this final episode.