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Math Horizons - February 2003

Articles

The Canadians Should Have Won!?

What determines an Olympic champion? Perhaps the tallying method.

Fitch Cheney's Five Card Trick

Amaze your friends with this telepathic trick.

The Professional Master's Degree

A new option for math majors.

The Card Game

Make friends and influence people with a little logic.

Curious Counts

Are you puppeteer or puppet in these perplexing puzzles?

The Isoperimetric Problem

A twelve-step program to solve the isoperimetric problem.

Tracking in Virtual Reality

Nearly-forgotten nineteenth-century mathematics provides the key to twenty-first-century problems.

Problem Section

S-73.

Proposed by Sidney Kung, University of North Florida. Let S denote the sum of the positive real numbers a_1, a_2,..., a_n. Prove that the sum as i goes from 1 to n of sqrt((S-a_i)/a_i) is >= n*sqrt(n-1).

S-74.

Proposed by Sidney Kung, University of North Floridy. The perpendicular bisector of the side BC of triangle ABC meets the bisector of the interior angle A at P and the bisector of the exterior angle A at Q. Prove that P and Q lie on the circumcircle of ABC.

S-75.

Proposed by Andris Cibulis, University of Latvia, Bob Wainwright, New Rochelle, and the editor. Find a region with minimum area which can be tiled by each of the following two polyominoes. (See Math Horizons for the picture of the polyominoes.)

The Final Exam: Puzzle me this


This month's Final Exam features half a dozen great puzzles, the kind of puzzles that your mom might have used to keep you quiet (at least for a little while) on a long trip. The cigarette butt puzzle was sent to us by Professor Ali R. Amir-Moez of Texas Tech University and may be original. We heard the cooling coffee and three light switches puzzles on National Public Radio's Cartalk many years ago. Versions of the others have been circulating in the mathematics community for, in some cases, centuries. 1. There are three light switches on the wall and you are told that one of them controls a lamp that is upstairs (and currently off). You cannot see the light come on when it does, there are too many corners and walls between you and it. You must walk upstairs to check whether the light is on or off. What is the fewest number of times you must walk up the stairs to determine which switch controls the lamp? 2. You are on an island populated by two kinds of folks, knights and knaves. Knights always tell the truth, knaves always lie, they are otherwise indistinguishable. The inhabitants of this island always travel in pairs consisting of one knight and one knave. You come to a fork in the road, standing there is a pair of inhabitants, you wish to determine whether to go left or right at this fork to get back to your hotel. What one question can you ask that will tell you which choice to make? 3. You have a balance scale and twelve coins. Exactly one of the coins is counterfeit, and therefore, either heavier or lighter than a genuine coin, but you don't know which. Using only three weighings, can you identify the counterfeit coin and decide whether it is too heavy or too light? 4. You have an inexhaustible supply of water and two pails, one holds nineteen gallons, the other seven. How do you use them to measure one gallon of water? 5. I like my morning coffee to cool a little before I drink it, so I usually pour it and then I come back and drink it. Will my coffee be cooler if I add the cream before I shave, or after? 6. A hobo can make a new cigarette out of three butts. Suppose a hobo has ten butts, is it possible for him to smoke five cigarettes? Here are our solutions. We make no claims of uniqueness and only mild claims of accuracy! 1. This puzzle is hard for a mathematician and trivial for a physicist. The required task can be accomplished with only one trip up the stairs. Label the switches A, B, and C. Flip switch A and wait a minute. Then turn off switch and turn on switch B and go upstairs. If the lamp is on, then B controls it. If the lamp is off, feel the bulb: if it's warm, then it's controlled by switch A; if it's cold, then switch C. 2. The trick here is to get your answer to pass through both guys, then you know that whatever answer you get, it must be false, because it passed through the knave. Ask one of them something like, "If I were to ask your companion whether the left fork would take me back to my hotel, what would his answer be?" 3. This is a truly difficult puzzle, pat yourself on the back if you managed to solve it. Begin by dividing the coins into four sets of three coins each, call them A={1,2,3}, B={4,5,6}, C={7,8,9}, and D={10,11,12}. First weigh A and coin 10 against B and coin 11. Record the result. Then weigh A and coin 11 against C and coin 10. At this point you will know which set contains the fake coin and if the set is one of A, B or C, you will know whether the fake is too heavy or too light. It's easy to see this by just listing all possible outcomes, which I won't do here, but imagine, for example, that on your first weighing A+10 out weighed B+11 and on your second that A+11 outweighed C+10. You now know that A contains the fake and that it is too heavy. Now it's easy to finish in one weighing, just weigh any coin in A against any other. If the false coin is in D, then your infornation will be different, but still sufficient to decide in one more weighing. Consider, for example, what you would know if your first two weighings both balanced: coin 12 must be the counterfeit, but you don't know whether it is too heavy or too light, this is easily determined by weighing it against any other coin. For another example, if A+10 outweighs B+11 and C+10 outweighs A+11, then either 10 or 11 is counterfeit. If 10 is fake, it's too heavy. If 11 is the fake it's too light. But it's easy to determine which is true by just weighing, e.g., 10 against 12, which is known to be real. 4. This is a puzzle with nice number-theoretic pedigree. Notice that 3*19-8*7=1. So, you want to fill up the 19-gallon bucket three times and empty out the 7-gallon bucket eight times. Is it obvious, from here? Fill the 19-gallon bucket, use it to fill the 7-gallon bucket. Empty out the 7-gallon bucket, refill it from the 19 and empty again. Now pour the remaining five gallons from the 19 into the 7. Refill the 19. Use it to top off the 7, which you then empty, keep filling the 7-gallon bucket from the 19-gallon bucket and pouring it off (and refilling the 19-gallon when it's empty) until you've filled the 19-gallon three times and dumped the 7-gallon eight times. 5. Thinking like a physicist helps here. The total amount of "heat energy" that you put into the cup is the same no matter when you add the cream. So, the question really is: under which scenario does more heat radiate away from the cup? According to Newton's Law of Cooling (and common sense) a hotter object radiates more heat than a cooler one; thus, you should refrain from adding cream until it's time to drink the coffee, it will be cooler. 6. This is not really a math problem, maybe sociology, or human relations-but it's charming. Our hobo can make and smoke three cigarettes from nine of the butts he's got. After that he'll have four butts, one from each cigarette plus his original tenth. So he can use three of them to make another cigarette, smoke it, and have two butts left. To get his fifth smoke he should borrow a butt from a fellow hobo, make and smoke a cigarette, and then give the butt back to his friend.