π数学杂志4

Life and Travel in 4D Life and Travel in 4D Life and Travel in 4DTic-Tetris-Toe Tic-Tetris-Toe Tic-Tetris-Toe S h a r k A t t a c k s S h a r k A t t a c k s S h a r k A t t a c k s December 2001 pi in the Sky is a semi-annual publication of PIMS is supported by the Natural Sciences and Engineer- ing Research Council of Canada, the British Columbia Information, Science and Technology Agency, the Al- berta Ministry of Innovation and Science, Simon Fraser University, the University of Alberta, the University of British Columbia, the University of Calgary, the Univer- sity of Victoria, the University of Washington, the Uni- versity of Northern British Columbia, and the University of Lethbridge. This journal is devoted to cultivating mathematical rea- soning and problem-solving skills and preparing students to face the challenges of the high-technology era. Editors in Chief Nassif Ghoussoub (University of British Columbia) Tel: (604) 822–3922, E-mail: director@pims.math.ca Wieslaw Krawcewicz (University of Alberta) Tel: (780) 492–7165, E-mail: wieslawk@v-wave.com Associate Editors John Bowman (University of Alberta) Tel: (780) 492–0532 E-mail: bowman@math.ualberta.ca Dragos Hrimiuc (University of Alberta) Tel: (780) 492–3532 E-mail: hrimiuc@math.ualberta.ca Volker Runde (University of Alberta) Tel: (780) 492–3526 E-mail: runde@math.ualberta.ca Editorial Board Peter Borwein (Simon Fraser University) Tel: (640) 291–4376, E-mail: pborwein@cecm.sfu.ca Florin Diacu (University of Victoria) Tel: (250) 721–6330, E-mail: diacu@math.uvic.ca Klaus Hoechsmann (University of British Columbia) Tel: (604) 822–5458, E-mail: hoek@math.ubc.ca Michael Lamoureux (University of Calgary) Tel: (403) 220–3951, E-mail: mikel@math.ucalgary.ca Ted Lewis (University of Alberta) Tel: (780) 492–3815, E-mail: tlewis@math.ualberta.ca Copy Editor Barb Krahn & Associates (11623 78 Ave, Edmonton AB) Tel: (780) 430–1220, E-mail: bkrahn@v-wave.com Addresses: pi in the Sky pi in the Sky Pacific Institute for Pacific Institute for the Mathematical Sciences the Mathematical Sciences 449 Central Academic Blg 1933 West Mall University of Alberta University of British Columbia Edmonton, Alberta Vancouver, B.C. T6G 2G1, Canada V6T 1Z2, Canada Tel: (780) 492–4308 Tel: (604) 822-3922 Fax: (780) 492–1361 Fax: (604) 822-0883 E-mail: pi@pims.math.ca http://www.pims.math.ca/pi Contributions Welcome pi in the Sky accepts materials on any subject related to mathematics or its applications, including articles, problems, cartoons, statements, jokes, etc. Copyright of material submit- ted to the publisher and accepted for publication remains with the author, with the understanding that the publisher may re- produce it without royalty in print, electronic and other forms. Submissions are subject to editorial revision. We also welcome Letters to the Editor from teachers, students, parents or anybody interested in math education (be sure to include your full name and phone number). Cover Page: In October 2001, pi in the Sky was invited by the principal of Edmonton’s Tempo School, Dr. Kapoor, to meet with students in grades 10 and 11. The picture on the cover page was taken by Henry Van Roessel at Tempo School during our visit. More photos from that visit are published on page 28. If you would like to see your school on the cover page of pi in the Sky , please invite us for a short visit to meet your students and staff. CONTENTS: The Language of Mathematics Timothy Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Tic-Tetris-Toe Andy Liu, University of Alberta . . . . . . . . . . . . . . . . . . . . . . . 4 Weierstraß Volker Runde, University of Alberta . . . . . . . . . . . . . . . . . . . . 7 Life and Travel in 4D Tomasz Kaczynski, Universite´ de Sherbrooke . . . . . . . . . . . . . 10 Shark Attacks and the Poisson Approximation Byron Schmuland, University of Alberta . . . . . . . . . . . . . . . . 12 The Rose and the Nautilus A Geometric Mystery Story Klaus Hoechsmann, University of British Columbia . . . . . . . . 15 Three Easy Tricks Ted Lewis, University of Alberta . . . . . . . . . . . . . . . . . . . . . 18 Inequalities for Convex Functions (Part I) Dragos Hrimiuc, University of Alberta . . . . . . . . . . . . . . . . . 20 Math Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Math Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 Our School Visit Don Stanley, University of Alberta . . . . . . . . . . . . . . . . . . . . . 28 c©Copyright 2001 Sidney Harris 2 This column is an open forum. We welcome opin- ions on all mathematical issues: research; educa- tion; and communication. Please feel free to write. Opinions expressed in this forum do not necessarily reflect those of the editorial board, PIMS, or its sponsors. The Language of Mathematics by Timothy Taylor I’m a writer. I write stories—novels, short stories, novel- las. I’m aware that sometimes the creative arts (like my fiction) and the hard sciences (like mathematics) are con- sidered uncomfortable companions. People tend to imag- ine themselves as being attracted to one or the other, but not to both. In my case, to be truthful, I found mathe- matics difficult in school and for a long time I thought I didn’t like the subject. But I was wrong. Not only have I surprised myself in recent years by discovering an interest in mathematics and its applications, I have surprised my- self more by discovering this interest through my creative writing. Somebody once asked me, “What’s the most important skill required to write fiction?” I told them that you had to be able to sit alone in a room and type on a computer for long periods of time. This is partly true. When you have your story idea and you know your characters, there comes a point when you simply have to sit down and write for as long as it takes to finish. But that’s not the only requirement, of course. There is also a lot of research required to prepare yourself. In my writing, some of this research might be considered incidental. If my character visits Rome, I make it my business to learn about the city. I don’t just write, “He went to Rome.” I write something like, “He stayed at the Albergo Pomezia in Campo di’Fiori not far from the old Jewish ghetto.” This detail might not help us understand the character better, but it serves to create a sense of reality. Ultimately, you do want the reader to understand the characters however, and here’s where a more integral kind of research comes into play. Characters in my stories tend to have a fairly clear set of desires and objectives in life. These can range from grand to mundane. But in any case, there will be a particular set of issues that concern a char- acter, some variety of problem that he or she must solve. For example, a character arriving at a new school might wish to meet new friends. My character in Rome (he’s an art critic) is consumed with the work of a particular painter. The crucial thing is that the set of issues, or the problems that confront a character, determine which language they use. When I say “language,” I don’t mean “tongue,” where Chinese, Russian, English, or French might be examples. Instead, I mean the set of words and concepts that a char- acter is inclined to use within a given tongue. These arise directly from the issues that concern that character. And so, there is a unique language of art criticism (colour, com- position, theme, culture, aesthetic etc.), just as there is a language of business, of church, of personal relationships, and—drum roll please—of mathematics! I hadn’t really considered this until I wrote a story called Silent Cruise a few years back. In that story, I introduce Dett and Sheedy. Sheedy is a businessman and thinks only in those terms. Dett is a young man who is consumed by his own way of calculating probabilities (he does this, in part, because he likes betting at the racetrack). In order to put words into Dett’s mouth that make sense given his very peculiar obsessions, I had to re-acquaint myself with a language I hadn’t thought about in some time: mathematics. I should emphasize that Dett’s way of making calculations is not rigourous. A math student reading the story would see this right away. But the point is that he thinks not in a literate language (like Sheedy), but in a numerate one. And Dett’s way of expressing himself, to a large extent, defines who he is, how well he communicates with Sheedy, and what kinds of problems he is able or unable to solve. As I wrote the story, I enjoyed trying to “think” through Dett in his numerate way, even though I had a hard time doing so at school. In the process, I came to think that of all the languages I had researched for characters over the years, mathematics is very special. I would even go so far as to say that it is a precious language. It’s difficult to learn and, as a result, it is rare and valuable. But it is also very powerful, and perhaps this interests me more. As I wrote Dett’s story— even though he applied his numeracy in an unconventional way—he had the tools to solve many, many problems that Sheedy did not. And that fact, boiled right down, was the essence of my story. An editor once commented on Silent Cruise in a news- paper article. He said that I had drawn a picture of a character who thought primarily with numbers and how that was very unusual in Canadian fiction. I take these words as a compliment beyond any other that I have re- ceived in connection with my writing. They mean that I had not only done my research well enough to convince this editor, but that I had communicated some of what is precious—rare and valuable—in the language of math- ematics. I only wish I spoke it better. Timothy Taylor is the author of the national bestseller Stanley Park, a novel. His book, Silent Cruise and Other Stories, will be published next year. The short story, Silent Cruise, was short-listed for the Journey Prize 2000. Taylor won the prize for a different story, where the lan- guage spoken was concerned mostly with cheese. We should also mention that an article written by Timo- thy Taylor for Saturday Night Magazine won a Gold Medal at the National Magazine Awards. 3 Tic-Tetris-Toe by Andy Liu Part I: Introduction Tic-Tetris-Toe is very much like Tic-Tac-Toe. The classic game is played on a 3 by 3 board, taking a square in each turn. Whoever is first to get 3 squares in a row or 3 on a diagonal wins. However, in this new game, we make two changes. First, while we still play on a square board, it does not have to be 3 by 3. Tic-Tetris-Toe is actually five different games, each with a board of a different size. Second, we try to get different shapes. We use those from the popular video game Tetris, as shown in Figure 1. Figure 1 There are actually seven different pieces, but since they are allowed to turn over, we have only five Tic-Tetris- Toe games. We call these pieces N4, L4, T4, I4 and O4, because they each have four squares and look like the let- ters N, L, T, I and O, respectively. For N4, we play on a 3 by 3 board. For L4, we play on a 4 by 4 board. For T4, we play on a 5 by 5 board. For I4, we play on a 7 by 7 board. For O4, we play on a 9 by 9 board. Of course, the advantage is with the first player. Can you figure out a way for a sure win if I let you go first? Try these games with your friends, and then check below. Don’t peek—that will spoil the fun! Part II: The N4 Game Let us label the rows of the board 1, 2, and 3, and the columns a, b, and c. That way, each square will have a name. For example, the square at the bottom left corner will be called a1. You will need at least four moves to win, and you will have an extra fifth move that may come in handy in some scenarios. Mapping out a winning strategy requires that you look quite far ahead. On the other hand, I (the op- posing player) may be able to stop you from winning with one or two moves. Perhaps you should first consider what my strategy will be. a b c 1 2 3 O a b c 1 2 3 O O a b c 1 2 3 O O Figure 2 Figure 2 shows three ways in which I can stop you from winning. In each case, even if I give you all of the remain- ing squares, you still cannot complete N4. This tells you that you must take b2 in your first move, and make sure that you take at least one of b1 and b3, and at least one of a2 and c2. Note that once you have taken b2, you do not have to worry about me sneaking up on you for a surprise win. You will do no worse than a draw. It would be too em- barrassing to lose as the first player. Can I still stop you from winning? After you have taken b2, I really have two different choices: taking an edge square or a corner one. Suppose I take a2. You already know that you must take c2. Now I give up. In your third move, you can take b1 and create a double-threat at a3 and c1. If I prevent you from doing this by taking one of these three squares, you will take b3 and create a double-threat at a1 and c3. Figure 3 shows that the key to your success is the W5 shape. a b c 1 2 3 O1 X1 X2 a b c 1 2 3 O1 X1 X2 Figure 3 Am I better off if I start with a corner square, say a1? Suppose you still take c2. After all, it has worked once. Now I know that I must take one of a3, b3, and c1. You can force me to take c3 on my next move by taking b1 yourself. Then you can create a double-threat by taking one of c3, a2, and a3, depending on my move. a b c 1 2 3 O1 O2 O3 X1 X2 X3 X4 a b c 1 2 3 O1 O2 O3 X1 X2 X3 X4 a b c 1 2 3 O1 O2 O3 X1 X2 X3 X4 Figure 4 Can you remember all of this? You do not have to do that. Just understand that you must have b2, one of a2 and c2, and one of b1 and b3. Then look for double threats. With a little bit of practice, you will always win if you move first. 4 Part III: The L4, T4 and I4 Games Label the extra rows 4, 5 and so on, and the extra columns d, e and so on. You can have an easy win in the L4 game. Start by taking b2. You are guaranteed to get b3 or c2 in your next move. If both are still there and both b1 and b4 are still empty, take b3. Otherwise, a2 and d2 will be empty, so take c2. On your third move, complete 3 squares in a row, and I cannot stop you from completing L4 on your fourth move. Since I have only made three moves so far, I cannot beat you to it. You can win the T4 game by starting at the obvious place, c3. I can make one of five essentially different re- sponses, at a1, a2, a3, b2, or b3. On your second move, you take d4. On your third move, you take either c4 to create a double-threat at b4 and c5, or d3 to create a double-threat at d2 and e3. I can neither stop you nor beat you to it. 1 2 3 4 5 6 7 a b c d e f g O1 O2 O3 O4O5 X1 X2 X3 X4X5 X6 Figure 5 The I4 game is the only one of the five that can be played competitively. While you have a sure win, it cannot be forced until your eighth move. In trying for the win, it is possible that you may set up a double-threat for me. Figure 5 shows a sample game in which I put up a good fight. On the seventh move, you either take c6 for the double-threat at a6 and e6, or take b4 for the double- threat at b3 and b7. I cannot stop both. Part IV: The O4 Game This game holds a big surprise. Even though the board looks more than large enough, you will not be able to force a win. I have a very simple but effective counter strategy that will prevent you from winning, even if we play on an infinite board. It is an elegant idea which demonstrates the beauty of mathematics. Figure 6 I will combine pairs of adjacent squares into dominoes in the pattern of a brick wall, into which you will bash your head in vain. Whenever you take a square, I will take the other square of the same domino. As shown in Figure 6, no matter how you fit in O4, it must contain a complete domino. Since you can only have half of it, you cannot win! Part V: Further Projects Problem 1. Find four connected shapes of three squares or less, joined edge-to-edge. Remark: These shapes are called the monomino O1, the domino I2, and the trominoes I3 and V3. None of them provides much challenge as a game—the first player has an easy win if the board is big enough. This is because each of these pieces form parts of other pieces for which the first player can win. Our Tetris pieces are the tetrominoes. If we go to the pentominoes, you will find these games much more challenging. There are twelve such pieces, called F5, I5, L5, N5, P5, T5, U5, V5, W5, X5, Y5, and Z5, as shown in Figure 7. Pentomino is a registered trademark of Solomon Golomb, who has written a wonderful book called Polyominoes. This word means, “shaped or formed of many squares.” After the pentominoes come the 35 hexominoes, 108 heptominoes, 369 octominoes, and so on. Problem 2. Since P5 contains O4, and the domino strategy of Figure 6 works for the O4 game, the second player can also force a draw in the P5 game, even if it is played on an infinite board. On the other hand, there are four pentominoes that do not contain O4, but for which the domino strategy of Figure 6 also works. Which pentominoes are they? Problem 3. Show how the first player can force a win for the N5 game on a 6 by 6 board, and for the L5 and Y5 games on a 7 by 7 board. Figure 7 Problem 4. Match each of the other four pentominoes with one of the patterns in Figure 8 for a domino strategy. 5 Figure 8 Problem 5. How many of the hexominoes contain a pentomino for which the second player has a domino strategy? Figure 9 Problem 6. Find domino strategies for the second player in games us- ing the hexominoes in Figure 9. For one of them, you will have to find a new pattern. Figure 10 Problem 7. With the possible exception of the hexomino in Figure 10, no polyominoes formed of six or more squares offer the first player a sure win, even on an infinite board. Can a win be forced in a game using this hexomino? Problem 8. Returning to Tic-Tetris-Toe, it is easy to see that there is no win for the first player in the N4 game if it is played on a 2 by 2 board, because it is not even big enough to hold the piece. The L3 game on a 3 by 3 board is also a draw, if played properly. The classic Tic-Tack-Toe is still a draw even with additional winning configurations. Can the first player still force a win in the T4 games on a 4 by 4 board, or the I4 game on a 6 by 6 board? Part VI: Acknowledgement This article is based onMartin Gardner’s Mathemati- cal Games column in Scientific American magazine, April, 1979. It has since been collected into the anthology Frac- tal Music, Hypercards and More, as Chapter 13, under the title Generalized Tic-tac-toe. This book was published by W. H. Freeman and Company, New York, in 1992. The original work was done by the noted graph theorist Frank Harary. A mother of three is pregnant with her fourth child. One evening, her eldest daughter says to her dad, “Do you know, daddy, what I’ve found out?” “No.” “The new baby will be Chinese!” “What?!” “Yes. I’ve read in the paper that statistics show that every fourth child born nowadays is Chinese. . . .” A father who is very much concerned about his son’s poor grades in math decides to register him at a religious school. After his first term there, the son brings home his report card; he gets ‘A’s in math. The father is, of course, pleased, but wants to know, “Why are your math grades suddenly so good?” “You know,” the son explains, “when I walked into the class- room the first day and saw that guy nailed to a plus sign on the wall, I knew one thing—this place means business!” “What happened to your girlfriend, that really smart math student?” “She is no longer my girlfriend. I caught her cheating on me.” “I don’t believe that she cheated on you!” “Well, a couple of nights ago I called her on the phone, and she told me that she was in bed wrestlin