THIS IS MATH: Magic Squares, Circles, and Stars

If you have been following the opening of the windows in the Mathematical Awareness Month Poster, you might want to go back to window #1 and review Magic Squares. If you haven’t been there yet, please take a look at it. You will learn how to amaze your friends with your magical math abilities.

Magic squares come in many types, shapes, and sizes. Below you will see a magic square, a magic circle, and a magic star. If you would like to see hundreds more, you might want to check out The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions by Clifford Pickover.

Normal Magic Squares

This is a third-order normal magic square where all of the rows, columns, and diagonals add to 15.

 

squares

Is this the only solution to this magic square? Can you find others?

You could also have a 4 x 4 square or a 5 x 5 square and so on. How big of a square can you solve?

 

Magic Circles

Below you will see a magic circle composed of eight circles of four numbers each and the numbers on each circle all add to 18.  The thing that makes this magic circle special is that each number is at the intersection of four circles but no other point is common to the same four circles.

circles

Magic Stars

The magic star below is one of the simplest. They can get extremely complicated and also quite beautiful.

star

So, where’s the math?

Well, you should have noticed already that there are numbers on this page. However, there is more to math than numbers. Let’s add at least one equation.

If we go back to the normal magic square you should know that all these magic squares have the same number of rows and columns, they are n2. The constant that is the same for every column, row, and diagonal is called the magic sum and we will call it M.  Now we can figure out what that constant should be. If we use our 3 x 3 square above, we know that n = 3. If we plug our n into the given formula below we will find what our constant has to be.

equation

 Since our n = 3, the formula says M = [3 (32 + 1)]/2, which simplifies to 15. For normal magic squares of order n =  4, 5, and 6 the magic constants are, respectively: 34, 65, and 111. What would M be for n = 8? See if you can solve this square. (The figure for the normal square is from Wikipedia.)

 

 

 

 

 

THIS IS MATH!: Amaze your friends with The Baby Hummer card trick

Welcome to THIS IS MATH! a new series from math editor Vickie Kearn.

This is the first of a series of essays on interesting ways you can use math. You just may not have thought about it before but math is all around us. I hope that you will take away something from each of the forthcoming essays and that you will pass it on to someone you know.

3-28 Diaconis_MagicalApril is Math Awareness Month and the theme this year is Mathematics, Magic, and Mystery. There is a wonderful website where you will find all kinds of videos, puzzles, games, and interesting facts about math. The homepage has a poster with 30 different images. Each day of the month, a new window will open and reveal all of the wonders for that day.

Today I am going to elaborate on something behind window 3 which is about math and card magic. You will find more magic behind another window later this month. This particular trick is from Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks by Persi Diaconis and Ron Graham. It is a great trick and it is easy to learn. You only need any four playing cards. Take a look at the bottom card of your pack of four cards. Now remember this card and follow the directions carefully:

  1. Put the top card on the bottom of the packet.
  2. Turn the current top card face up and place it back on the top of the pack.
  3. Now cut the cards by putting any amount you like on the bottom of the pack.
  4. Take off the top two cards (keeping them together) and turn them over and place them back on top.
  5. Cut the cards again and then turn the top two over and place them back on top.
  6. Give the cards another cut and turn the top two over together and put them back on top.
  7. Give the cards a final cut.
  8. Now turn the top card over and put it on the bottom of the pack.
  9. Put the current top card on the bottom of the pack without turning it over.
  10. Finally, turn the top card over and place it back on top of the pack.
  11. Spread out the cards in your pack. Three will be facing one way and one in the opposite way.
  12. Surprise! Your card will be the one facing the opposite way.

This trick is called the Baby Hummer and was invented by magician Charles Hudson. It is a variation on a trick invented by Bob Hummer.

So where’s the math?
The math behind this trick covers 16 pages in the book mentioned above.

THIS IS MATH! will be back next week with an article on Math-Pickover Magic Squares!

 

PGS Exclusive: “On promoting Pythagoras’ Revenge in foreign lands” by Arturo Sangalli

I was delighted to be the editor for Arturo Sangalli’s book, Pythagoras’ Revenge: A Mathematical Mystery, which Princeton published in 2009. As far as anyone knows, Pythagoras did not leave behind any writings. But, suppose he did? What would he have said, and where would the writings be found? Would it have been possible for his writings to survive and in what kind of container would they have been preserved? During the course of his writing, Arturo checked all sorts of obscure facts to make sure that everything in the book would be viable. He even traveled near and far.

He received technical advice from specialists in various fields. He consulted a special collections paper conservator and an expert in the conservation of rare paper-based objects and another expert who had knowledge about the effects of time and environment on the structure of various metals that might have been used to store Pythagoras’ documents. The Canadian Conservation Institute provided valuable information on papyrus conservation. Arturo even went to Faversham, England to meet with an antiquarian book dealer. So, you can see that it takes much more than just sitting at your computer to make a really good book.

Since the publication of the book in English, Pythagoras’ Revenge has been translated into several languages. I asked Arturo to tell us what it has been like to promote his book in different languages.




On promoting Pythagoras’ Revenge in foreign lands

ARTURO SANGALLI

The Genoa Science Festival is a huge event: a ten-day annual feast of lectures, workshops, laboratory experiments, exhibitions, films, and other activities, over 300 of them, at all levels and for people of all ages with an interest in science. Attendance figures routinely exceed 200,000. So, when I received an invitation from the editor of the Italian translation of Pythagoras’ Revenge to present the book at the festival, I accepted without hesitation.

To be honest, the invitation didn’t just fall out of the Italian sky—I had to work for it. The promotion of a book does not come cheap. I was willing to actively participate in the process and bear some of the cost. Since I planned to be in Florence on holidays in the fall, I informed the Italian editor in May that I would be available for a reading, interview, or some other promotional event.

Three months later, I received an invitation to give a talk about the mathematics in the book at the Genoa festival on October 31, sponsored by Ponte alle Grazie, the Italian publisher of the book. They would cover transportation costs from Florence and accommodation for one night.

The Genoa Science Festival gets a lot of publicity, and it receives wide coverage in the local and regional press. Nonetheless, I was pleasantly surprised to find that the room at the FNAC bookstore was packed, with many people standing. I first presented and analyzed some of the mathematical ideas in the book and read a couple of passages, all this in Italian. There followed an interview with Giovanni Filocamo, the Italian mathematician who had introduced me, after which I answered a few questions from the audience. A handful of people came forward to have their copies of the book signed. Among these was a young boy, not older than thirteen, accompanied by his mother. She told me he had a passion for mathematics. I wrote, in Italian: ”To Joshua. This book was written for you. Now go and become a great mathematician! ”

Since I was going to be in Europe anyway, I figured “Why stop in Italy?” The book’s German translation was due out in October. I contacted the German publisher, Spektrum-Springer, mentioning in passing the Italian event. An exchange of emails followed and culminated with an invitation for a reading at the university bookstore in Heidelberg on November 3, and the offer to pay for part of the expenses. My European book tour was set up.

Heidelberg was the next stop. In the afternoon, I was given the rare opportunity to pitch the book to the Spektrum sales people, who happened to be holding their annual national meeting that day. The reading at the Ziehank bookstore in the evening had been announced through posters and fliers, with an entire window devoted exclusively to Pythagoras’ Rache. I had been told that people from the university, the likely audience, understood English. So I read some passages from the original English version, and Frank Wigger, the editor of the book, read other excerpts in German. The attendance was somewhat disappointing, though, but those who did come really appreciated the bilingual reading, judging from the comments afterward.

Frank told me that it was only the second time Spektrum had published a novel (the first one was seven years ago), and that sales figures were very promising: nearly 1,000 copies sold in the first three weeks.
All in all, it was a great experience, which I would be happy to repeat. Who knows? Two more translations of Pythagoras’ Revenge will be released shortly. But a trip to either Greece or Japan looks like a long shot at the moment.

Wait! This just in: the Paris-based Dunod Editions has bought the French translation rights for the book. Hmmm, Paris, that’s a different story.

Museum of Mathematics (@MoMath1) to Open Early 2012

Last spring my husband and I went to the World Science Festival in Manhattan. He is a high school science teacher and I am the math editor for Princeton so we thought this was a great way to spend a Sunday afternoon. The exhibits were wonderful but our favorite was the Math Midway. It also was the most crowded. There were all kinds of hands-on exhibits, puzzles, and even a tricycle with square wheels. We loved the NASA exhibit and the Virginia Tech robot, Charlie, but we kept going back to the Math Midway. Seeing the excited, and sometimes concentrated, looks on the faces of kids and adults alike was almost as much fun as being a part of the action.

Even better than this, we learned that the Math Midway was a preview of what will become the Museum of Mathematics.  It will be a unique and innovative institution which will strive to enhance public understanding and perception of mathematics as an evolving, creative, and aesthetic human endeavor. Slated to open in Spring 2012, the Museum’s hands-on, interactive exhibits will provide a place for children and adults to become excited about math. MoMath will be North America’s only museum devoted to the wonders of mathematics and its many connections, from art to science to finance.

In addition to the exhibits that we saw in the Math Midway, there will be lectures, classroom events, and a gift shop where you can purchase all kinds of great math toys. If you just can’t wait, go to http://momath.org/shop to find a great math gift. You will also be supporting the Museum. If you would like to know more about the museum, visit http://momath.org/. If you will be attending the Joint Mathematics Meetings in New Orleans this January, please stop by their exhibit.

PGS Exclusive: Flows in Networks by L. R. Ford and D.R. Fulkerson, Back in Print

Back by popular demand! This phrase may seem out of place when applied to an academic monograph titled Flows in Networks, but as you’ll read below, this title truly has been in demand for a while. Mathematics Editor Vickie Kearn relates her long history with this book and how it came to be back in print this Fall.




The story of this reprint began in the late 1980s. I was working at the Society for Industrial and Applied Mathematics and was getting quite a few requests to reprint the Ford & Fulkerson book on Flows in Networks in the Classics in Applied Mathematics series. This is a wonderful series that reprints books that are out of print. There was no Google at the time so finding the copyright holder was not as easy a task as it is today. The book was originally published in 1962 and had been out of print for quite a long time so finding a copy was also difficult. Due to many other duties, I pursued this off and on but did not have much time to devote to the search.

I moved to Princeton University Press in 2001 and was looking at the titles in the Princeton Landmarks in Mathematics series (also a series of classic books that had been published in another series or that had gone out of print) and thought again about the Ford & Fulkerson. It really bothered me that I had not followed through on finding the copyright holder. In 2004 I was visiting David Williamson at Cornell and he suggested putting the Ford & Fulkerson back in print. I started to tell him my long history with this book and he told me he didn’t think it would be a problem since Princeton University Press had published the book. Since the book was out of print, I had no record of it. What a wonderful surprise that was!

The next thing I had to do was actually find a copy of the book. Although this might sound ridiculous, we did not have one in our warehouse and there were no copies in the Princeton University library. I began to think it was just not meant for this book to be in print. Once again, I drifted away from the project. In 2008 I was back at Cornell and met Bob Bland who also suggested I put the Ford & Fulkerson back in print. I started to tell my long, sad story but he stopped me in mid-sentence and handed me a copy of the book. I had run out of excuses. We had a wonderful conversation and he told me that Fulkerson had been his advisor. Bob and James Orlin have written a wonderful new foreword which describes the continued importance of this book and its many applications, almost 50 years after it was originally published. If there are any other Princeton University Press books that you would like to see back in print, let us know.

PGS Mathematics: Rain Puddles and Speed

Here, Princeton University Press author John Adam presents two math problems you might encounter in your quest for fall foliage this month. The problems and solutions are similar to those found in his books A Mathematical Nature Walk and draw upon the skills you described in Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin.



The season of Autumn can be one of great beauty, especially where the foliage changes to a bright variety of reds, oranges and yellows. Why do leaves change their colors in the fall? There are in fact several different reasons, but the most important is the increasing length of night and cooler temperatures at night. Other factors are the amount of rainfall and the overall weather patterns in the preceding months. Just like sunsets, the weather before each fall is different. Basically the production of chlorophyll slows and stops in the autumn months, causing the green color of the leaves to disappear, and the colors remaining are mixtures of brown, red, orange and yellow, depending on the types of tree. To have any real chance of seeing the wonderful fall foliage, you have to go to the right places at the right time. Going to the beach in summer or even the fall won’t do! And going to the Blue Ridge Mountains or New England in the depths of winter will not enable you to see the fall foliage either, pretty though the snow-covered trees may be!

But there are some other aspects of this season that are present at anytime of the year, and do not depend on the leaves changing. Those aspects involve trees, rain and, in this case, my left foot!

Consider this: you are in the hills of New Hampshire, or West Virginia, or perhaps you are somewhere on the Appalachian Trail enjoying the glorious fall colors, when suddenly a rain squall appears out of nowhere, or so it seems. You run to take cover in a deserted shelter a hundred yards away near the trail. Playfully (there being nothing else to do) you stick your foot outside the shelter, and of all things, photograph the rain falling around it! After ten minutes of intense rainfall, it stops as suddenly as it started. You wonder how fast that rain must have been falling to create the scene you now survey: large puddles all along the trail, drops dripping from every available leaf above you, and the temporary dark-brown stains on the trunks of rain-soaked silver birches.

As you set out on your way again, you start to notice the wave patterns formed when the drops falling from the branches above you hit the surface of puddles. Question: which of the following patterns represents this situation?

Answer: the one on the left. Raindrops falling on the surface of a puddle generate wave patterns that are dominated by the effects of surface tension. The speed of these waves is inversely proportional to the square root of the wavelength; thus shorter waves travel faster and move out first. Note the expanding region of calm associated with, and inside these waves. The other pattern is dominated by gravity, which produces longer waves with speeds directly proportional to the square root of the wavelength, so the longer waves travel faster and move out first.

Once home, you upload your pictures onto the computer. On noticing the picture you took back at the shelter, you realize that knowing the exposure time of the shot, you can estimate the speed of the rain. Here is the picture: can you find the speed of the rain?

This rather pedestrian photograph was taken from a sheltered area outside the Mt. Washington Resort, Bretton Woods, NH on September 30th, courtesy of Tropical Storm Nicole. Because of the heavy rain, Mt. Washington was nowhere to be seen! Nevertheless, it seemed like fun to “guesstimate” the speed of the raindrops, given that the exposure time for the shot was 1/200 second and an estimate of the width of my sneaker (you don’t need to know my shoe size to do this.). My foot and the raindrops shown were about the same distance from the camera. You can assume that the foreshortening of the rain streaks (due to the downward angle of the camera) is not significant.
Answer: From the photograph, The width of my sneaker is approximately 5 inches, so the raindrops travelled approximately 5/3 inches in 1/200th of a second, or 1000/3 inches/s, or 1000/36 ft/s. Therefore

Since 1 m/s 2.2 mph, this is about 8.6 m/s.]
For those who don’t have access to my foot, anything in the range 4 inches to 6 inches wide is a reasonable estimate. This would result in a range of speeds 15-23 mph (7-10 m/s).


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PGS Editors: Vickie Kearn, Executive Editor in Mathematics, reflects on the acquisition of The Calculus of Friendship by Steve Strogatz

“Have you ever taken a moment to thank a teacher or someone else who has helped you become what you are today? I haven’t, but I really wish I could,” says mathematics editor Vickie Kearn.



I had an amazing high school math teacher. Because I was at a small boarding school, I had the same teacher for three years. Elsie Nunn was about 4’5” tall, with Einstein-like hair. She was a whirling dervish and taught me an amazing amount of math using a broomstick, chalk, and a coat hangar. Because of Ms. Nunn, I knew I would major in math and be a school teacher. I did exactly that and taught math for eight years before entering the world of publishing.

In November 2006 Steve Strogatz called me and asked if I would be interested in publishing his new book about his thirty-year correspondence with his high school math teacher, Don Joffray. He told me that the focus of the book was on math problems they discussed through the mail and he also planned to show how their relationship matured and changed through the years. That sounded wonderful and my immediate answer was, “Yes!”. Steve told me that a trade house was interested but only if he would take out the math. That seemed ridiculous to me and I said we would want the book only if the math stayed in. Princeton was a perfect match and we began to work on developing the manuscript right away. Steve had an incomplete set of correspondence because he had not saved all the letters from his teacher and there were large gaps when he did not respond to Joff’s letters. (You will have to read the book to learn why.) The big task was how to write a story around the letters so that everyone would enjoy it—not just mathematicians.

We talked about the book and how it should be developed for a long time. The letters were all about math, which Steve would have to explain to some readers. It was also important to explain what was not in the letters—the non-math things happening in Steve’s and Joff’s lives. Some were quite personal and we had to decide which to include. Joff illustrated his letters with drawings of birds and fish and we knew we had to add them in some way. There were a few false starts, but nothing we tried was just quite right. One day Steve called me and said, “I know how to do it.” He sent me a sample chapter and I agreed that his plan would work. Although we stayed in touch, I didn’t see anything further for a while. A few months later, I got a package from Steve. I decided to take a quick peak while I ate lunch. It is rare that an editor gets the chance to read a manuscript from beginning to end in one sitting, but that is exactly what I did. I could not stop reading until I got to the last page. I had seen sample letters but reading them all together within the context of the story that Steve had told was amazing. I saw a teacher become a student and a student become a teacher. I saw a self-centered young man become a compassionate teacher, husband, and father. I understood why Joff lived through and for his students and why he was so thrilled by their accomplishments. The bond of the love of math brought Steve and Joff together and no matter what was happening in their lives, the math was constant. But, there came a time when it wasn’t quite enough and that is when you discover than a stronger connection had probably been there for a long time but neither of them realized it.

I did eventually finish my lunch but my napkin was tear-soaked. I had never read a manuscript that had such a strong impact on me. I have tried to find my math teacher but have not been successful. I fear that I will not be able to thank her for instilling such a love of math in me. Because of her, I have had a wonderful 30+ year career as a math editor.

Thank you Ms. Nunn, and thank you Steve.


The Calculus of Friendship was published in September 2009 to much fanfare, including this insightful interview with Alan Alda.

More recently, Steve Strogatz wrote a popular weekly column for the New York Times Opinionator blog that was in essence a light, interesting introduction to big math ideas.

Math Awareness Month — Math editor Vickie Kearn on why math is so important in sports

Most people like at least one sport. Many of us are addicted to ESPN. But how many people think of math when they watch sports? Whether you are filling in your March Madness brackets or want to have the highest percentage of goals on your soccer team, math will help you out. Perhaps you are on the track team. How do you adjust your pace to win the race, depending on your lane?

It might be that safety is a big issue for your sport. If you are a NASCAR fan, you might be interested to find that math plays a big part in the manufacture of tires as well as tire pressure. If the design of the car changes, the tires need to change as well. For me, I want to make sure my son’s lacrosse helmet is made of the strongest materials possible and that it fits properly. A new mouth guard is now available that is much safer and increases performance by 25%.

Mathematics is also involved in accurately assessing a team’s chance of winning a particular game, whether or not it is on a winning streak. Baseball managers and owners of teams use math to determine the value of a particular player. Coaches crunch numbers to determine whether they go for a touchdown on fourth and 3. If you have ever volunteered to schedule a tournament for your town’s Little League, you already know the value of math. Can you imagine scheduling the MLB season?

If you are a college football fan, you know that the championship game is not determined by a playoff system but by a combination of human polls and computer ranking methods. Now, you might not like this system, but you have to admit it would be impossible to assign the bowl games without math.

I often hear comments like “I hate math!” or “What is it good for anyway?” It is good for lots of things, and one of the most interesting is its applications in sports. If you are a “math-hater”, pick a sport you like and commit yourself to finding the math behind it during Math Awareness Month. Then, share your new really neat math fact with a friend.

The Joint Mathematics Meetings

Math editor Vickie Kearn reports on the goings on at the Joint Mathematics Meetings held in Washington, DC, January 5-8.

The Joint Mathematics Meetings, held in Washington DC this January, had its largest attendance ever—over 6,000 professors, practitioners, and students from high school on up. This was an increase of more than 500 over the last meeting in San Diego in January 2008. And there’s even more good news for math. During the meeting, JobsRated.com announced that the most satisfying job is that of mathematician.  This top ranking was based on stress, work environment, physical demands, income and outlook. The meeting had something for everyone. There were plenty of talks on quantum computation, differential geometry, and operator theory. There also were sessions that connected math to biology, finance, and medicine.

The education sessions were filled with innovative ways to bring excitement to the classroom: “Using mime to see the remainder”, “Football rankings using linear algebra”, and “Using game theory to get a date” are just a few examples. Using sports and art to teach mathematics has become very popular with both professors and students. There are two new journals devoted to math and art. Even after a long day of talks, the evening sessions are always well attended. Who would want to miss the CNN United States of Mathematics Debate, Who Wants to Be a Mathematician, or Alice in Numberland?  These are all events that showed the lighter and very entertaining side of math.

The biggest event at the meeting for Princeton University Press was the release of The Princeton Companion to Mathematics, edited by Tim Gowers. The champagne reception celebrated the culmination of five years of hard work by more than 150 people. The celebration continued later in the week with a reception at the math department back at Princeton.

One of the most active areas for presentations at the meeting is the history of mathematics. The talks are always interesting and the evening skits and movies are a guaranteed source of entertainment. Princeton released four new titles just in time for the meeting.  These include Plato’s Ghost by Jeremy Gray, The Mathematics of the Heavens and the Earth by Glen Van Brummelen, Mathematics in Ancient Iraq by Eleanor Robson, and Mathematics in India by Kim Plofker. These titles, along with The Mathematics of Egypt, Mesopotamia, China, India, and Islam, edited by Victor Katz provide a very comprehensive library for the history of math.

A highlight of every meeting is the book signings for new popular math titles. David Richeson was on hand to sign copies of his newly released Euler’s Gem: The Polyhedron Formula and the Birth of Topology. This is the latest release in a long line of books that bring the excitement of math to readers outside the field.

One of the best attended events of the meeting is the awards ceremony.  We would like to congratulate our author, Jeremy Gray, for his outstanding achievement in receiving the Albert Leon Whiteman Memorial Prize for notable exposition on the history of mathematics.

This meeting always has a high level of excitement about it. It is a great place to renew friendships with professors and classmates and make new friends, or to look for a first, or new, job. Each year, hundreds of people wait anxiously for the ribbon cutting and the grand opening of the exhibit hall, filled with book and software vendors as well as puzzle makers and mathematical sculptors. If you so desired, you could also get a foot massage, buy jewelry, or get a new tie (with or without math on it). The exhibit hall is a place to gather informally and have a cup of coffee. It is also a fantastic library where browsers can sit and read all of the new titles and talk with editors about forthcoming titles and new writing projects. It is the ultimate opportunity to find a new text for the next semester. For four days each January, mathematicians at all levels and with varied interests come together to share their ideas and learn of new ones from their colleagues.

Here are some additional photos from the meetings. Click on each picture to see captions, etc.