Games, Puzzles, Magic, and Mathematics

People love thinking as a pastime – you'll often find someone playing minesweeper or solitaire on their computer or cell-phone. Newspapers and airline magazines have puzzles: Sudoku, cross-words, and letter scrambles. Throughout the ages people invented and played intricate games like chess, poker, Scrabble, and Pokemon. Mathematicians create algorithms to solve puzzles like the Rubik's Cube and games like Connect-Four. Magicians create paradoxes that defy intuition. Party goers play detective and solve murder mysteries. Gamers play Civilization for days on end.

Why? What drives us? The Mathematician's Brain and How People Think are some of the many titles sitting in your neighborhood bookstore's mathematics section. Why do so many pastime-puzzle-solvers say “I was never good at math”? In the article What Makes the Puzzler Tick, ([4]) Rick Irby sums it up:

“Puzzles Intrigue, challenge, amuse, and quite often even aggravate. Historically, almost every culture has developed puzzles of some kind, often making them serve as locks or intelligence tests. While the difficulty of these puzzles can vary greatly, each requires ingenuity and impatience to discover its secret.” ([4])

When one becomes too aggravated, he or she either gives up or gets help. For puzzles, you look up the solution. For games, you play again. In magic, you're awed and keep thinking about the solution. In Mathematics, its no different – you can give up or keep thinking.

Why study mathematics? For the same reason that people are awed by magic. The Puzzler's Tribute said it best: “Puzzle aficionados and magicians are experts in the unsolvable, the undoable, and the unbelievable. Their currency is paradox.” Just the fact that something is seemingly undoable is enough to spur you on to figure out why. Even finding the solution is only half the problem; it remains to find all solutions or prove there are none. Such feats seem intangible, but with a little guidance you can do it. The difference between science and mathematics lies within this seeming impossibility that mathematical proof is devoid of doubt and infallible:

“Scientific proof is inevitably fickle and shoddy. On the other hand, mathematical proof is absolute and devoid of doubt. Pythagoras died confident in the knowledge that his

[Pythagorean Theorem], which was true in 500 B.C, would remain true for eternity.

Science is operated according to the judicial system. A theory is assumed to be true if there is enough evidence to prove it 'beyond all reasonable doubt.' On the other hand, mathematics does not rely on evidence from fallible experimentation, but it is built on infallible logic. This is demonstrated by the problem of the 'mutilated chessboard' ...”

([3]), which can be found below.

Enough about awe, what does understanding do for me? Polya wrote a book about mathematical problem solving, in which he starts out by saying

“A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.” [2]

The skills you learn far out-reach the realm of mathematics. Real-world puzzles may not directly involve math, but will require a reasoning process best isolated by mathematics. In your job, you can't just find something else to do; an answer is required. Mathematics gives the structure to reason through these problems to a complete-and-total solution; guess-and-check is insufficient. In the words of Einstein ([1]),

“But the creative principle lies in Mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.”

Perhaps the most important piece of the picture is grabbing one's attention and proving that thinking about math is a valid pastime. Some people have events that change their lives, like Sophie Germain, while others merely follow their dreams, like John Conway's friend. Below are a few case studies from history to wet your appetite – enjoy!

CASE STUDIES

Sophie Germain “The event that changed her life occurred the day when she was browsing in her father's library and chanced upon Jean-Etienne Montucla's book History of Mathematics. The chapter that caught her imagination was Montucla's essay on the life of Archimedes. His account of Archimedes' discoveries was undoubtedly interesting, but what particularly kindled her fascination was the story surrounding his death. Archimedes has spent his life at Syracuse, studying mathematics in relative tranquility, but when he was in his late seventies the peace was shattered by the invading Roman army. Legend has it that during the invasion Archimedes was so engrossed in the study of a geometric figure in the sand that he failed to respond to the questioning of a Roman soldier. As a result he was speared to death. Germain concluded that if somebody could be so consumed by a geometric problem that it could lead to their death, then mathematics must be the most captivating subject in the world.” After years dedicated to mathematics and “as a result of [her research on elasticity] and her work on Fermat's Last Theorem, she received a medal from the Institut de France.” [3].

Pythagoras observed that the Egyptians and Babylonians conducted each calculation in the form of a recipe that could be followed blindly. The recipes, which would have been passed down through the

generations, always gave the correct answer and so nobody bothered to question them or explore the logic underlying the equations. What was important for these civilizations was that the calculation worked – why it worked was irrelevant. After twenty years of travel Pythagoras had assimilated all the mathematical rules in the known world. ... He wanted to understand numbers, not merely exploit them. ... Pythagoras founded the Pythagorean Brotherhood – a band of six hundred followers who were capable not only of understanding his teachings, but who could add to them by creating new ideas and proofs.”

At the Olympic Games he was asked how he would describe himself. In response he coined the word “Philosopher,” stating

“Life ... may well be compared with these public Games for in this vast crowd assembled here some are attracted by the acquisition of gain, others are led on by the hopes of ambitions and glory. But among them there are a few who have come to observe and to understand all that passes here. It is the same with life. Some are influenced by the love of wealth while others are blindly led on by the mad fever for power and domination, but the finest man gives himself up to discovering the meaning and purpose of life itself. He seeks to uncover the secrets of nature. This is the man I call a philosopher for although no man is completely wise in all respects, he can love wisdom as the key to nature's powers.”

John Conway is a legendary Princeton game theorist. I once had the pleasure of attending a talk of his, or should I say a competition... He began half an hour early to compete in the game of Dots and Boxes with us Undergraduates. The game was simple: ten rounds of a 3 x 3 grid of dots and boxes. We win the game if we beat him in one round of the ten. He wins otherwise. One student was able to beat him, on his second try... He even let us choose who went first. Given time, anyone could of course go through all the possibilities. But we had no time, and he had strategies and tactics that worked for not only 3 x 3 grids but also n x m grids, for any integers n and m. In 3 x 3 we at least had a chance! Then Professor Conway told us about a book he wrote describing these strategies, and that he is only second place in the world championships of dots and boxes. First place was his co-author, who from an early age studies all possibilities in this game and, with John Conway, developed the mathematics required to do it. A true story of passion about problem solving!

Legendary Arthur Benjamin, Mathematics Professor and Magician. An excerpt from his website:

“Dr. Arthur Benjamin is both a professor of mathematics and a magician. He has combined his two loves to create a dynamic presentation called "Mathemagics," suitable for all audiences, where he demonstrates and explains his secrets for performing rapid mental calculations faster than a calculator. Reader's Digest calls him "America's Best Math Whiz". He has presented his high energy talk on over a thousand occasions to audiences throughout the world.

Dr. Benjamin has appeared on many television and radio programs, including: The Today Show, CNN, Amazing Discoveries! and National Public Radio. He has been profiled in The New York Times, Los Angeles Times, USA Today, Scientific American, Discover Magazine, Omni Magazine, Esquire Magazine, People Magazine, and Reader's Digest.” (reference: http://www.math.hmc.edu/~benjamin/mathemagics.htm )

Albert Einstein loved “thought experiments,” what we call daydreaming. As a patent clerk he questioned everything. When analyzing patents for ways to synchronize clocks, he even questioned what “simultaneous” means. Does it take time for light to reach your eyes? If so, then a man standing on a train moving through a station would judge a lightning strike as being at a different time than someone standing still outside the train. While Einstein was no mathematician, he was able to describe several physical theories and work with the mathematician Minkowski to prove their mathematical consequences. The result was a generalization of Newton's Laws, viewed until then as describing all of physics. Today no one doubts his brilliance.

Paul Erdos was a Hungarian Mathematician that could, after reading some stories, be confused with a bum. He traveled the world working with various mathematicians everywhere, saying that every waking moment of his life was an opportunity to solve problems. The most prolific mathematician ever, a number was named after him: the Erdos Number. Everyone has their own Erdos number: Erdos himself has an Erdos number of 0, while all his collaborators have an Erdos number of 1, all of theirs have an Erdos number of 2, and so on. It has been conjectured that every mathematician in the world has an Erdos number of at most 5. Many want an Erdos number less than one...

Note that mathematics is not a solitary subject. Einstein worked closely with the mathematician Minkowski; Germain worked with Gauss; Pythagoras had a whole brotherhood of 600. Today this is even more true: with the wealth of mathematics out there it is difficult to be an expert in every field, so the “modern mathematician” is one who specializes in one small area and finds paper co-writers in the areas he or she needs to learn about.

APPLIED MATHEMATICS

Applied mathematics is the mathematical study of real world principles. Though this subject has many cross-overs with physics, the defining difference is that physicists primarily use experiments to discover important physical principles (the earth is round, an object in motion stays in motion unless acted upon, the force of gravity is inversely proportional to the distance, etc), while mathematicians seek use these principles as axioms to prove deeper theorems derived from them (the aerodynamics of flying, that black holes are places of no return, etc). The two go hand in hand and must work together. Without physics, mathematics would be completely unrelated to life. Without mathematics, physics would be more of a religion than a science.

Is applied mathematics as fun as pure mathematics? Definitely. After taking a course in fluid dynamics, I was always awed when seeing the unexplainable (that is, not until we solve a certain differential equation) effect of spinning vortices every time I took a paddle stroke or saw an eddy line. Let me explain what those are: a simple experiment is to put a cylinder in moving water. Directly downstream of the cylinder there is little or no movement; this is called an eddy. The “eddy line” is the distinctive line between the current and the eddy. One wouldn't guess it, but along this line there are often small spinning whirlpools called vortices. These vortices have some yet unexplained push to them, which is could be harnessed for airplanes and boats.

Maybe I've peaked your interest, so I'll go on about the theory. The Milne-Thompson circle theorem says that everything happening outside of the cylinder is a mirror image of what could be happening inside the cylinder; that is, we could study an almost identical (though much more compacted) flow with well defined boundaries. The mathematics of a circle is much simpler than that of the whole world! The Navier-Stokes differential equation describes all two-dimensional flows. Its solution is so important that the Clay Mathematics Institute is offering one million dollars; here's what they have to say about it:

“Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.” (reference: http://www.claymath.org/millennium/Navier-Stokes_Equations/ )

Applied mathematics is a broad subject, involving more than just fluids. Jobs are plentiful, from designing airplanes at Boeing to designing robots to walk on Mars.

CAREERS IN MATHEMATICS

While few there are few grand-masters of pure mathematics, there's still a lot that you can do as a profession or a pastime. Here's a short list of viable career options:

• K-12 Teachers put the spark of problem solving into youngsters.

• Actuaries work for Insurance firms to calculate insurance rates.

• Applied Mathematicians research applications of mathematics to real world industry or to physics.

• Math Education Professors research teaching methods.

• Math Research Professors research mathematics to broaden the general theory for future application.

ILLUSTRATIVE PROBLEMS

Problem 1 includes a descriptive solution. Problem 2 may be tricky – the answers will be obvious if you see them but may not be apparent beforehand. A background of probability may be necessary for problem 3. Problem 4 may be aggravating, but the reference is stimulating.

1) Mutilated Chessboard (referred to in the Introduction).

“We have a chessboard with the two opposing corners removed, so that there are only 62 squares remaining. Now we take 31 dominoes shaped such that each domino covers exactly two squares. The question is: Is it possible to arrange the 31 dominoes so that they cover all the 62 squares on the chessboard?

There are two approaches to the problem:

(1) The scientific approach

The scientist would try to solve the problem by experimenting, and after trying out a few dozen arrangements would discover that they all fail. Eventually the scientist believes that there is enough evidence to say that the board cannot be covered. However, the scientist can never be sure that this is truly the case, because there might be some arrangement that has not been tried that might do the trick. There are millions of different arrangements, and it is possible to explore only a small fraction of them. The conclusion that the task is impossible is a theory based on experiment, but the scientist will have to live with the prospect that one day the theory may be overturned.

(2) The mathematical approach

The mathematician tries to answer the question by developing a logical argument that will derive a conclusion that is undoubtedly correct and that will remain unchallenged forever. One such argument is the following:

• The corners that were removed from the chessboard were both white. Therefore there are now 32 black squares and only 30 white squares.

• Each domino covers two neighboring squares, and neighboring squares are always different in color, i.e, one black and one white.

• Therefore, no matter how they are arranged, the first 30 dominoes laid on the board must cover 30 white squares and 30 black squares.

• Consequently, this will always leave you with one domino and two black squares remaining.

• But remember, all dominoes cover two neighboring squares, and neighboring squares are opposite in color. However, the two squares are the same color and so cannot both be covered by the one remaining domino. Therefore, covering the board is impossible.” ([3])

2) Mel Stover (from an article by Martin Gardner in [4])

“I always looked forward to a letter from Mel because it usually contained a new puzzle, often a puzzle that Mel had invented. Here are ... typical brainteasers that first came my way in a letter from Mel:

1. In the equation 26 – 63 = 1, change the position of just one digit to make the equation correct.

2. Form the figure of a giraffe, as shown below, with matches or toothpicks.

Change the position of just one piece so as to leave the giraffe exactly where it was before, except possibly for a rotation or reflection of the original figure.”

3) The Birthday Problem (a classic): There are 23 people in the room. You have to bet your entire fortune on whether (a) nobody shares the same birthday, or (b) two or more people share the same birthday, or. Which is more probable? I would bet (b)...

4) Friday the Thirteenth: In John Conway and Fred Kochman's excitingly written article Calendrical Conundrums in [4], they describe the evolution of our calendar and why, in its current state, the thirteenth of a month is more likely to be a Friday than any other day!

Want more puzzles? Here's a sheet I made for the day after an exam: puzzles

Here's one I made for the first day of Linear Algebra and Differential Equations:puzzles

The following links are to sites concerning puzzles, games, and other diversions that will test your reasoning skills.

• http://www.daily-sudoku.com/

• Math, Puzzles, Magic, Mazes, Illusions: Gathering for Gardner

• Mathematical Games

• American Math Society Careers and Employment page

REFERENCES

[1] Hyperspace, M. Kaku

[2] How to Solve It, G. Polya

[3] Fermat's Enigma, S. Singh

[4] Puzzler's Tribute, D. Wolfe & T. Rodgers