Friday, May 28, 2004

Angels in Katana Ya

Sitting with an empty wallet in a restaurant that only accepts cash, I thought of calling a friend, who was at that moment desperately looking for his missing cell phone.

It happens to us all. Seeing that "Cash Only" sign hanging on the restaurant door at the end of our solitary meal, and finding to our great dismay that, despite those "accepted everywhere" major credit cards and except for a few loose pennies, we are out of cash. We blush, anticipating the embarrassing moment when the friendly waitress brings the bill with a big smile; we panic, knowing that the nearest ATM machine is at least twenty minutes away; we regret not having checked our wallets before entering and not having come with a friend; we ask ourselves, what are you gonna do?

Well, I could call my friend Yu Chen, I thought as I regained my composure. At that time, I was eating by myself in the restaurant Katana Ya. I had almost finished my meal before I opened my wallet and found no cash in it. I contemplated my way out of this dilemma, and Yu Chen came into my mind. After all, he was partially responsible for my situation. Had he let me use my credit card to pay for the meal that we had together the previous night with a few other friends, I would not have depleted my cash reserve. No, he had to make those award points, he explained complacently as he collected cash from us. Of course, the other benefit of paying the group meal with one's credit card is to get cash without a visit to a bank ATM. Yu Chen took advantage of us all.

Yu Chen was my instinctive choice of rescue for other reasons too. He lived close by, no further away than the closest ATM of my knowledge. He was always ready to help his friends in need. He knew the location of the restaurant, since we had dined in it together. (He had a funny way to say the name of the restaurant -- "Kata--naya" -- making the name sound Polish instead of Japanese.) He had cash in his pocket, those very twenty dollar bills that he had harvested from us the night before; therefore he would not be delayed by a detour to his bank.

The thought of Yu Chen gave me back comfort and appetite. With ease I ingested the last few bites of my meal. Having washed down the food with ice water, I took out my cell phone to summon help. I dialed, and eagerly listened to the long ring tones. Three rings without answer. My waitress gracefully put down my bill tray and took away my plate. My anxiety grew. Five rings, still no answer. Six rings, I hang up. Where was Yu Chen?

Yu Chen was, at that very moment, pressing his face against the restaurant window to assure my presence at the table. Presently he pushed open the door and entered.

“I am in a hurry.” he said, “I could not find my cell phone. I am really worried that it might be stolen.”

“No wonder”, I smiled. But my inappropriate comment was lost on Yu Chen, who continued with urgency.

“I was at work. Then I realized that my cell phone was not with me. So I took the bus back immediately. I must go home now and check if I have left it there." he said, and was on his way out again.

Confronted with his distress, I had not for a single second forgot my own. "Hold on a second", I grabbed the back of his shirt, explained my problem, and asked for a loan. As Yu Chen opened his wallet, I had a glimpse of all the greenbacks that he had happily taken from us the night before. He offered me a twenty, the same bill, I suspected, that had been transferred from my wallet to his. My waitress witnessed our transactions with a tinge of relief on her face. I offered Yu Chen to call his cell phone to see if anybody had it. He said that was the first thing he had done once he found his phone missing. I bid him good luck, and told him that I would stop by his home later to return his money and to see if he would have found his phone. This I did. Expectedly, Yu Chen had left his cell phone at home and had found it. He thanked me for calling it, not knowing the true purpose of my call.

But is there someone to whom I should be grateful? When my call went to a cell phone far away from its owner, when my intended rescuer was not at home nearby as I had hoped, who made him pass by the restaurant, and who made him turn his head to see me in there? Angels in Katana Ya.

Monday, May 10, 2004

The Chinese Language Exam

I had one night to fear my imminent devastation.

I spent half of my lived life preparing for and taking exams. I excelled in them. They gave me instantaneous satisfaction and self-assurance. From primary school to college, I enjoyed the feeling of triumph after scoring higher than most of my classmates in each exam. Although I was scarcely the top of the class, I was invariably close enough to earn my teachers' favor and my classmates' respect. Through what other convenient means can an ordinary teenager achieve self-confidence, if he does not do well in sports and is not dazzling-looking? Mid-term, final, TOEFL, GRE, bring them on. I was the master of them all.

Not only did my excellence in exams give me confidence, they also let me learn modesty.

Friend: "I heard that you got 2300 in GRE, that's amazing!"
I: "Well, that's just a pleasant accident."

Friend: "How did you get such a high mark in the math final?"
I: "I happened to have reviewed the right problem sets."

This time, of course, should be no different. I have already finished three exams, and I have done well in them. Only the math exam remains, and math is my forte. Time to relax and think of the fun things to do in the summer. Then a classmate comes to me and asks me if I am prepared for the final on the Chinese language.

"You are kidding me, right? I don't remember we have a final on that."

"Hmm, no. The exam is on tomorrow morning."

Panic seizes me. Out of my scrambled memory I retrieve the terrifying truth. I now remember clearly that the teacher has mentioned the final in the class, but I have somehow forgot it altogether. I have not once touched the textbook, nor have I read a single reading assignment. As it is, I owe a reading to three novels, five essays, two monograms, half a dozen proses in ancient Chinese, and scores of poems. I cannot even Xerox all the pages before tomorrow morning, let alone reading and memorizing them. I am going to flunk. My mind goes blank.

This is invariably when I wake up from the dream.

I have had this same dream innumerable times, always at dawn. The scenario is identical: falsely thinking that I have only one more exam to take, I am caught off guard with a second exam. The one exam that I remember is on either math or physics, and the one I forget is always, always on the Chinese language. Details may vary. I may remember the Chinese exam with a friend's reminder, I may walk into the classroom and find that everyone is working on the exam that I am completely unprepared for, or I may remain unaware of the exam until I see a big red zero on my score sheet on the first day of school after the summer holiday.

It was the last semester of college when this dream first invaded, shortly after I had taken what I thought was going to be the last exam in my life. In the beginning, the dream was more lenient, and I was given a week's time to cram for the neglected subject. Like a virus that mutates into more virulent strains, however, the dream increased in malice each time it visited, leaving me less and less time to make up, until it gave me only one evening, not to study, since it would be useless, but to dread my imminent and inevitable failure.

To me, it is no coincidence that the dream came on the heels of my supposedly last exam in life. Until then, doing well in exams had been my primary means of establishing my identity. Sure there were other things too, like having a couple of hobbies. But how else does one demonstrate superiority unequivocally? The exams carry scores, and these numbers can be compared. A tall guy says that he is 6'1, a rich man says that he earns half-million a year, and a basketball player says that he scores 20 points average a game. The height, the earning power, and the sportsmanship can all be measured by numbers. My number was my scores on the exams, and it got me ahead in school. Now I had to adapt to a life without exams, and to search for a new means to measure my success. The dream came, I think, to fill in the void. I might no longer have the exams in life, but at least I could still cherish them in my dreams.

Yet it is not a happy dream -- I flunk in it. It seems to reflect my subconscious and persistent fear of failure in the exams. Excellence in the exams was all I had, and how tenuous was my success. Everything hinged on my performance in an hour or so. It resembles in this sense a game, just one shot, and chance happens to us all.

Monday, April 26, 2004

The Backside of a Puzzle

How some classic puzzles can be solved in different ways.

I am fond of puzzles, and often trade them with my friends of similar propensities. Over the years I have come across a large number of puzzles. As a result, the stream of new ones that flows my way has dwindled to a trickle. More often than not, I will have recognized a puzzle of past acquaintance before my friends are only half way through the narrative. From time to time, I will delight in a "new" puzzle only to realize a few minutes later that it is merely a variant of a puzzle that I have encountered before but have failed to see through the disguise of its new form at first sight. This situation may have given my friends some disappointment, as many of them are waiting to expound the clever solutions they know after I fail to devise my own. But it is more frustrating to me than to my friends, because whereas they are only denied the pleasure of reviewing a familiar cleverness, I lose the thrill of a new discovery, and the joy of an intellectual odyssey through a fresh vista of imagination.

But even the familiar puzzles can give me the pleasure of surprises. I continue to get unexpected solutions to some classic puzzles from my friends. It seems to contradict a popular belief among mathematicians and physicists: The truth is often so simple and elegant that it cannot be otherwise. I will show you two puzzles, whose canonical solutions are so simple and elegant that they make you doubt if any other solutions are possible.

Puzzle #1. Construct a mathematical expression containing four zeros but no other numerals such that it is equal to 24.

The standard solution to this puzzle, when I first hit upon it, convinced me that it had to be THE solution. (For readers with a desire to solve every puzzle by themselves, I place the standard solution and the alternative solution at the end of this article, so as not to deprive them of the pleasure of an independent discovery, which is so often taken away from me.) With its simple symmetry, and its inevitable and apparent connection with the number 24, the standard solution seems to have left no room for any alternative to fit in. Well, I fell off my chair when my friend Dave showed me his beastly solution. His solution is a monster that has all four zeros in hideous arrangements, and its equality to 24 cannot be easily verified without the aid of a scientific calculator. Nonetheless, with the rules of the puzzle relented slightly, his solution is every bit as valid as the beautiful, standard one. Appearance may be misleading.

Puzzle #2. Consider the following equation, formed by 14 matchsticks.

2 2 / 2 = 11

(Because of the limitations of the display, the matchsticks are not drawn here. Each number "2" is formed by three matchsticks: one horizontal on the top, one horizontal at the bottom, and one diagonal leaning right in the middle; the division sign "/" is made of one diagonal matchstick leaning right; the equal sign "=" is made of two parallel horizontal matchsticks; the number "1"s are just one vertical matchstick each.)

You are to move one and only one matchstick to a different place in the equation, and form a different equality. The emphasis is on DIFFERENT and EQUALITY, and therefore the new equation should be different from 22 divided by 2 equals 11, and it must have on both sides of an EQUAL sign two entities that are EQUAL.

As for the first puzzle, I reveal the standard and the alternative solutions only at the end of this article. Again, the standard solution is clever, exact, and cute. Anyone who discovers the standard solution will find in it harmony and enjoy the graceful twist of his or her brain. But when I told this puzzle to my former colleague and friend Dimitris, as we were drinking a few beers in a noisy restaurant, he gave me a jaw-dropping answer. Admittedly, his solution is not exact, but it happens to be one of the most famous approximations to probably the most famous number of all. It is almost devilish. A couple of years later, the same Dave who took me by surprise with his monstrous solution to the first puzzle gave the same answer as Dimitris's to this second puzzle. Coincidence, or a trick of the devil?

G. H. Hardy once remarked: "Beauty is the first test: there is no permanent place in this world for ugly mathematics." For the theories of physical sciences, beauty is almost as important a criterion of judgement as experimental veracity. We all readily embrace the Greek faith of the unity of beauty and truth. But we must be wary that truth can sometimes be ugly.

Finally, here are the solutions to the puzzles.

Puzzle #1, standard solution: (cos(0)+cos(0)+cos(0)+cos(0))!=24, where n! is the factorial of n.

Dave's beast: floor[ - (0!)0 / cos(0! + 0!) ]=24
where floor(x) is the largest integer smaller than x. (0!)0 should be read 10, the number ten.

Now you wonder how anyone can think of 1/cos(2)=-2.403...!

Puzzle #2. standard solution: move one vertical matchstick from 11 on the right of the "=" sign to a horizontal "-" between the two's in 22, and the new equation reads 2-2 / 2 = 1. Remember the precedence of arithmetics!

Dimitris and Dave's monster: move the bottom horizontal matchstick from the third 2 to over 11 and make it the Greek letter "pi". The new equation is 2 2 / 7 = pi -- the famous approximation to the ratio of the circumference to the diameter of a circle discovered by the ancient Chinese mathematician Chongzhi Zu.