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.