Martin Gardner: The man who made maths fun

The American amateur mathematician, who died last month, explored and popularised the surprisingly significant puzzles that lie at the fringe of mathematics.

Martin Gardner's Mathematic Games column was a decades-long effort to debunk pseudoscience.

Can a simple sum prove there is no such thing as free will? That appears to be the astonishing implication of the following experiment. Empty your mind, and write down a completely random number (two or three digits long will do). Double this number and add 10 to it. Now halve the result, subtract the original number and convert the outcome to a letter of the alphabet, using 1 = A, 2 = B and so on.

Once you have worked out the letter, imagine an animal which starts with that letter. Now see if it's the same as the one named at the end of this column. If you've done the arithmetic right, you'll be staring at proof that you were destined to choose the random number you did. Anyone with a bit of algebra will see straight through this bit of hokum. But it brings together the twin themes that ran through the life of the American amateur mathematician Martin Gardner, the most influential writer on mathematics of the last 50 years, who died last month.

From the mid-1950s to the 1980s, Gardner wrote the Mathematical Games column of Scientific American, whose accounts of old paradoxes, emerging concepts and conundrums was read avidly by professionals and no-hopers alike. In the process, he introduced countless readers to what many would regard as an oxymoron: the pleasures of mathematics. At the same time he began a crusade aimed at debunking pseudoscience and quackery. Few would doubt the value of Gardner's pioneering efforts in sceptical inquiry. There has never been a greater need for critical thinking - and not just in the traditional areas of fringe science and medicine.

Over recent months it has become clear that claims made in mainstream science, from evidence of the supposed ravages of climate change to the unveiling of new "breakthroughs" in medicine, cannot be taken at face value. By comparison, Gardner's writings on recreational mathematics seem almost trivial. He introduced the world to the delights of "hexaflexagons", folded pieces of paper that produce different patterns when flexed at their corners, and popularised "non-periodic tiles", shapes that will cover a floor like ordinary tiles, but won't produce a repeating pattern. All very amusing, perhaps, but hardly the stuff of which revolutions are made.

Yet, as mathematicians are the first to admit, there is no telling what bit of "trivia" can turn out to be anything but. Gardner's own columns provide many examples, a case in point being his writings about so-called fractals. In the 1920s, the English mathematician Lewis Fry Richardson noticed something odd about coastlines. Checking various textbooks, he found each gave different values for the lengths of the coastlines of European countries - in some cases varying by 20 per cent.

Richardson found that the cause was not sloppiness on the part of the authors: rather, it was the scale of the maps. Specifically, the smaller the scale, the longer the coastline - simply because more detail was being revealed. Richardson went further, however, and found the mathematical law governing the dependence of coastline length on scale. This was the starting point for what is now known as the study of fractals - roughly speaking, shapes with infinite layers of detail.

Gardner was also keen on paradoxes and puzzles, many of which centred on probability, arguably the richest source of counter-intuitive results in mathematics. While no one would doubt the importance of probability, it too has "trivial" origins. In the mid-1600s, a shady French aristocrat called Chevalier de Méré needed some help figuring the odds for a game of dice, and approached the brilliant French mathematician Blaise Pascal for advice. Pascal called in his no less brilliant friend Pierre de Fermat (of Last Theorem fame) and the two developed the foundations of probability theory.

Of all the puzzles described by Gardner, perhaps the most striking example of the piffling proving profound is the so-called Seven Bridges of Konigsberg problem. First posed about 300 years ago, this asks whether it was possible to stroll around the eponymous city crossing each of its seven bridges only once. The solution was published in 1735 by the renowned Swiss mathematician Leonhard Euler - and the answer was no, it was not possible. But Euler's 12-page analysis leading up to that answer laid the foundations for two new areas of applied mathematics: graph theory and topology. The former is now used to design microprocessors, route internet traffic and plan airline routes, while topology has found applications in everything from molecular biology to the search for the Theory of Everything in physics.

Gardner was convinced of the power of mathematics to cut through to the truth. He was equally convinced of the need to chop down the thickets of pseudoscience wherever they spring up. It would have been interesting to know what he made of the increasing use of mathematics itself for bamboozlement. This is not a new phenomenon. In the 1720s, the French atheist philosopher Denis Diderot took part in a debate on the existence of God with a renowned mathematician at the court of Catherine the Great. It is said that the mathematician - thought to be Euler of the seven bridges fame, a devout Catholic - attempted to wrong-foot his opponent by writing down a meaningless equation that was said to prove the existence of God.

Whether Diderot saw through the ruse is not clear; he is said to have simply packed his bags and left. But it is undoubtedly the case that the same strategy has been used many times since. Indeed, some financiers - Warren Buffett among them - allege that mathematical befuddlement lies at the centre of the current global economic crisis, in the form of hideously complex financial instruments that no-one could understand.

In Buffett's evocative phrase, mathematics can produce "weapons of financial mass destruction". But in the right hands it can illuminate and delight us all. And, by the way, you were thinking of an elephant. Robert Matthews is Visiting Reader in Science at Aston University, Birmingham, England. His website is www.robertmatthews.org