In 1975, a San Diego woman named Ms Marjorie Rice read in her son's Scientific American magazine that there were only eight known pentagonal shapes that could entirely tile, or tessellate, a plane. Despite having had no mathematics lessons beyond high school, she resolved to find another. By 1977, she had discovered not just one, but four new tessellations - a result noteworthy enough to be published the following year in a maths journal.
The article that turned Ms Rice into an amateur researcher was by the legendary polymath Martin Gardner. His "Mathematical Games" series, which ran in Scientific American for more than 25 years, introduced millions worldwide to the joys of recreational maths. I read him in Mumbai as an undergraduate, and even dug up his original 1956 column on "hexaflexagons" (folded paper hexagons that can be flexed to reveal different flower-like faces) to construct some myself.
"Recreational maths" might sound like an oxymoron to some, but the term can broadly include immensely popular puzzles such as Sudoku and KenKen, in addition to various games and brain teasers. The qualifying characteristics are that no advanced mathematical knowledge such as calculus be required, and that the activity engages enough of the same logical and deductive skills used in maths.
Unlike Sudoku, which always has the same format and gets easier with practice, the disparate puzzles that Gardner favoured required different, inventive techniques to crack. The solution in such a puzzle usually pops up in its entirety, through a flash of insight, rather than emerging steadily via step-by-step deduction as in Sudoku. An example: How can you identify a single counterfeit penny, slightly lighter than the rest, from a group of nine, in only two weighings?
Gardner's great genius lay in using such basic puzzles to lure readers into extensions requiring pattern recognition and generalisation, where they were doing real maths. For instance, once you solve the nine-coin puzzle above, you should be able to figure it out for 27 coins, or 81, or any power of three, in fact. This is how maths works, how recreational questions can quickly lead to research problems and striking, unexpected discoveries.
A famous illustration of this was a riddle posed by the citizens of Konigsberg, Germany, on whether there was a loop through their town traversing each of its seven bridges only once. In solving the problem, mathematician Leonhard Euler abstracted the city map by representing each land mass by a node and each bridge by a line segment. Not only did his method generalise to any number of bridges, but it also laid the foundation for graph theory, a subject essential to Web searches and other applications.
With the diversity of entertainment choices available nowadays, Gardner's name may no longer ring a bell. The few students in my current batch who say they still do mathematical puzzles seem partial to a website called Project Euler, whose computational problems require not just mathematical insight but also programming skill.
This reflects a sea change in maths itself, where computationally intense fields have been gaining increasing prominence in the past few decades.
Also, Sudoku-type puzzles, so addictive and easily generated by computers, have squeezed out one-of-a-kind "insight" puzzles, which are much harder to design - and solve.
Yet, Gardner's work lives on through websites that render it in the visual and animated forms favoured by today's audiences, through a constellation of his books that continue to sell, and through the biannual "Gathering 4 Gardner" recreational maths conferences.
In his final article for Scientific American, in 1998, Gardner lamented the "glacial" progress resulting from his efforts to have recreational maths introduced into school curriculums "as a way to interest young students in the wonders of mathematics". Indeed, a paper this year in the Journal Of Humanistic Mathematics pointed out that recreational maths can be used to awaken maths-related "joy," "satisfaction," "excitement" and "curiosity" in students, which the educational policies of several countries (including China, India, Finland, Sweden, England, Singapore and Japan) call for in writing. In contrast, the Common Core in the United States does not explicitly mention this emotional side of the subject, regarding maths as only a tool.
Of course, the Common Core lists only academic standards, and leaves the curriculum to individual districts - some of which are indeed incorporating recreational maths. For instance, maths lesson plans in Baltimore County public schools now usually begin with computer-accessible game and puzzle suggestions that teachers can choose to adopt, to motivate their classes.
The body of recreational maths that Gardner tended to and augmented is a valuable resource for mankind. He would have wanted no greater tribute, surely, than to have it keep nourishing future generations.
NEW YORK TIMES
•The writer isa mathematics professor at the University of Maryland
