Dominic Olivastro wrote the 'Object Lessons' column for *The Sciences*, the publication of The New York Academy of Sciences. That little stint led to Ancient Puzzles, a book that looked at the history of Mathematics through the history of puzzles. Here Dominic writes about Martin Gardner.

**Dominic Olivastro on Martin Gardner**

In 'Tonight at Seven-Thirty', W.H. Auden plans a friendly dinner party:

For authentic

comity the gathering should be small...

... six lenient semble sieges,

none of them perilous,

is now a Perfect Social Number.

One critic wrote of this: 'Auden's ideals are again as temporal, local, and pragmatic as they are self-deflating', which is the sort of stuff only critics can write. I believe Auden was making a good pun. Six really is a perfect number, since it is the sum of its proper positive divisors – that is 6 = 1 + 2 + 3. Perfect numbers figure prominently in ancient numerology. Creation was finished in 6 days, the moon circles the earth in 28 days, and so on. After 6 and 28, the next perfect numbers are 496 and 8128. All perfect numbers are also triangular, so if we generalized this puzzle and asked, 'In a party with N people, how many handshakes occur if everyone shakes hands once with everyone else', then perfect numbers will be among the answers – for example, at a party of 8192, there are a perfect number of handshakes. Are there any odd perfect numbers? That is one of the unanswered questions in Number Theory. None are known, and they have checked all numbers up to 10^{300}, but no one has proved that none exist.

It is easy to become hooked on this sort of thing, and soon you see puzzles everywhere. Auden again:

Lovers of Big numbers go horridly mad,

Would have the Swiss abolished, all of us

Well purged, somatotyped, baptized, taught baseball:

They empty bars, spoil parties, run for Congress.

How's this for spoiling parties? There are 12 coins in front of you, 5 heads and 7 tails. Close your eyes and divide the coins into two piles, such that the number of heads in each pile is the same. You are free to flip any or all coins.

I got hooked because once, in high school, I picked up an old issue of *Scientific American* and discovered Martin Gardner's monthly column, 'Mathematical Games'. In that issue I found 'Perfect, Amicable and Sociable Numbers'. It is reprinted in Mathematical Magic Show. When I finished, I asked myself, 'Did I really read an article about Mathematics and not yawn?' I did. And it was an experience I repeated every month without fail throughout high school, college, graduate school, and beyond. I have many favourite authors, but of no one else can I truthfully say, 'I read something new by this author every month for much of my adult life, and I was never disappointed.'

It is difficult to pinpoint what Gardner has achieved, but I am certain it has never been duplicated either before or since. In the world of Recreational Mathematics (a small world, I know), there is Gardner, then a huge gap, then everyone else. There is, of course, his elegantly simple prose, always achieving a perfect balance between light-hearted chattiness, which is needed for Recreations, and dry precision, which is needed for Mathematics. Then, too, there is the sheer scope of his knowledge. I remember the column, now justly famous among Gardner-philes, titled simply 'Nothing'. It takes you through Set Theory, mathematicians like Frege and Russell, and John Horton Conway, Sam Loyd's Sliding Block Puzzle, novelists like Lewis Carroll and L. Frank Baum and Jean Paul Sartre, and finally the great question itself – 'Why is there something and not nothing?' All of it tied together by the single concept of Nothingness.

But I think what I enjoyed most was the sense that Gardner never knew, or cared, who was reading his columns. Housewives and auto-workers, mathematicians and physicists, all were among his admirers. And then, of course, there was O'Gara, the Mathematical Mailman, the mail carrier who created puzzles to idle away the long hours of his postal duties. Here is one of his simpler puzzles: If you write 4 letters and address 4 envelopes, then insert the letters into the envelopes at random, what is the probability that exactly one letter goes to the right person? What is the probability that exactly three letters go to the right person?

When one looks over his columns, it is obvious now that he should not be classed with other writers in Mathematical Recreations, because no one else is like him. Not Sam Loyd, who lacked Gardner's depth. Not H.E. Dudeney, who lacked Gardner's broad range. It is better to say that he created a whole new genre, the popular exposition of Mathematical Research. The simple prose belies the fact that, sooner or later, he will get around to deeply profound truths. Without Gardner, I don't think we would have Douglas Hofstadter's magnificent *Gödel, Escher, Bach*, or any other example of a mathematician writing a simple account of his work. Gardner created the audience for this.

Over the years, Gardner branched out into other areas. Fads and Fallacies in the Name of Science is still a classic debunking of pseudo-science, The Annotated Alice and The Annotated Snark are must reads for anyone who enjoys Lewis Carroll's classic nonsense, and Relativity for the Million brings his simple style to a complicated subject in Physics. But it is the columns in *Scientific American* that remain his most original contributions. By the time he retired from the column in 1981, there were 15 collections in all. Whatever you are reading at the moment, finish it, then buy a good sample of them. Get some from the early, middle and late years. For added pleasure, read them in order. You will not be disappointed.

**Answers:** At a party of 8192 people, there are 33550336 handshakes, the fifth perfect number.

Concerning the twelve coins, divide the coins into two piles of 7 and 5 coins each. If there are *h* heads in the large pile, then there must be 5-*h* heads in the small pile, and therefore 5-(5-*h*) tails (or simply *h* tails) in the small pile. It is easy to miss this, but no matter which coins go where, the number of heads in the large pile must equal the number of tails in the small pile. Now just flip each coin in the small pile.

To answer O'Gara's questions, the probability of exactly one letter going to the right person is 1/3. The probability of exactly three letters going to the right person is, of course, 0. If three letters go to the right person, then the last one must also.

[All the pieces that have appeared to date in this series, with the links to them, are listed here, here, here, here and here.]