david klarner: a memorial tribute

This essay is reprinted with the kind permission of Solomon W. Golomb and AK Peters, Ltd.. It originally appeared in the book Puzzler's Tribute: a feast for the mind, edited by David Wolfe and Tom Rodgers, AK Peters, 2001. ISBN 1-56881-121-7.

David A. Klarner—A Memorial Tribute
Solomon W. Golomb

I may have been the Prophet of Polyominoes, but David Klarner was their most faithful apostle.

It was Christmastime in 1959 (or thereabouts) when I received a large oblong rectangular wooden box in the mail. The return address was one D. A. Klarner from the far north in California. The box had a sliding lid, which I opened carefully, and dumped the wooden contents out on a tabletop. These turned out to be the 29 "pentacubes," and it took several hours to get them back in the box as neatly as they had arrived. David's enclosed letter revealed that he was a student at Humboldt State College who had learned about polyominoes from Martin Gardner's columns in Scientific American.

Several years ago, when I had gone from JPL to USC, I was invited by Professor Leo Moser of the University of Alberta, in Edmonton, Canada, to be the "outside reader" of the Ph.D. dissertation of one his students. The student was David Klarner, and the doctoral thesis contained a proof that the number P(n) of n-ominoes lies between aⁿ and bⁿ, where a=2 and b=8 was an acceptable choice. It was a day in mid-March 1964 (or thereabouts) when I flew from Los Angeles around noon (80° F), and with several intermediate stops arrived in Edmonton late at night (— 40° F) and met David for the first time. It had been a severe winter, and he had been ill with pneumonia much of the time. The next day was his successful thesis defense.

David had an important article, "Packing a Rectangle with Congruent N-ominoes," published in Volume 7 of the Journal of Combinatorial Theory, in 1969, where he introduced the concept of the order of a polyomino, defined as the minimum number of congruent copies which can be assembled to form a rectangle. (If the given polyomino does not tile any rectangle, its order is undefined). In the same article, he defined the odd-order of a polyomino to be the smallest odd number (if any) of congruent copies that can be assembled to form a rectangle. He had several beautiful illustrative examples, and the subject has inspired important research ever since.

When David was on the faculty at Stanford, he invited me to give a seminar talk (on polyominoes, of course). The date of my talk should be easy to establish, because the news of the day was the death of former president Lyndon B. Johnson.

After Stanford, David was at SUNY-Binghamton, and also spent more than one sabbatical year visiting the Technical University of Eindhoven, in the Netherlands, where he interacted with N. G. de Bruijn and L. E. J. Bouwkamp, among others. From Binghamton, his next academic position was at the University of Nebraska at Lincoln.

David had suffered since childhood from "type 1" diabetes (formerly called "juvenile onset" diabetes), and had a lifelong battle with the many complications of this ailment. Driving from New York to Nebraska, around the time he arrived in Lincoln he had a near-fatal heart attack. He was also the recipient of a kidney transplant.

It was in 1993 (or thereabouts) that I was invited (no doubt at David's suggestion) to be a member of a team of visiting experts to evaluate the progress of the University of Nebraska in the several areas of science and technology that the State Legislature had identified for special funding. I spent most of the free time I had during that period with David Klarner—as it turned out, for the last time.

Ever since the first edition of Polyominos appeared in 1965 (published by Charles Scribner and Sons), I was compiling material for a second edition, which finally appeared in 1994 (with Princeton University Press). I had considerable help from David in the preparation of the final manuscript for the new edition.

David had been the editor of the Mathematical Gardner, to which I contributed a chapter; and he was asked to edit the proceedings of the first "Gathering for Gardner", but by that time his health problems seriously intervened.

In mid-March of 1999, I was visiting the University of Waterloo, in Ontario, Canada, for several days. Dougles Stinson had been at the University of Nebraska when I had visited there, but was now at Waterloo. I asked him what he knew of David's whereabouts. He told me that David had retired, for health reasons, and had relocated, with his wife, back to Humboldt, California. I resolved to contact David when I returned to Los Angeles, but I never got the chance. Shortly after coming home, I learned the sad news of his demise.

David Klarner made an important and very distinctive contribution to literature of combinatorial mathematics in general, and to polyominoes in particular. He will be long remembered by many mathematicians who never actually met him for the quality of his work, and he will be sorely missed by those of us who knew him.