# The Search for a Finite Projective Plane of Order 10***

Clement W. H. Lam
Computer Science Department
Concordia University
Montréal Québec

### Abstract:

Projective planes are special cases of a class of combinatorial objects called symmetric block designs. We are not going to discuss block designs, except to mention that Chowla and Ryser have generalized the Bruck-Ryser theorem to symmetric block designs [10], which it is now known as the Bruck-Ryser-Chowla theorem. Here again, a partial converse exists, providing more credence to the hope that the conditions in the Bruck-Ryser-Chowla theorem are both necessary and sufficient. This hope is now shattered by the non-existence of the finite projective plane of order 10.
***Previously appeared in the American Mathematical Monthly 98, (no. 4) 1991, 305 - 318.

## Prologue

When I was a graduate student looking for a thesis topic, Herbert Ryser advised me not to work on the projective plane of order 10. Even though he was extremely interested in this subject, he believed that it was too difficult and that I might get nowhere with it. I took his advice and chose another problem. Somehow, this problem has a beauty that fascinates me as well as many other mathematicians. Finally in 1980, I succumbed to the temptation and started working on it with some of my colleagues. We eventually managed to get somewhere, but unfortunately, Dr. Ryser is no longer with us to hear of the final result. This is an expository article describing the evolution of the problem and how computers were used to solve it.

## Epilogue

While we were tracing the origin of the existence problem of the plane of order 10, we asked Dan Hughes, who has worked in this area for a long time and is famous for the Hughes planes which are named after him. He recounted the following story. In about 1957, at a Chinese restaurant in Chicago, Reinhold Baer, another mathematician well known for his work in group theory and projective planes, was trying to impress the younger Hughes by remarking that if the plane of order 10 was settled by a computer, he hoped not be alive to see it. Baer got his wish but I do not think Herb Ryser shared this opinion. Ryser was happy that the weight 12 case was settled by a computer. I can only extrapolate and hope that he would also be happy that the whole problem has been ``settled'', even if by a computer.