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 BruckRyser theorem to symmetric block designs [10], which it is now known as the
BruckRyserChowla theorem. Here again, a partial converse exists, providing more credence to the
hope that the conditions in the BruckRyserChowla theorem are both necessary and sufficient. This
hope is now shattered by the nonexistence of the finite projective plane of order 10.
***Previously appeared in the American Mathematical Monthly
98, (no. 4) 1991, 305  318.
Author's Reflections:
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.