Using a Singer cycle in Desarguesian planes of order q = 1 (mod 3), q a prime power, Brouwer [2] ave a construction of sets such that every line of the plane meets them in one of three possible intersection sizes. These intersection sizes x, y, and z satisfy the system of equations x + y + z = q + 1 x(2) + y(2) + z(2) = 1/3 (q(2) + 4q + 1). Brouwer claimed that this system has a unique solution in integers. Further, Brouwer noted that for q a perfect square, this system has a solution for which two of the variables are equal, ostensibly implying that when q is a square the constructed set has only two intersection numbers. In this paper, we perform a detailed analysis which shows that this system does not in general have a unique solution. In particular, the constructed sets when q is a square might have three intersection numbers. The cases for which this occurs are completely determined. (C) 2001 Academic Press.
