  A Costas array of order $n$ is an arrangement of dots and blanks into $n$
rows and $n$ columns, with exactly one dot in each row and each column, the
arrangement satisfying certain specified conditions. A dot occurring in such an
array is even/even if it occurs in the $i$-th row and $j$-th column, where $i$
and $j$ are both even integers, and there are similar definitions of odd/odd,
even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas
and Welch-Costas arrays, can be defined using finite fields. When $q$ is a
power of an odd prime, we enumerate the number of even/even odd/odd, even/odd
and odd/even dots in a Golomb-Costas array. We show that three of these numbers
are equal and they differ by $\pm 1$ from the fourth. For a Welch-Costas array
of order $p-1$, where $p$ is an odd prime, the four numbers above are all equal
to $(p-1)/4$ when $p\equiv 1\pmod{4}$, but when $p\equiv 3\pmod{4}$, we show
that the four numbers are defined in terms of the class number of the imaginary
quadratic field $\mathbb{Q}(\sqrt{-p})$, and thus behave in a much less
predictable manner.
