17. In the $7 \times 7$ unit square grid shown in Figure 10, there are 64 grid points, and there are many squares with these grid points as vertices. How many different values of the areas of these squares are there?
17. Since the vertices of the square are grid points in the grid, as shown in Figure 13, the area of the shaded square is $a^{2}+b^{2}$.
Where, $0 \leqslant a+b \leqslant 7$.
Without loss of generality, let $a \geqslant b$. We can enumerate all possible values of $(a, b)$:
$$
\begin{array}{l}
(0,0),(1,0),(2,0),(3,0),(4,0),(5,0), \\
(6,0),(7,0) ; \\
(1,1),(2,1),(3,1),(4,1),(5,1),(6,1) ; \\
(2,2),(3,2),(4,2),(5,2) ; \\
(3,3),(4,3) .
\end{array}
$$

Among them, $(0,0)$ does not meet the requirements and is discarded.
In addition, $5^{2}=4^{2}+3^{2}$, that is, $(5,0)$ and $(4,3)$ give the same area.

Upon inspection, it is known that the other area values are all different, so there are 18 different values in total.