3. In a $13 \times 13$ square grid, select the centers of $k$ small squares such that no four of these points form the vertices of a rectangle (with sides parallel to those of the original square). Find the maximum value of $k$ that satisfies this condition.
(Let the $i$-th column have $x_{i}$ points $(i=1,2, \cdots, 13)$, then $\sum_{i=1}^{13} x_{i}=k$, the $x_{i}$ points in the $i$-th column form $\mathrm{C}_{x_{i}}^{2}$ different point pairs (if $x_{1}<2$, then $\mathrm{C}_{x_{1}}^{2}=0$). Add a column to the side of the $13 \times 13$ square, and each point pair projects onto this column. Since any 4 different points are not the vertices of a rectangle, the projections of different point pairs on the newly drawn column are different. There are $\mathrm{C}_{13}^{2}$ different point pairs on the newly drawn column, thus, $\sum_{i=1}^{13} C_{x_{i}}^{2} \leqslant C_{13}^{2}$, hence $\sum_{i=1}^{13} x_{i}^{2} \leqslant 156+k$. Since $\sum_{i=1}^{13} x_{i}^{2} \geqslant \frac{1}{13}\left(\sum_{i=1}^{13} x_{i}\right)^{2}=\frac{k^{2}}{13}$, then $k^{2} \leqslant 13 \times 156+13 k$, which gives $k \leqslant 52$. When $k=52$, a graph that meets the conditions can be constructed.)