To solve the problem, we first visualize the \(4 \times 4\) grid of points represented in the coordinate system:
\[ (0,0), (0,1), (0,2), (0,3), \ (1,0), (1,1), (1,2), (1,3), \ (2,0), (2,1), (2,2), (2,3), \ (3,0), (3,1), (3,2), (3,3) \]
This grid contains 16 points, placed at coordinates \( (i, j) \) where \( i, j \) take values from \(0\) to \(3\).
We need to choose four points such that the distance between any two chosen points is at most \(\sqrt{2}\). The distance between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For \(d \leq \sqrt{2}\), we need:
\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 \leq 2 \]
This condition implies either the points are adjacent (distance 1) or diagonal neighbors (distance \(\sqrt{2}\)). In other words, any pair of selected points can be at most one unit apart in either the horizontal or vertical direction, or at most one unit diagonalized.
Based on this understanding, we examine the configurations of choosing four points:
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Single \(2 \times 2\) block: The most straightforward approach is to select points from a \(2 \times 2\) block. Each block will be of the following configurations (there are 9 such blocks):
\[ \begin{array}{|c|c|} \hline (i, j) & (i, j+1) \ \hline (i+1, j) & (i+1, j+1) \ \hline \end{array} \] with \(i,j \in {0, 1, 2}\). Clearly, selecting any four points from such a block will meet the distance condition.
The number of \(2 \times 2\) blocks is:
\[ 3 \times 3 = 9 \]
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Non-contiguous configurations: There is no way to select points that are not in a \(2 \times 2\) block while satisfying the maximum distance condition. If a point is chosen from outside a selected \(2 \times 2\) block, it will exceed the distance limit of \(\sqrt{2}\) from at least one of the points in the block.
Hence, the valid configurations solely consist of selecting all four points from each of the \(2 \times 2\) blocks.
Thus, the total number of ways to choose four points such that the distance condition is satisfied is exactly the number of \(2 \times 2\) blocks:
\[ \boxed{9} \]