3. In each cell of a $2017 \times 2017$ grid, there is a lamp, and each lamp has only two states: on or off. A lamp is called "bad" if and only if there is an even number of lamps that are on among its adjacent lamps. Find the minimum possible number of bad lamps in this $2017 \times 2017$ grid.

Note: If two lamps are in cells that share an edge, then these two lamps are considered adjacent.
3. Color each cell of a $2017 \times 2017$ grid in a checkerboard pattern, with the cell in the first row and first column being black. A lamp in a black cell is called a black lamp, and a lamp in a white cell is called a white lamp. Assign a value of 1 to a lamp that is on, and 0 to a lamp that is off.

Using the center of the central cell of the grid as the origin, and the directions parallel and perpendicular to the grid lines as the $x$-axis and $y$-axis respectively, with the side length of each cell as the unit length, establish a Cartesian coordinate system. Assume that each lamp is located at the center of its cell, and use the coordinates of the cell to represent its position.

Thus, the lamp at position $(i, j)$ is a black lamp if and only if $i+j$ is even.

If there are no faulty lamps, i.e., each lamp is adjacent to an odd number of lamps that are on.

For the lamp at position $(i, j)$, let $a_{i, j}$ be the sum of the values of the adjacent lamps, and let $S$ be the sum of all $a_{i, j}$ where both $i$ and $j$ are even.

On one hand, note that each $a_{i, j}$ is odd, and there are an odd number of $(i, j)$ where both $i$ and $j$ are even, so $S$ is odd.

On the other hand, for a black lamp at position $(i, j)$ where both $i$ and $j$ are even, each adjacent white lamp is adjacent to exactly two black lamps at positions where both coordinates are even, meaning each such white lamp contributes twice its value to $S$, which is even. Therefore, $S$ is the sum of several even numbers, leading to a contradiction.
Thus, there must be at least one faulty lamp.
Consider the following scenario.
For a black lamp at position $(i, j)$:
If $\max \{|i|,|j|\} \equiv 0,1(\bmod 4)$, then assign a value of 0;

If $\max \{|i|,|j|\} \equiv 2,3(\bmod 4)$, then assign a value of 1.
For a white lamp at position $(i, j)$:
If $\max \{|i|,|j|-1\} \equiv 0,1(\bmod 4)$, then assign a value of 0;

If $\max \{|i|,|j|-1\} \equiv 2,3(\bmod 4)$, then assign a value of 1.

In this case, for a white lamp at position $(i, j)$, since $i$ and $j$ have different parities, assume $|i| < |j|$. Similar to the analysis for white lamps above, this lamp is not faulty.
In summary, the minimum possible number of faulty lamps is 1.