Example 1 A $6 \times 6$ chessboard has 36 squares. Placing chess pieces in the squares, if there are 4 pieces of the same color in a straight line (horizontal, vertical, or at a $45^{\circ}$ diagonal), it is called a "four-in-a-row". Player A places white pieces, and Player B places black pieces. If A goes first, what is the minimum number of pieces A needs to place to ensure that B cannot subsequently form a four-in-a-row ${ }^{[1]}$?
Solution: At least 10 chess pieces need to be placed.
This is because, to prevent the four-in-a-row formation by the black pieces placed subsequently by player B, player A must place at least one white piece in each $1 \times 4$ rectangle, meaning that at least 8 white pieces must be placed in the $2 \times 4$ rectangles on the four sides as shown in Figure 1.

If at least 2 white pieces are placed in the 4 central squares, then the number of white pieces is no less than 10.

If no white pieces are placed in the 4 central squares, then at least 2 white pieces must be placed in each of the two middle columns, at least 2 white pieces in each of the two middle rows, and at least 2 white pieces on each of the two main diagonals, and these white pieces are distinct, so at least $2 \times 6 = 12 (>10)$ white pieces must be placed.

If exactly 1 white piece is placed in the 4 central squares, by symmetry, assume it is placed at position $C$ in Figure 2, with no white pieces at $A$, $B$, or $D$. Then at least 1 white piece must be placed at $E$ and $F$, at least 1 white piece at $G$ and $H$, at least 2 white pieces on the diagonal of $A$ and $D$, and at least 1 white piece on the diagonal $M Q$, and these white pieces are distinct (at least 5 in total). Thus, by the pigeonhole principle, at least one of the two shaded areas in Figure 2 must contain 3 white pieces (assume the upper left $2 \times 4$ square contains 3 pieces), and by Figure 1, the part outside $A$, $B$, $C$, and $D$ in Figure 2 must contain at least $3 + 3 \times 2 = 9$ pieces, plus $C$, making a total of at least 10 white pieces.

Figure 3 shows one possible placement (shaded areas indicate where white pieces are placed).