11.46 On an $8 \times 8$ chessboard, some squares are marked with stars, such that
(1) no two squares marked with stars share a common edge or vertex;
(2) every unmarked square shares a common edge or vertex with at least one marked square.
What is the minimum number of squares that need to be marked with stars? Explain your reasoning.
[Solution] The right figure is marked with 9 asterisks, which obviously meets the requirements of the problem. And in the figure, 9 checkmarks are marked. No matter which square on the chessboard is marked with an asterisk, it can share at most one side or one vertex with one of these 9 squares. Therefore, to meet the requirements of the problem, at least 9 squares must be marked with asterisks. In summary, to meet the requirements of the problem, the minimum number of asterisks that need to be marked is 9.