11.6. A $10 \times 10$ chessboard has $k$ rooks. A square on the board that can be attacked by a rook is called "dangerous" (the square occupied by the rook itself is also considered dangerous). If removing any rook results in at least one dangerous square becoming safe, find the maximum possible value of $k$.
11. 6. $k_{\max }=16$.

Consider a chessboard with $k$ rooks satisfying the problem's conditions. There are two cases to consider.
(1) Each row (column) has a rook.
In this case, all squares are dangerous. If there is a row (column) with at least two rooks, removing one of these rooks will still leave all squares dangerous. This is a contradiction.
Therefore, each row (column) has only one rook. Thus, $k=10$.
(2) At least one row and one column have no rooks.
For each rook, in its row (or column), it is the only rook (otherwise, removing this rook does not reduce the number of dangerous squares). Associate this rook with this row (or column). If all rooks are associated with 9 rows (or columns), then $k=9$; if the total number of rows and columns associated with all rooks is no more than 8, then $k \leqslant 8+8=16$.
An example for $k=16$.
The 16 rooks are located at
$(1, i) 、(i, 1)(i=3,4, \cdots, 10)$.

Removing the rook from square $(1, i)$ makes square $(2, i)$ change from dangerous to safe; removing the rook from square $(i, 1)$ makes square $(i, 2)$ change from dangerous to safe.