we call the central $22$ columns(rows) middle columns(rows) and the rest $78$ columns(rows) outside columns(rows) for a column or grid, we call the central $22$grids middle grids, and the rest $78$ grids outside grids. Observation 1 in each column(row), there are at most two rooks. Observation 2 if a column(row) has two rooks, then these $100-2\times(100-61)=22$ middle grids are empty. Now, if there are at least $179$ rooks, then at least $79$ columns that has two rooks. Thus, At least $1$ of these middle columns has two rooks. Thus, the two rooks in this column are in the [middle grids] of the two rows they are in, and that two rows both only has one rook. Also, note that there are at least $79$ rows that has two rooks, and two of the $78$ outside rows have only one rook in each. Thus, At least $3$ of these middle rows have two rooks. Thus, the two rooks in each of these rows are in the [middle grids] of the two columns they are in, and that two columns both only has one rook. Also, note that there are at least $79$ columns that has two rooks, and $2\times 3=6$ of the $78$ outside columns have only one rook in each. Thus, At least $7$ of these middle columns have two rooks. ... By induction, there will be too many middle columns/rows have two rooks, which is impossible.