The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?

[i]S. Berlov[/i]
1. **Proving the minimality:**
   - To achieve a chessboard coloring, we need to ensure that the cells alternate in color such that no two adjacent cells (horizontally or vertically) share the same color.
   - Consider the set \( S \) of cells on the main diagonal, i.e., cells \((i, i)\) for \( i = 1, 2, \ldots, 100 \). In a chessboard coloring, these cells must all be of the same color (either all black or all white).
   - Each cell in \( S \) lies in a different row and column from every other cell in \( S \). Therefore, to change the color of each cell in \( S \), we need at least one move per cell, resulting in a minimum of \( |S| = 100 \) moves.

2. **Construction for 100 moves:**
   - We need to find a sequence of moves that will achieve the desired chessboard coloring in exactly 100 moves.
   - Consider the cells \((2i, 2(51-i))\) for \( i = 1, 2, \ldots, 50 \). These cells are chosen such that each cell is in a different row and column.
   - For each cell \((2i, 2(51-i))\), perform the following:
     - Select the 99 other cells in the same row and recolor them.
     - Select the 99 other cells in the same column and recolor them.
   - This special move for each cell \((2i, 2(51-i))\) requires 2 moves (one for the row and one for the column).
   - Since there are 50 such cells, the total number of moves is \( 50 \times 2 = 100 \).

Thus, the smallest number of moves needed to achieve a chessboard coloring is 100.

The final answer is \(\boxed{100}\).