Question 34
In an $n \times n$ table some of the cells are black and the rest of them are white. *Alice* and *Bob* each have a copy of this table and are trying to make the whole table red in the following ways:
If Alice finds a cell that is the only black cell in its row, she changes the color of all the cells in its column to red. If Bob finds a cell that is the only black cell in its column, he changes the color of all the cells in its row to red.
Prove that Alice can make the whole table red if and only if Bob can.
If Alice finds a cell that is the only black cell in its row, she changes the color of all the cells in its column to red. If Bob finds a cell that is the only black cell in its column, he changes the color of all the cells in its row to red.
Prove that Alice can make the whole table red if and only if Bob can.