Is it possible to remove one square from a 5 × 5 board so
that the remaining 24 squares can be covered by eight 3 × 1
rectangles? If yes, find all such squares
(Hint: A domino is a 2 × 1 rectangle. As you may know,
if two diagonally opposite squares of an ordinary 8 × 8–
chessboard are removed, the remaining 62 squares cannot
be covered by 31 non-overlapping dominos. The reason
being, after removing the two corners 32 squares of one
colour and 30 of the other are left. No matter how you place
a domino it will cover one white and one black square.)
I found that if the central square is removed, the remaining squares can be covered. I don't think there are any other squares which apply.
How do you show that you cannot cover the board if you remove any other square?