Question 90
A $9 \times 9$ square was divided into 81 unit squares $1 \times 1$, and 8 of them were painted black and the rest white.
a) Show that, no matter where they were painted, it is always possible to find a $3 \times 3$ square, formed by juxtaposing unit squares and with sides parallel to the sides of the board, all of whose squares are white.
b) Give an example of a painting of the board in which it is not possible to find any rectangle with more than 9 unit squares, also formed by juxtaposing unit squares and with sides parallel to the sides of the board, that contains no black unit square.
a) Show that, no matter where they were painted, it is always possible to find a $3 \times 3$ square, formed by juxtaposing unit squares and with sides parallel to the sides of the board, all of whose squares are white.
b) Give an example of a painting of the board in which it is not possible to find any rectangle with more than 9 unit squares, also formed by juxtaposing unit squares and with sides parallel to the sides of the board, that contains no black unit square.