(a) Find the minimum number of lines required to hit every vertex of a 10 ×10 square grid of dots,
if no line is allowed to be horizontal or vertical. (Prove it is the minimum.)
(b) Prove that if 8 2 ×2 blocks of squares are removed from an 8 ×8 chessboard, then there is at
least one 2 ×2 block in the remaining squares. Is the same true if 9 2 ×2 blocks are removed?
(c) Suppose squares of an 8 ×8 chessboard are covered in grass, which spreads as follows: Grass
spreads to a square when two adjacent squares (i.e. squares that share an edge) are covered.
Find the minimum number of squares which must initially be covered in grass to ensure that
the whole chessboard is eventually covered in grass. (Prove it is the minimum.)
Hints:
(a) How many points are on the edge of the square grid?
(b) Represent the 49 2 ×2 blocks of the chessboard as a 7 ×7 grid.
(c) Consider the perimeter of the grassy area
