Question 1
Some checkers on an $n\times n$ board satisfy:
(a) every empty square shares a side with an occupied square;
(b) between any two occupied squares there is a sequence of occupied squares joining them, with consecutive squares sharing a side.
Prove that at least $(n^2-2)/3$ checkers have been placed.
(a) every empty square shares a side with an occupied square;
(b) between any two occupied squares there is a sequence of occupied squares joining them, with consecutive squares sharing a side.
Prove that at least $(n^2-2)/3$ checkers have been placed.