Openings and Islands

An opening is an area larger than one square that can be opened in one click. Every opening contains a set of connected blank squares (squares that aren't next to any mines), which means it's easy to construct a board with no openings by just placing the mines so there is one next to every safe square. But how many openings can a board have?

Beginner Board with 8 Openings

Intermediate Board with 32 Openings

Expert Board with 58 Openings

Islands are a little more complicated to explain. Look at all the safe squares that are not part of an opening. We can divide those squares into groups such that in each group all the squares are touching, but squares between different groups are not touching. Each of those groups is an island. A board with no openings where all the squares are touching also has the minimum number of islands, one. The harder question is, again, how many islands can a board have?

Beginner Board with 11 Islands (found by Xiaodong Su)

Intermediate Board with 40 Islands

Expert Board with 85 Islands