1. In the game of Minesweeper, a number on a square denotes the number of mines that share at least one vertex with that square. A square with a number may not have a mine, and the blank squares are undetermined. How many ways can the mines be placed in this configuration?
Answer: 95 Let $A$ be the number of mines in the first two columns. Let $B, C, D, E$ be the number of mines in the third, fourth, fifth, and sixth columns, respectively. We need to have $A+B=2$, $B+C+D=1$, and $D+E=2$. This can happen in three ways, which are $(A, B, C, D, E)=$ $(2,0,1,0,2),(2,0,0,1,1),(1,1,0,0,2)$. This gives $(10)(2)(1)+(10)(3)(2)+(5)(3)(1)=95$ possible configurations.