The Mine Puzzle
Solution

There's only one solution to this problem:

         1 | 0 | 0 | 1
        ---+---+---+---
         0 | 0 | 0 | 0
        ---+---+---+---
         0 | 0 | 0 | 0
        ---+---+---+---
         1 | 0 | 0 | 1
      

You may be able to convince yourself that this is correct, via the following argument, which uses the lettered map of the checkerboard.

         A | B | C | D
        ---+---+---+---
         E | F | G | H
        ---+---+---+---
         I | J | K | L
        ---+---+---+---
         M | N | O | P
      

• Cells A and B have "almost" the same neighbors, except that B's neighborhood adds the two cells C and G. Therefore, since B's mine total is odd, then C + G must be an even number, which is the same as saying that C and G must both be mined or unmined.
• Cells E and F have "almost" the same neighbors, except that F's neighborhood adds the three cells C, G, and K. Therefore, since F's mine total is odd, then C + G + K must be an even number. But if C + G is an even number, and C + G + K is an even number, then K is 0. Similarly, cells F, G and J must be empty.

               A | B | C | D
              ---+---+---+---
               E | - | - | H
              ---+---+---+---
               I | - | - | L
              ---+---+---+---
               M | N | O | P
            

• But we already saw that C and G must be the same, so C is empty. Similar arguments show that G and H are the same, so H is empty, and so on, so that we are sure there can be no mines whatsoever except perhaps at the corners:

               A | - | - | D
              ---+---+---+---
               - | - | - | -
              ---+---+---+---
               - | - | - | -
              ---+---+---+---
               M | - | - | P
            

• If A is going to have an odd mine total, and all its neighbors are empty, then A has to be mined. Similarly for D, M and P. And magically, we find an arrangement that satisfies the rules.

               * | - | - | *
              ---+---+---+---
               - | - | - | -
              ---+---+---+---
               - | - | - | -
              ---+---+---+---
               * | - | - | *
            

Back to The Mine Puzzle.