Problem 1304

For $k = 0$ this is impossible, since at least one non-mine field will be adjacent to the bombs unless the entire field is covered with bombs.

* For $k = 1$ we take $a = 1, b = 3 \cdot 1011 - 1$, and 1011 mines spaced two-spaces apart, with one on one of the endpoints, so the total number of fields marked 1 is 2021.

* For $k = 2$ we again take $a = 1, b$ large enough, and place 2021 empty fields in isolation of each other, and not on any of the endpoints.

* For $k = 3$, we take $a = 2, b$ large enough, and $2019/3$ mutually isolated L-triminoes as empty spaces, with two corner points on opposite sides of the longer edge also empty.

* For $k = 4$, we take $a = 3, b$ large enough, and 1008 pairs of adjacent empty fields on the edge, with one empty field on the middle row surrounded on all diagonals by 4 empty fields on the edge, resembling an X shape.

* For $k = 5$ we take $a = 3, b$ large enough, and place 2021 isolated empty fields on the edge.

* For $k = 6$ we take $a$ and $b$ large enough, and place $2013/3$ isolated empty L-triminoes and 2 diamond-shaped regions of 4 fields each.

* For $k = 7$ it is impossible to place an empty field on the edge and since each empty field in the interior is adjacent to exactly one other empty field it follows that the empty fields are paired and thus their total number cannot be 2021, an odd number.

* For $k = 8$ we simply take 2021 isolated empty fields in the interior.

Thus, the set of solutions is: $\{1, 2, 3, 4, 5, 6, 8\}$. $\square$