Minesweeper Queens

Question

_{Based off of: Ken Duisenberg's Puzzle of the Week Oct. 15, 2003: Minesweeper Queens}

For the puzzle, in the given diagram, the numbered squares are empty, from which the given mines can be attacked by a chess queen. As in standard chess, a queen can attack all eight directions, and only one mine in a single direction.

For example:

Place 5 mines:

1				2
			4
			3

has 4 solutions, one of them being (x represents a mine)

+---+---+---+---+---+
| 1 | x | x | x | 2 |
+---+---+---+---+---+
|   |   | x | 4 |   |
+---+---+---+---+---+
|   |   |   | 3 | x |
+---+---+---+---+---+

---

---

---

To get the $\color{green}✓$:

• Place 13 mines in this grid
• Show the steps you took.

+---+---+---+---+---+---+
|   | 4 |   | 4 |   |   |
+---+---+---+---+---+---+
| 4 |   |   |   |   |   |
+---+---+---+---+---+---+
|   |   | 6 |   |   |   |
+---+---+---+---+---+---+
| 5 |   |   |   |   |   |
+---+---+---+---+---+---+
|   |   |   | 5 |   |   |
+---+---+---+---+---+---+
| 3 |   | 4 |   |   |   |
+---+---+---+---+---+---+

Answer 1

Two numbers in the grid are forcing 3 mines:

From this point, I feel like there are many possible paths, just by trying to satisfy each number with the closest mines that don't contradict any other already cleared number. Here's a possible next step with 10 mines:

Now, 5 spots on the right are not seen by any number. Just pick your favourite 3 and place mines in them to reach the required number of 13 mines.