1. The first hint (A1) tells us that B5 and C5 are safe, so mark those safe.
2. B5’s hint tells us that every column contains exactly one pair of vertically adjacent mines. For column 5, the pair of vertically adjacent mines can only fit in D5/E5 or E5/F5; either way, E5 must be a mine. C5’s hint tells us that columns 4 and 5 don’t have any mines on the same row, so E4 must be safe.
3. E4 tells us A4 is safe, so mark A4 safe.
4. The pair of mines that column 4 must have can only be either D4/C4, or C4/B4; either way, C4 must be a mine.
5. A4 tells us that column 4 contains 3 mines, but B5 says there can’t be 3 vertically adjacent mines. So the third mine in column 4 must be in F4.
6. Columns 4 and 5 don’t have any horizontally adjacent mines (C5’s hint). There is a mine in F4, so F5 must be safe. E5 is a mine and must be adjacent to another mine, so this means that D5 is a mine, and D4 is safe. C4 also must be adjacent to another mine, so B4 must be a mine.
7. D4 says that B3 is a mine, so mark B3 as a mine. F5 tells us that no 2×2 square is all safe or all mines. B3, B4, and C4 are all mines, so the fourth square–C3–must be safe.
8. C3 tells us that no row has 3 consecutive mines. B3 and B4 are both mines, so B2 must be safe.
9. B2 tells us that C1 is safe, so mark C1 safe.
10. C1 tells us that E2 is a mine, so mark E2 as a mine.
11. Every column must have a pair of vertically adjacent mines in it (B5’s hint), and in column 1 that pair of mines can only fit in either D1/E1 or E1/F1. Therefore, E1 must be a mine. E1 and E2 are both mines, and we can’t have three adjacent mines in a row (C3’s hint), so E3 must be safe.
12. E3 tells us that row F has one fewer mine than row D. Right now, both rows have one mine each (D5 and F4). We know that column 1 and column 2 must each contain a vertical pair of mines, and those pairs both include row E (the pair cannot be C2/D2 in the second column because then we would have 3 vertically adjacent mines, which violates hint B5). But, the second mine in each of those pairs cannot be on the same row (both D1 and D2, or both F1 and F2), because then we would have a 2×2 square of mines, which is not allowed (F5’s hint). Therefore, row D and row F are both going to get one more mine from those pairs, though we don’t know yet which ones. This means that row D and row F each have 2 guaranteed mines. Row D must have one more mine than row F has, and the only place this third mine can be is in D3. Therefore, D3 must be a mine, and F3 must be safe. (See graphic below)
13. Column 3 must contain a vertical pair of mines (B5’s hint), so A3 must be a mine, to make a pair with B3.
14. F3 tells us that B1 is safe, so mark B1 safe.
15. Now we have two 2×2 squares where 3 out of 4 squares are safe– A1/B1/B2, and B1/B2/C1. No 2×2 square can be all safe or all mines (F5’s hint), so this means that A2 and C2 must both be mines.
16. B5 tells us that we can’t have 3 vertical mines in a column, so D2 can’t be a mine (C2 and E2 are already mines). So, D2 must be safe.
17. Column 2 still needs a vertical mine pair, and the only remaining space next to a mine is F2. So, F2 is a mine.
18. E1, E2 and F2 are all mines, so F1 must be safe, so that we don’t violate F5’s rule.
19. B1 tells us that F1 and A5 are either both safe or both mines. F1 is safe, so A5 must also be safe.
20. Finally, column 1 needs its vertical pair of mines, so D1 must be a mine, to pair it with E1.