The Hard Sudoku
Here is a hard Sudoku. We have solved this puzzle step-by-step by using various techniques.
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Show me the Final Solution directly.
To begin with, we need to write down all the candidate numbers for all the empty cells.
Step 1: Naked Single in cell ( F3 )
The cell has a single candidate with the value 6. Hence this cell is solved with the value 6.
Candidate 6 removed from cells (F1, F2, F5, A3, B3, G3, E2).
Step 2: Naked Single in cell ( G6 )
The cell has a single candidate with the value 3. Hence this cell is solved with the value 3.
Candidate 3 removed from cells (G4, G8, A6, B6, D6, H4, H5).
Step 3: Naked Single in cell ( E2 )
The cell has a single candidate with the value 3. Hence this cell is solved with the value 3.
Candidate 3 removed from cells (E4, E7, F2).
Step 4: Naked Single in cell ( E7 )
The cell has a single candidate with the value 4. Hence this cell is solved with the value 4.
Candidate 4 removed from cells (E4, A7, B7, F7, H7, F8, F9).
Step 5: Naked Single in cell ( F2 )
The cell has a single candidate with the value 8. Hence this cell is solved with the value 8.
Candidate 8 removed from cells (F1, F7, F8, A2, C2, D1).
Step 6: Naked Single in cell ( E4 )
The cell has a single candidate with the value 6. Hence this cell is solved with the value 6.
Candidate 6 removed from cells (A4, C4).
Step 7: Naked Single in cell ( F1 )
The cell has a single candidate with the value 7. Hence this cell is solved with the value 7.
Candidate 7 removed from cells (F7, F9, D1).
Step 8: Hidden Single in cell ( D7 )
The cell is the only cell in row D with the candidate value 8. Hence this cell is solved with the value 8. Candidate 8 removed from cell (I7).
Step 9: Hidden Single in cell ( D9 )
The cell is the only cell in row D with the candidate value 7. Hence this cell is solved with the value 7.
Step 10: Hidden Single in cell ( D6 )
The cell is the only cell in row D with the candidate value 5. Hence this cell is solved with the value 5.
Step 11: Hidden Single in cell ( F5 )
The cell is the only cell in row F with the candidate value 4. Hence this cell is solved with the value 4. Candidate 4 removed from cells (A5, B5, C5).
Step 12: Hidden Single in cell ( G1 )
The cell is the only cell in row G with the candidate value 6. Hence this cell is solved with the value 6. Candidate 6 removed from cells (A1, B1, I1, I2).
Step 13: Hidden Single in cell ( I4 )
The cell is the only cell in row I with the candidate value 7. Hence this cell is solved with the value 7. Candidate 7 removed from cell (A4).
Step 14: Hidden Single in cell ( A1 )
The cell is the only cell in column 1 with the candidate value 8. Hence this cell is solved with the value 8. Candidate 8 removed from cells (A4, A5).
Step 15: Hidden Single in cell ( A3 )
The cell is the only cell in column 3 with the candidate value 5. Hence this cell is solved with the value 5. Candidate 5 removed from cell (A2).
Step 16: Hidden Single in cell ( D3 )
The cell is the only cell in column 3 with the candidate value 1. Hence this cell is solved with the value 1. Candidate 1 removed from cell (D1).
Step 17: Naked Single in cell ( D1 )
The cell has a single candidate with the value 2. Hence this cell is solved with the value 2. Candidate 2 removed from cells (B1, H1, I1).
Step 18: Hidden Single in cell ( C6 )
The cell is the only cell in column 6 with the candidate value 1. Hence this cell is solved with the value 1. Candidate 1 removed from cell (C5).
Step 19: Pointing Pair in cells ( B1 , B3 )
These cells are the only cells in box 1 with the candidate value 4. The candidate must be in one of these cells and can be removed from other cells in row B. Candidates 4 removed from cells (B8).
Step 20: Pointing Pair in cells ( A6 , B6 )
These cells are the only cells in column 6 with the candidate value 9. The candidate must be in one of these cells and can be removed from other cells in box 2. Candidates 9 removed from cells (A4, A5, B5, C4, C5).
Step 21: Forcing Chain (B1)
Cell (B1) has 2 possibilities “4” and “9”. The following results agree in both cases and therefore can be confirmed:
(1) Cell (A7) cannot be 3
(2) Cell (A7) cannot be 6
Case 1
Step A: Suppose (B1) is 4
Step B: (H1) cannot be 4
Step C: (H9) must be 4
Step D: (H9) cannot be 3, (C9) cannot be 4
Step E: (C9) must be 6, (H7) must be 3
Step F: (A7) cannot be 3, (A7) cannot be 6
Case 2
Step a: Suppose (B1) is 9
Step b: (B6) cannot be 9
Step c: (B6) must be 7
Step d: (A6) cannot be 7
Step e: (A7) must be 7
Step f: (A7) cannot be 3, (A7) cannot be 6
Step 22: Forcing Chain (B1)
Cell (B1) has 2 possibilities “4” and “9”. In either case, digit “2” cannot be a candidate for cell (H2).
Case 1
Step A: Suppose (B1) is 4
Step B: (H1) cannot be 4
Step C: (H9) must be 4
Step D: (H9) cannot be 5
Step E: (H2) must be 5
Step F: (H2) cannot be 2
Case 2
Step a: Suppose (B1) is 9
Step b: (B1) cannot be 4
Step c: (B3) must be 4
Step d: (B3) cannot be 2
Step e: (G3) must be 2
Step f: (H2) cannot be 2
Step 23: Forcing Chain (B1)
Cell (B1) has 2 possibilities “4” and “9”. In either case, digit “3” cannot be a candidate for cell (F7).
Case 1
Step A: Suppose (B1) is 4
Step B: (H1) cannot be 4
Step C: (H9) must be 4
Step D: (H9) cannot be 3
Step E: (H7) must be 3
Step F: (F7) cannot be 3
Case 2
Step a: Suppose (B1) is 9
Step b: (C2) cannot be 9
Step c: (C8) must be 9
Step d: (F8) cannot be 9
Step e: (F7) must be 9
Step f: (F7) cannot be 3
Step 24: Naked Single (F7)
The cell has a single candidate with the value 9. Hence this cell is solved with the value 9.
Candidate 9 removed from cells (A7, B7, F8).
Step 25: Forcing Chain (I8)
Cell (B3) has 2 possibilities “2” and “4”. In either case, digit “2” cannot be a candidate for cell (I8).
Case 1
Step A: Suppose (B3) is 2
Step B: (A2) cannot be 2, (C2) cannot be 2
Step C: (I2) must be 2
Step D: (I8) cannot be 2
Case 2
Step a: Suppose (B3) is 4
Step b: (G3) cannot be 4
Step c: (G8) must be 4
Step d: (G8) cannot be 8
Step e: (I8) must be 8
Step f: (I8) cannot be 2
Step 26: Forcing Chain (A2)
Cell (C9) has 2 possibilities “4” and “6”. In either case, digit “9” cannot be a candidate for cell (A2).
Case 1
Step A: Suppose (C9) is 4
Step B: (H9) cannot be 4
Step C: (H1) must be 4
Step D: (B1) cannot be 4
Step E: (B1) must be 9
Step F: (A2) cannot be 9
Case 2
Step a: Suppose (C9) is 6
Step b: (C2) cannot be 6
Step c: (A2) must be 6
Step d: (A2) cannot be 9
Step 27: Forcing Chain (H9)
Cell (C9) has 2 possibilities “4” and “6”. In either case, digit “3” cannot be a candidate for cell (H9).
Case 1
Step A: Suppose (C9) is 4
Step B: (C9) cannot be 6
Step C: (B7) must be 6
Step D: (B7) cannot be 3
Step E: (H7) must be 3
Step F: (H9) cannot be 3
Case 2
Step a: Suppose (C9) is 6
Step b: (C9) cannot be 4
Step c: (H9) must be 4
Step d: (H9) cannot be 3
Step 28: Hidden Single in cell ( H7 )
The cell is the only cell in row H with the candidate value 3. Also in the 9th block (lower-right) 3 appears only in cell H7. Hence this cell is solved with the value 3. Candidate 3 removed from cell (B7).
Step 29: Hidden Single in cell ( F9 )
The cell is the only cell in column 9 with the candidate value 3. Hence this cell is solved with the value 3. Candidate 3 removed from cell (F8).
Step 30: Naked Single in cell ( F8 )
The cell has a single candidate with the value 5. Hence this cell is solved with the value 5. Candidate 5 removed from cell (I8).
Step 31: Naked Single in cell ( I8 )
The cell has a single candidate with the value 8. Hence this cell is solved with the value 8. Candidate 8 removed from cells (G8, I5).
Step 32: Hidden Single in cell ( G4 )
The cell is the only cell in row G with the candidate value 8. Also in the 8th block (lower-middle) 8 appears only in cell G4. Hence this cell is solved with the value 8. Candidate 8 removed from cell (C4).
Step 33: Hidden Single in cell ( C5 )
The cell is the only cell in row C with the candidate value 8. Also in the 2nd block (upper-middle) 8 appears only in cell C5. Hence this cell is solved with the value 8.
Step 34: Locked Candidates ( H4 & H5)
In row H, candidate 2s, marked in green, also falls in block 8 (lower-middle). So, in the 8th block, the digit “2” must be one of those marked in green and cannot be a candidate for any other cells in the same block.
Step 35: Forcing Chain (C8)
Cell (B1) has 2 possibilities “4” and “9”. In either case, digit “2” cannot be a candidate for cell (C8).
Case 1
Step A: Suppose (B1) is 4
Step B: (B3) cannot be 4
Step C: (B3) must be 2
Step D: (G3) cannot be 2
Step E: (G8) must be 2
Step F: (C8) cannot be 2
Case 2
Step a: Suppose (B1) is 9
Step b: (C2) cannot be 9
Step c: (C8) must be 9
Step d: (C8) cannot be 2
Step 36: Naked Pair (I1, I5)
In row I, the two cells I1 and I5 have the same two possibilities 1 and 9, so these two digits each will occupy one of those two green cells and cannot be possibilities for any other cells in the same row. Candidate 9 removed from cell (I2).
Step 37: Forcing Chain (A4)
Cell (B1) has 2 possibilities “4” and “9”. In either case, digit “2” cannot be a candidate for cell (A4).
Case 1
Step A: Suppose (B1) is 4
Step B: (B3) cannot be 4
Step C: (B3) must be 2
Step D: (C2) cannot be 2
Step E: (C4) must be 2
Step F: (A4) cannot be 2
Case 2
Step a: Suppose (B1) is 9
Step b: (C2) cannot be 9
Step c: (H2) must be 9
Step d: (H4) cannot be 9
Step e: (H4) must be 2
Step f: (A4) cannot be 2
Step 38: Nishio (A4)
If cell (A4) is “3”, cell (B3) will have no digits as candidates. Therefore, cell (A4) cannot be “3”.
Step A: Suppose (A4) is 3
Step B: (A4) cannot be 4
Step C: (A8) must be 4, (C4) must be 4
Step D: (G8) cannot be 4, (C4) cannot be 2
Step E: (C2) must be 2, (G3) must be 4
Step F: (B3) cannot be 2, (B3) cannot be 4
Step 39: Naked Single in cell ( A4 )
The cell has a single candidate with the value 4. Hence this cell is solved with the value 4.
Candidate 4 removed from cells (A8, C4).
Step 40: Naked Single in cell ( C4 )
The cell has a single candidate with the value 2. Hence this cell is solved with the value 2.
Candidate 2 removed from cells (A5, B5, C2, H4).
Step 41: Naked Single in cell ( H4 )
The cell has a single candidate with the value 9. Hence this cell is solved with the value 9.
Candidate 9 removed from cells (D4, H1, H2, H5, I5).
Step 42: Naked Singles
Naked Single of value 3 in cell (D4), and candidate 3 removed from cell D5.
Naked Single of value 5 in cell (H2), and candidate 5 removed from cells H9, I2.
Naked Single of value 1 in cell (I5), and candidate 1 removed from cells H5, I1.
Step 43: Naked Singles
Naked Single of value 9 in cell (D5).
Naked Single of value 2 in cell (H5).
Naked Single of value 4 in cell (H9), and candidate 4 removed from cells C9, G8, H1.
Naked Single of value 9 in cell (I1), and candidate 9 removed from cell B1.
Naked Single of value 2 in cell (I2), and candidate 2 removed from cells A2, G2, I7.
Step 44: Naked Singles
Naked Single of value 6 in cell ( A2 ), and candidate 6 removed from cells A5, C2.
Naked Single of value 4 in cell ( B1 ), and candidate 4 removed from cell B3.
Naked Single of value 6 in cell ( C9 ), and candidate 6 removed from cells B7, I9.
Naked Single of value 4 in cell ( G3 ).
Naked Single of value 2 in cell ( G8 ), and candidate 2 removed from cells A8, B8.
Naked Single of value 1 in cell ( H1 ).
Naked Single of value 6 in cell ( I7 ), and candidate 6 removed from cells B7, I9.
Step 45: Naked Singles
Naked Single of value 3 in cell ( A5 ), and candidate 3 removed from cells A8, B5.
Naked Single of value 2 in cell ( B3 ), and candidate 2 removed from cell B7.
Naked Single of value 9 in cell ( C2 ), and candidate 9 removed from cell C8.
Naked Single of value 5 in cell ( I9 ).
Step 46: Naked Singles
Naked Single of value 9 in cell ( A8 ), and candidate 9 removed from cell B8.
Naked Single of value 6 in cell ( B5 ).
Naked Single of value 7 in cell ( B7 ), and candidate 7 removed from cells A7, B6.
Naked Single of value 4 in cell ( C8 ).
Step 47: Naked Singles
Naked Single of value 2 in cell ( A7 ).
Naked Single of value 9 in cell ( B6 ), and candidate 9 removed from cell A6.
Naked Single of value 3 in cell ( B8 ).
Step 48: Naked Single
Naked Single of value 7 in cell (A6).
Final Solution
