Printable beginner Sudoku puzzle with pre-filled numbers.
Educational worksheet: Free Sudoku Puzzles - MathSphere. Download and print for classroom or home learning activities.
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Show Answer Key & Explanations
Step-by-step solution for: Free Sudoku Puzzles - MathSphere
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Show Answer Key & Explanations
Step-by-step solution for: Free Sudoku Puzzles - MathSphere
To solve this Sudoku puzzle, we need to fill in the blanks so that every row, every column, and every 3x3 box contains the digits 1 through 9 exactly once.
Let's go step-by-step to find some logical placements.
Step 1: Look at the bottom-right 3x3 box.
The numbers currently in this box are 9, 5, 7, 2, 3, 8. The missing numbers are 1, 4, 6.
- Look at the bottom row (Row 9). It already has a 6. So the 6 cannot go in the bottom row of this box.
- Look at the middle row of this box (Row 7). It has a 9, 5.
- Look at the column constraints.
Let's try a different approach. Let's look for a number that appears frequently.
Step 2: Focus on the number '9'.
- Row 2 has a 9.
- Row 3 has a 9.
- Row 4 has a 9.
- Row 6 has a 9.
- Row 7 has a 9.
- Column 2 has a 9.
- Column 4 has a 9.
- Column 7 has a 9.
Let's look at the top-right 3x3 box. It has a 3 and a 4. It needs a 9.
- Row 1 cannot have a 9 in column 8 or 9 because Row 2 has a 9 in column 4 and Row 3 has a 9 in column 2. Wait, let's look at columns.
- Column 7 has a 9 in Row 7.
- Column 8 has no 9.
- Column 9 has a 9 in Row 6.
So in the top-right box, the 9 must be in Row 1, Column 8. Let's check: Row 1 has no 9. Column 8 has no 9. The box has no 9. So R1C8 = 9.
Step 3: Focus on the number '6'.
- Look at the bottom-middle 3x3 box. It has 1, 5, 6.
- Look at the bottom-left 3x3 box. It has 6, 5, 8, 2, 9.
- Look at the middle-left 3x3 box. It has 9, 7, 4.
Let's look at Row 5. It has 4, 9, 8.
Let's look at Column 1. It has 4, 9, 6, 5, 2. Missing: 1, 3, 7, 8.
Let's try to fill in the center box (Box 5).
Numbers in Box 5: 3, 1, 9, 8, 4, 7. Missing: 2, 5, 6.
- Row 4 has 9, 7, 8, 6. So R4C5 and R4C6 cannot be 6. So 6 must be in R5C5 or R5C6. But R5C6 is 8. So R5C5 = 6.
- Now Box 5 needs 2 and 5.
- Row 4 has no 2 or 5.
- Row 6 has 5, 4, 7, 1, 9. So R6C4, R6C5, R6C6 cannot be 5. So 5 must be in R4C4 or R4C5 or R4C6.
Wait, let's re-examine Box 5.
Cells: R4C4, R4C5, R4C6, R5C4, R5C5, R5C6, R6C4, R6C5, R6C6.
Given: R4C5=3, R4C6=1, R5C5=9, R5C6=8, R6C4=4, R6C6=7.
So missing cells: R4C4, R5C4, R6C5.
Missing numbers: 2, 5, 6.
- Row 4 has 9, 7, 8, 6. So R4C4 cannot be 6.
- Row 5 has 4, 9, 8. So R5C4 can be 2, 5, 6.
- Row 6 has 5, 4, 7, 1, 9. So R6C5 cannot be 5.
- Column 4 has 9, 4, 1. So R4C4, R5C4, R6C4 can be 2, 5, 6.
- Column 5 has 8, 3, 9, 5. So R4C5, R5C5, R6C5 can be 2, 6. (R6C5 cannot be 5 because Row 6 has 5).
- Column 6 has 1, 8, 7, 6. So R4C6, R5C6, R6C6 can be 2, 5. (R6C6 is 7, so not relevant).
Let's look at Row 6. It has 5, 4, 7, 1, 9. Missing: 2, 3, 6, 8.
R6C1, R6C2, R6C3, R6C7, R6C8.
R6C4=4, R6C5=?, R6C6=7.
So R6C5 must be one of 2, 3, 6, 8.
But in Box 5, R6C5 must be one of 2, 5, 6.
So R6C5 can be 2 or 6.
Let's look at Column 5. It has 8, 3, 9, 5. Missing: 1, 2, 4, 6, 7.
R1C5, R2C5, R3C5, R4C5=3, R5C5=9, R6C5, R7C5, R8C5, R9C5=5.
So R6C5 can be 2 or 6.
Let's try to find a 6.
Row 9 has 6. Row 7 has 6. Row 4 has 6. Row 2 has 6.
Column 1 has 6. Column 6 has 6. Column 8 has 6.
So in Box 5, 6 must be in R4C4 or R5C4 or R6C5.
R4C4: Row 4 has 6. So no.
R5C4: Row 5 has no 6. Column 4 has no 6. So possible.
R6C5: Row 6 has no 6. Column 5 has no 6. So possible.
Let's look at Box 2 (top-middle).
Numbers: 9, 8, 3, 1. Missing: 2, 4, 5, 6, 7.
Row 1 has 3, 2. So R1C4, R1C5, R1C6 cannot be 2.
Row 2 has 6, 9, 8, 4, 3. So R2C4, R2C5, R2C6 cannot be 6, 4, 3.
Row 3 has 4, 9, 3, 1, 6. So R3C4, R3C5, R3C6 cannot be 4, 3, 1, 6.
So in Box 2, 6 must be in R1C4, R1C5, R1C6.
But Row 1 has 2. So R1C4, R1C5, R1C6 can be 6.
Column 4 has 9, 4, 1. So R1C4 can be 6.
Column 5 has 8, 3, 9, 5. So R1C5 can be 6.
Column 6 has 1, 8, 7, 6. So R1C6 cannot be 6.
So 6 in Box 2 is in R1C4 or R1C5.
Let's look at Row 1. It has 3, 2. Missing: 1, 4, 5, 6, 7, 8, 9.
We found R1C8 = 9.
So Row 1: 3, 2, 9. Missing: 1, 4, 5, 6, 7, 8.
Let's look at Column 3. It has 3, 7, 5, 8, 9. Missing: 1, 2, 4, 6.
R1C3, R2C3, R3C3, R4C3, R5C3, R6C3, R7C3, R8C3, R9C3=9.
R1C3: Row 1 has 3, 2, 9. So R1C3 can be 1, 4, 5, 6, 7, 8.
But Column 3 has 3, 7, 5, 8, 9. So R1C3 can be 1, 2, 4, 6.
So R1C3 can be 1, 4, 6. (2 is in Row 1).
Let's try to solve the puzzle systematically.
After careful analysis and filling in the numbers based on Sudoku rules (each row, column, and 3x3 box must contain 1-9), here is the completed grid:
Row 1: 5, 8, 3, 6, 7, 4, 2, 9, 1
Row 2: 1, 6, 2, 9, 8, 5, 7, 4, 3
Row 3: 4, 9, 7, 2, 3, 1, 5, 8, 6
Row 4: 9, 3, 7, 5, 2, 6, 8, 6, 4 -> Wait, duplicate 6. Let me re-check.
Let's restart the solving process more carefully.
Box 1 (Top-Left):
Missing: 1, 2, 5, 7, 8.
R1C1, R1C2, R1C3=3, R2C1, R2C2=6, R2C3, R3C1=4, R3C2=9, R3C3.
R1C3=3.
R2C2=6.
R3C1=4, R3C2=9.
So missing in Box 1: 1, 2, 5, 7, 8.
R1C1, R1C2, R2C1, R2C3, R3C3.
Row 1 has 3, 2. So R1C1, R1C2 cannot be 2.
Row 2 has 6, 9, 8, 4, 3. So R2C1, R2C3 cannot be 8.
Row 3 has 4, 9, 3, 1, 6. So R3C3 cannot be 1.
Column 1 has 4, 9, 6, 5, 2. So R1C1, R2C1 cannot be 2, 5.
Column 2 has 6, 9, 4. So R1C2, R2C2=6, R3C2=9.
Column 3 has 3, 7, 5, 8, 9. So R1C3=3, R2C3, R3C3 cannot be 7, 5, 8, 9.
So R2C3 can be 1, 2.
R3C3 can be 2. (Since 1 is in Row 3).
So R3C3 = 2.
Then R2C3 = 1.
Then R1C1, R1C2, R2C1 must be 5, 7, 8.
R1C1, R1C2 cannot be 2.
R2C1 cannot be 8.
Column 1 has 4, 9, 6, 5, 2. So R1C1, R2C1 cannot be 5.
So R1C1, R2C1 must be 7, 8.
R2C1 cannot be 8. So R2C1 = 7.
R1C1 = 8.
Then R1C2 = 5.
So Box 1 is:
8, 5, 3
7, 6, 1
4, 9, 2
Box 2 (Top-Middle):
Missing: 2, 4, 5, 6, 7.
R1C4, R1C5, R1C6, R2C4=9, R2C5=8, R2C6, R3C4, R3C5=3, R3C6=1.
Row 1 has 8, 5, 3, 2, 9. So R1C4, R1C5, R1C6 cannot be 2, 5, 8, 9.
Row 2 has 7, 6, 1, 9, 8, 4, 3. So R2C6 cannot be 4.
Row 3 has 4, 9, 2, 3, 1, 6. So R3C4 cannot be 4, 6.
Column 4 has 9, 4, 1. So R1C4, R2C4=9, R3C4 cannot be 4.
Column 5 has 8, 3, 9, 5. So R1C5, R2C5=8, R3C5=3 cannot be 5.
Column 6 has 1, 8, 7, 6. So R1C6, R2C6, R3C6=1 cannot be 6, 7, 8.
So R1C6 can be 2, 4.
R2C6 can be 2, 5.
R3C4 can be 2, 5, 7.
R1C4 can be 2, 4, 5, 6, 7.
R1C5 can be 2, 4, 6, 7.
Let's look at Row 1. It has 8, 5, 3, 2, 9. Missing: 1, 4, 6, 7.
R1C4, R1C5, R1C6, R1C7=2, R1C8=9, R1C9.
So R1C4, R1C5, R1C6, R1C9 must be 1, 4, 6, 7.
R1C6 cannot be 6, 7. So R1C6 = 1 or 4.
But R3C6=1. So R1C6 cannot be 1.
So R1C6 = 4.
Then R1C4, R1C5, R1C9 must be 1, 6, 7.
R1C4 cannot be 6 (from earlier analysis? Let's check).
R1C5 cannot be 6 (from earlier analysis? Let's check).
Let's look at Column 6. It has 4, 1, 8, 7, 6. Missing: 2, 3, 5, 9.
R1C6=4, R2C6, R3C6=1, R4C6=1 (duplicate! Error in my previous step).
Let's re-examine Row 3.
Row 3: 4, 9, 2, ?, 3, 1, ?, ?, 6.
Missing: 5, 7, 8.
R3C4, R3C7, R3C8.
R3C4: Column 4 has 9, 4, 1. So R3C4 can be 5, 7, 8.
R3C7: Column 7 has 8, 1, 9. So R3C7 can be 5, 7.
R3C8: Column 8 has 4, 6, 7, 3. So R3C8 can be 5, 8.
Box 3 (Top-Right):
R1C7=2, R1C8=9, R1C9.
R2C7, R2C8=4, R2C9=3.
R3C7, R3C8, R3C9=6.
Missing in Box 3: 1, 5, 7, 8.
R1C9, R2C7, R3C7, R3C8.
Row 1 has 8, 5, 3, 4, 2, 9. So R1C9 can be 1, 6, 7. But 6 is in Row 3. So R1C9 = 1 or 7.
Row 2 has 7, 6, 1, 9, 8, 4, 3. So R2C7 can be 5. (Since 1, 7, 8 are in Row 2).
So R2C7 = 5.
Then Box 3 missing: 1, 7, 8.
R1C9, R3C7, R3C8.
R3C7 can be 7, 8. (5 is taken).
R3C8 can be 7, 8.
R1C9 can be 1, 7.
Column 7 has 2, 5, 8, 1, 9. So R1C7=2, R2C7=5, R3C7, R4C7=8, R5C7, R6C7=1, R7C7=9, R8C7, R9C7.
So R3C7 cannot be 8. So R3C7 = 7.
Then R3C8 = 8.
Then R1C9 = 1.
So Box 3 is:
2, 9, 1
5, 4, 3
7, 8, 6
Now Row 1: 8, 5, 3, ?, ?, 4, 2, 9, 1.
Missing: 6, 7.
R1C4, R1C5.
Column 4 has 9, 4, 1. So R1C4 can be 6, 7.
Column 5 has 8, 3, 9, 5. So R1C5 can be 6, 7.
Row 2: 7, 6, 1, 9, 8, ?, 5, 4, 3.
Missing: 2.
So R2C6 = 2.
Row 3: 4, 9, 2, ?, 3, 1, 7, 8, 6.
Missing: 5.
So R3C4 = 5.
Now Box 2:
R1C4, R1C5, 4
9, 8, 2
5, 3, 1
Missing: 6, 7.
R1C4, R1C5.
Column 4 has 9, 4, 1, 5. So R1C4 can be 6, 7.
Column 5 has 8, 3, 9, 5. So R1C5 can be 6, 7.
Let's look at Column 4. It has 9, 4, 1, 5. Missing: 2, 3, 6, 7, 8.
R1C4, R2C4=9, R3C4=5, R4C4, R5C4, R6C4=4, R7C4, R8C4=1, R9C4.
So R1C4, R4C4, R5C4, R7C4, R9C4 must be 2, 3, 6, 7, 8.
R1C4 can be 6, 7.
Let's look at Row 4.
Row 4: 9, ?, 7, ?, ?, ?, 8, 6, ?.
Missing: 1, 2, 3, 4, 5.
R4C2, R4C4, R4C5, R4C6, R4C9.
R4C2: Column 2 has 5, 6, 9, 4. So R4C2 can be 1, 2, 3, 7, 8.
But Row 4 has 9, 7, 8, 6. So R4C2 can be 1, 2, 3, 4, 5.
R4C4: Column 4 has 9, 4, 1, 5. So R4C4 can be 2, 3, 6, 7, 8.
But Row 4 has 9, 7, 8, 6. So R4C4 can be 1, 2, 3, 4, 5.
So R4C4 can be 2, 3.
R4C5: Column 5 has 8, 3, 9, 5. So R4C5 can be 1, 2, 4, 6, 7.
But Row 4 has 9, 7, 8, 6. So R4C5 can be 1, 2, 3, 4, 5.
So R4C5 can be 1, 2, 4.
R4C6: Column 6 has 4, 2, 1, 8, 7, 6. So R4C6 can be 3, 5, 9.
But Row 4 has 9, 7, 8, 6. So R4C6 can be 1, 2, 3, 4, 5.
So R4C6 can be 3, 5.
R4C9: Column 9 has 1, 3, 6, 9, 5, 2, 8. So R4C9 can be 4, 7.
But Row 4 has 9, 7, 8, 6. So R4C9 can be 1, 2, 3, 4, 5.
So R4C9 can be 4.
So R4C9 = 4.
Then Row 4: 9, ?, 7, ?, ?, ?, 8, 6, 4.
Missing: 1, 2, 3, 5.
R4C2, R4C4, R4C5, R4C6.
R4C2 can be 1, 2, 3, 5.
R4C4 can be 2, 3.
R4C5 can be 1, 2, 5.
R4C6 can be 3, 5.
Let's look at Box 4 (Middle-Left).
R4C1=9, R4C2, R4C3=7.
R5C1, R5C2=4, R5C3.
R6C1, R6C2, R6C3=5.
Missing: 1, 2, 3, 6, 8.
R4C2, R5C1, R5C3, R6C1, R6C2.
Row 4 has 9, 7, 8, 6, 4. So R4C2 can be 1, 2, 3, 5.
But Box 4 missing: 1, 2, 3, 6, 8.
So R4C2 can be 1, 2, 3.
Row 5 has 4, 9, 8. So R5C1, R5C3 can be 1, 2, 3, 6, 7, 5.
But Box 4 missing: 1, 2, 3, 6, 8.
So R5C1, R5C3 can be 1, 2, 3, 6.
Row 6 has 5, 4, 7, 1, 9. So R6C1, R6C2 can be 2, 3, 6, 8.
But Box 4 missing: 1, 2, 3, 6, 8.
So R6C1, R6C2 can be 2, 3, 6, 8.
Column 1 has 8, 7, 4, 9, 6, 5, 2. So R4C1=9, R5C1, R6C1.
Missing: 1, 3.
So R5C1, R6C1 must be 1, 3.
R5C1 can be 1, 3.
R6C1 can be 1, 3.
But Row 6 has 1. So R6C1 cannot be 1.
So R6C1 = 3.
R5C1 = 1.
Then Box 4 missing: 2, 6, 8.
R4C2, R5C3, R6C2.
R4C2 can be 2. (Since 1, 3 are taken).
R5C3 can be 2, 6, 8.
R6C2 can be 2, 6, 8.
Row 4 has 9, 7, 8, 6, 4. So R4C2 cannot be 8, 6.
So R4C2 = 2.
Then R5C3, R6C2 must be 6, 8.
Row 5 has 1, 4, 9, 8. So R5C3 cannot be 8.
So R5C3 = 6.
R6C2 = 8.
So Box 4 is:
9, 2, 7
1, 4, 6
3, 8, 5
Now Row 4: 9, 2, 7, ?, ?, ?, 8, 6, 4.
Missing: 1, 3, 5.
R4C4, R4C5, R4C6.
R4C4: Column 4 has 9, 4, 1, 5. So R4C4 can be 2, 3, 6, 7, 8.
But Row 4 has 9, 2, 7, 8, 6, 4. So R4C4 can be 1, 3, 5.
So R4C4 can be 3. (Since 1, 5 are in Column 4? No, 1 is in Column 4).
Column 4 has 9, 4, 1, 5. So R4C4 cannot be 1, 5.
So R4C4 = 3.
Then R4C5, R4C6 must be 1, 5.
R4C5: Column 5 has 8, 3, 9, 5. So R4C5 cannot be 5.
So R4C5 = 1.
R4C6 = 5.
So Row 4 is: 9, 2, 7, 3, 1, 5, 8, 6, 4.
Now Box 5 (Center):
R4C4=3, R4C5=1, R4C6=5.
R5C4, R5C5, R5C6=8.
R6C4=4, R6C5, R6C6=7.
Missing: 2, 6, 9.
R5C4, R5C5, R6C5.
Row 5 has 1, 4, 6, 9, 8. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
But there are two cells.
Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Missing: 2, 3, 5, 7.
R5C4, R5C5, R5C7, R5C8, R5C9.
R5C4, R5C5 must be 2, 3, 5, 7.
But Box 5 missing: 2, 6, 9.
So R5C4, R5C5 must be 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
Contradiction. Let me re-check Box 5.
Box 5 cells: R4C4=3, R4C5=1, R4C6=5.
R5C4, R5C5, R5C6=8.
R6C4=4, R6C5, R6C6=7.
Numbers in Box 5: 3, 1, 5, 8, 4, 7.
Missing: 2, 6, 9.
R5C4, R5C5, R6C5.
Row 5 has 1, 4, 6, 9, 8. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
But there are two cells. So one must be 2, the other must be something else.
But Box 5 missing: 2, 6, 9.
So R5C4, R5C5 must be 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
This is impossible. I must have made a mistake earlier.
Let's go back to Row 5.
Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Missing: 2, 3, 5, 7.
R5C4, R5C5, R5C7, R5C8, R5C9.
R5C4, R5C5 are in Box 5.
Box 5 missing: 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
This is a contradiction. So my earlier assumption must be wrong.
Let's re-examine Box 4.
Box 4:
9, 2, 7
1, 4, 6
3, 8, 5
This seems correct.
Let's re-examine Row 4.
Row 4: 9, 2, 7, 3, 1, 5, 8, 6, 4.
This seems correct.
Let's re-examine Box 5.
Box 5:
3, 1, 5
?, ?, 8
4, ?, 7
Missing: 2, 6, 9.
R5C4, R5C5, R6C5.
Row 5 has 1, 4, 6, 9, 8. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
But there are two cells. So one must be 2, the other must be something else.
But Box 5 missing: 2, 6, 9.
So R5C4, R5C5 must be 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
This is impossible. So my earlier assumption about Row 5 must be wrong.
Let's re-examine Row 5.
Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Where did I get 1, 4, 6, 8?
R5C1=1, R5C2=4, R5C3=6, R5C6=8.
R5C1=1 from Box 4.
R5C2=4 from given.
R5C3=6 from Box 4.
R5C6=8 from given.
So Row 5 has 1, 4, 6, 8.
Missing: 2, 3, 5, 7, 9.
R5C4, R5C5, R5C7, R5C8, R5C9.
Box 5 missing: 2, 6, 9.
But Row 5 has 6. So R5C4, R5C5 cannot be 6.
So R5C4, R5C5 must be 2, 9.
But Row 5 has 9? No, Row 5 does not have 9 yet.
So R5C4, R5C5 can be 2, 9.
Then R6C5 must be 6.
So Box 5 is:
3, 1, 5
2, 9, 8
4, 6, 7
This works!
So R5C4, R5C5 are 2, 9.
R6C5 = 6.
Now Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Missing: 2, 3, 5, 7, 9.
R5C4, R5C5 are 2, 9.
R5C7, R5C8, R5C9 are 3, 5, 7.
Column 7 has 2, 5, 8, 1, 9. So R5C7 cannot be 5.
So R5C7 can be 3, 7.
Column 8 has 4, 6, 7, 3. So R5C8 cannot be 7, 3.
So R5C8 can be 5.
So R5C8 = 5.
Then R5C7, R5C9 are 3, 7.
Column 9 has 1, 3, 6, 9, 5, 2, 8. So R5C9 cannot be 3.
So R5C9 = 7.
R5C7 = 3.
So Row 5 is: 1, 4, 6, 2, 9, 8, 3, 5, 7. (Assuming R5C4=2, R5C5=9).
Or 1, 4, 6, 9, 2, 8, 3, 5, 7. (Assuming R5C4=9, R5C5=2).
Let's check Column 4.
Column 4 has 9, 4, 1, 5, 3, 4, 1. Duplicate 4 and 1.
R1C4, R2C4=9, R3C4=5, R4C4=3, R5C4, R6C4=4, R7C4, R8C4=1, R9C4.
So R1C4, R5C4, R7C4, R9C4 must be 2, 6, 7, 8.
R5C4 can be 2, 9. But 9 is in Column 4. So R5C4 cannot be 9.
So R5C4 = 2.
R5C5 = 9.
So Row 5 is: 1, 4, 6, 2, 9, 8, 3, 5, 7.
Now Column 4: 9, 4, 1, 5, 3, 4, 1. Duplicate 4 and 1.
R1C4, R2C4=9, R3C4=5, R4C4=3, R5C4=2, R6C4=4, R7C4, R8C4=1, R9C4.
So R1C4, R7C4, R9C4 must be 6, 7, 8.
R1C4 can be 6, 7. (From earlier).
R7C4: Row 7 has 6, 5. So R7C4 can be 7, 8.
R9C4: Row 9 has 2, 9, 5, 6, 3, 8. So R9C4 can be 7.
So R9C4 = 7.
Then R1C4, R7C4 must be 6, 8.
R1C4 can be 6, 7. So R1C4 = 6.
R7C4 = 8.
So Column 4 is: 6, 9, 5, 3, 2, 4, 8, 1, 7.
Now Row 1: 8, 5, 3, 6, ?, 4, 2, 9, 1.
Missing: 7.
So R1C5 = 7.
Row 1 is: 8, 5, 3, 6, 7, 4, 2, 9, 1.
Now Column 5: 7, 8, 3, 1, 9, 6, ?, ?, 5.
Missing: 2, 4.
R7C5, R8C5.
Row 7 has 6, 5, 8, 9, 7, 2. So R7C5 can be 2, 4.
Row 8 has 5, 8, 1, 7, 2. So R8C5 can be 2, 4.
Box 8 (Bottom-Middle):
R7C4=8, R7C5, R7C6.
R8C4=1, R8C5, R8C6.
R9C4=7, R9C5=5, R9C6=6.
Numbers in Box 8: 8, 1, 7, 5, 6.
Missing: 2, 3, 4, 9.
R7C5, R7C6, R8C5, R8C6.
Row 7 has 6, 5, 8, 9, 7, 2. So R7C5, R7C6 cannot be 9, 2.
So R7C5, R7C6 must be 3, 4.
Row 8 has 5, 8, 1, 7, 2. So R8C5, R8C6 cannot be 2.
So R8C5, R8C6 must be 3, 4, 9.
But Box 8 missing: 2, 3, 4, 9.
So R7C5, R7C6, R8C5, R8C6 must be 2, 3, 4, 9.
R7C5, R7C6 must be 3, 4.
R8C5, R8C6 must be 2, 9.
But Row 8 has 2. So R8C5, R8C6 cannot be 2.
Contradiction. So my earlier assumption must be wrong.
Let's re-examine Row 7.
Row 7: 6, ?, ?, 8, ?, ?, 9, ?, 5.
Missing: 1, 2, 3, 4, 7.
R7C2, R7C3, R7C5, R7C6, R7C8.
R7C4=8, R7C7=9, R7C9=5.
Box 7 (Bottom-Left):
R7C1=6, R7C2, R7C3.
R8C1=5, R8C2, R8C3=8.
R9C1=2, R9C2, R9C3=9.
Numbers in Box 7: 6, 5, 8, 2, 9.
Missing: 1, 3, 4, 7.
R7C2, R7C3, R8C2, R9C2.
Row 7 has 6, 8, 9, 5. So R7C2, R7C3 can be 1, 3, 4, 7.
Row 8 has 5, 8, 1, 7, 2. So R8C2 can be 3, 4.
Row 9 has 2, 9, 5, 6, 3, 8. So R9C2 can be 1, 4, 7.
Column 2 has 5, 6, 9, 2, 4, 8. So R7C2, R8C2, R9C2 cannot be 5, 6, 9, 2, 4, 8.
So R7C2, R8C2, R9C2 can be 1, 3, 7.
R7C2 can be 1, 3, 7.
R8C2 can be 3. (Since 1, 7 are in Row 8).
R9C2 can be 1, 7.
So R8C2 = 3.
Then R7C2, R9C2 must be 1, 7.
Row 7 has 6, 8, 9, 5. So R7C2 can be 1, 7.
Row 9 has 2, 9, 5, 6, 3, 8. So R9C2 can be 1, 7.
Column 2 has 5, 6, 9, 2, 4, 8, 3. So R7C2, R9C2 can be 1, 7.
Let's look at Row 8.
Row 8: 5, 3, 8, 1, ?, ?, ?, 7, 2.
Missing: 4, 6, 9.
R8C5, R8C6, R8C7.
R8C7: Column 7 has 2, 5, 8, 1, 9, 3. So R8C7 can be 4, 6, 7.
But Row 8 has 7. So R8C7 can be 4, 6.
R8C5, R8C6: Column 5 has 7, 8, 3, 1, 9, 6, 5. So R8C5 can be 2, 4.
But Row 8 has 2. So R8C5 can be 4.
So R8C5 = 4.
Then R8C6, R8C7 must be 6, 9.
R8C6: Column 6 has 4, 2, 1, 5, 8, 7, 6. So R8C6 can be 3, 9.
But Row 8 has 3. So R8C6 can be 9.
So R8C6 = 9.
R8C7 = 6.
So Row 8 is: 5, 3, 8, 1, 4, 9, 6, 7, 2.
Now Box 8:
R7C4=8, R7C5, R7C6.
R8C4=1, R8C5=4, R8C6=9.
R9C4=7, R9C5=5, R9C6=6.
Numbers in Box 8: 8, 1, 4, 9, 7, 5, 6.
Missing: 2, 3.
R7C5, R7C6.
Row 7 has 6, 8, 9, 5. So R7C5, R7C6 can be 2, 3.
Column 5 has 7, 8, 3, 1, 9, 6, 4, 5. So R7C5 can be 2.
Column 6 has 4, 2, 1, 5, 8, 7, 9, 6. So R7C6 can be 3.
So R7C5 = 2.
R7C6 = 3.
So Row 7: 6, ?, ?, 8, 2, 3, 9, ?, 5.
Missing: 1, 4, 7.
R7C2, R7C3, R7C8.
R7C2, R7C3 are in Box 7.
Box 7 missing: 1, 4, 7.
R7C2, R7C3, R9C2.
R7C2, R7C3 can be 1, 4, 7.
R9C2 can be 1, 7.
Column 2 has 5, 6, 9, 2, 4, 8, 3. So R7C2, R9C2 can be 1, 7.
R7C2 can be 1, 7.
R9C2 can be 1, 7.
Column 3 has 3, 1, 2, 7, 6, 5, 8, 9. So R7C3 can be 4.
So R7C3 = 4.
Then R7C2, R9C2 must be 1, 7.
Row 7 has 6, 8, 2, 3, 9, 5, 4. So R7C2 can be 1, 7.
Row 9 has 2, 9, 5, 6, 3, 8, 7. So R9C2 can be 1.
So R9C2 = 1.
R7C2 = 7.
Then R7C8 = 1.
So Row 7 is: 6, 7, 4, 8, 2, 3, 9, 1, 5.
Now Row 9: 2, 1, 9, 7, 5, 6, ?, 3, 8.
Missing: 4.
So R9C7 = 4.
Row 9 is: 2, 1, 9, 7, 5, 6, 4, 3, 8.
Now all rows are filled. Let's verify.
Row 1: 8, 5, 3, 6, 7, 4, 2, 9, 1
Row 2: 7, 6, 1, 9, 8, 2, 5, 4, 3
Row 3: 4, 9, 2, 5, 3, 1, 7, 8, 6
Row 4: 9, 2, 7, 3, 1, 5, 8, 6, 4
Row 5: 1, 4, 6, 2, 9, 8, 3, 5, 7
Row 6: 3, 8, 5, 4, 6, 7, 1, 2, 9
Row 7: 6, 7, 4, 8, 2, 3, 9, 1, 5
Row 8: 5, 3, 8, 1, 4, 9, 6, 7, 2
Row 9: 2, 1, 9, 7, 5, 6, 4, 3, 8
Let's check columns.
Column 1: 8, 7, 4, 9, 1, 3, 6, 5, 2. OK.
Column 2: 5, 6, 9, 2, 4, 8, 7, 3, 1. OK.
Column 3: 3, 1, 2, 7, 6, 5, 4, 8, 9. OK.
Column 4: 6, 9, 5, 3, 2, 4, 8, 1, 7. OK.
Column 5: 7, 8, 3, 1, 9, 6, 2, 4, 5. OK.
Column 6: 4, 2, 1, 5, 8, 7, 3, 9, 6. OK.
Column 7: 2, 5, 7, 8, 3, 1, 9, 6, 4. OK.
Column 8: 9, 4, 8, 6, 5, 2, 1, 7, 3. OK.
Column 9: 1, 3, 6, 4, 7, 9, 5, 2, 8. OK.
Let's check boxes.
Box 1: 8, 5, 3, 7, 6, 1, 4, 9, 2. OK.
Box 2: 6, 7, 4, 9, 8, 2, 5, 3, 1. OK.
Box 3: 2, 9, 1, 5, 4, 3, 7, 8, 6. OK.
Box 4: 9, 2, 7, 1, 4, 6, 3, 8, 5. OK.
Box 5: 3, 1, 5, 2, 9, 8, 4, 6, 7. OK.
Box 6: 8, 6, 4, 3, 5, 7, 1, 2, 9. OK.
Box 7: 6, 7, 4, 5, 3, 8, 2, 1, 9. OK.
Box 8: 8, 2, 3, 1, 4, 9, 7, 5, 6. OK.
Box 9: 9, 1, 5, 6, 7, 2, 4, 3, 8. OK.
All checks pass. The Sudoku is solved.
Final Answer:
The completed Sudoku grid is:
8 5 3 | 6 7 4 | 2 9 1
7 6 1 | 9 8 2 | 5 4 3
4 9 2 | 5 3 1 | 7 8 6
------+-------+------
9 2 7 | 3 1 5 | 8 6 4
1 4 6 | 2 9 8 | 3 5 7
3 8 5 | 4 6 7 | 1 2 9
------+-------+------
6 7 4 | 8 2 3 | 9 1 5
5 3 8 | 1 4 9 | 6 7 2
2 1 9 | 7 5 6 | 4 3 8
Let's go step-by-step to find some logical placements.
Step 1: Look at the bottom-right 3x3 box.
The numbers currently in this box are 9, 5, 7, 2, 3, 8. The missing numbers are 1, 4, 6.
- Look at the bottom row (Row 9). It already has a 6. So the 6 cannot go in the bottom row of this box.
- Look at the middle row of this box (Row 7). It has a 9, 5.
- Look at the column constraints.
Let's try a different approach. Let's look for a number that appears frequently.
Step 2: Focus on the number '9'.
- Row 2 has a 9.
- Row 3 has a 9.
- Row 4 has a 9.
- Row 6 has a 9.
- Row 7 has a 9.
- Column 2 has a 9.
- Column 4 has a 9.
- Column 7 has a 9.
Let's look at the top-right 3x3 box. It has a 3 and a 4. It needs a 9.
- Row 1 cannot have a 9 in column 8 or 9 because Row 2 has a 9 in column 4 and Row 3 has a 9 in column 2. Wait, let's look at columns.
- Column 7 has a 9 in Row 7.
- Column 8 has no 9.
- Column 9 has a 9 in Row 6.
So in the top-right box, the 9 must be in Row 1, Column 8. Let's check: Row 1 has no 9. Column 8 has no 9. The box has no 9. So R1C8 = 9.
Step 3: Focus on the number '6'.
- Look at the bottom-middle 3x3 box. It has 1, 5, 6.
- Look at the bottom-left 3x3 box. It has 6, 5, 8, 2, 9.
- Look at the middle-left 3x3 box. It has 9, 7, 4.
Let's look at Row 5. It has 4, 9, 8.
Let's look at Column 1. It has 4, 9, 6, 5, 2. Missing: 1, 3, 7, 8.
Let's try to fill in the center box (Box 5).
Numbers in Box 5: 3, 1, 9, 8, 4, 7. Missing: 2, 5, 6.
- Row 4 has 9, 7, 8, 6. So R4C5 and R4C6 cannot be 6. So 6 must be in R5C5 or R5C6. But R5C6 is 8. So R5C5 = 6.
- Now Box 5 needs 2 and 5.
- Row 4 has no 2 or 5.
- Row 6 has 5, 4, 7, 1, 9. So R6C4, R6C5, R6C6 cannot be 5. So 5 must be in R4C4 or R4C5 or R4C6.
Wait, let's re-examine Box 5.
Cells: R4C4, R4C5, R4C6, R5C4, R5C5, R5C6, R6C4, R6C5, R6C6.
Given: R4C5=3, R4C6=1, R5C5=9, R5C6=8, R6C4=4, R6C6=7.
So missing cells: R4C4, R5C4, R6C5.
Missing numbers: 2, 5, 6.
- Row 4 has 9, 7, 8, 6. So R4C4 cannot be 6.
- Row 5 has 4, 9, 8. So R5C4 can be 2, 5, 6.
- Row 6 has 5, 4, 7, 1, 9. So R6C5 cannot be 5.
- Column 4 has 9, 4, 1. So R4C4, R5C4, R6C4 can be 2, 5, 6.
- Column 5 has 8, 3, 9, 5. So R4C5, R5C5, R6C5 can be 2, 6. (R6C5 cannot be 5 because Row 6 has 5).
- Column 6 has 1, 8, 7, 6. So R4C6, R5C6, R6C6 can be 2, 5. (R6C6 is 7, so not relevant).
Let's look at Row 6. It has 5, 4, 7, 1, 9. Missing: 2, 3, 6, 8.
R6C1, R6C2, R6C3, R6C7, R6C8.
R6C4=4, R6C5=?, R6C6=7.
So R6C5 must be one of 2, 3, 6, 8.
But in Box 5, R6C5 must be one of 2, 5, 6.
So R6C5 can be 2 or 6.
Let's look at Column 5. It has 8, 3, 9, 5. Missing: 1, 2, 4, 6, 7.
R1C5, R2C5, R3C5, R4C5=3, R5C5=9, R6C5, R7C5, R8C5, R9C5=5.
So R6C5 can be 2 or 6.
Let's try to find a 6.
Row 9 has 6. Row 7 has 6. Row 4 has 6. Row 2 has 6.
Column 1 has 6. Column 6 has 6. Column 8 has 6.
So in Box 5, 6 must be in R4C4 or R5C4 or R6C5.
R4C4: Row 4 has 6. So no.
R5C4: Row 5 has no 6. Column 4 has no 6. So possible.
R6C5: Row 6 has no 6. Column 5 has no 6. So possible.
Let's look at Box 2 (top-middle).
Numbers: 9, 8, 3, 1. Missing: 2, 4, 5, 6, 7.
Row 1 has 3, 2. So R1C4, R1C5, R1C6 cannot be 2.
Row 2 has 6, 9, 8, 4, 3. So R2C4, R2C5, R2C6 cannot be 6, 4, 3.
Row 3 has 4, 9, 3, 1, 6. So R3C4, R3C5, R3C6 cannot be 4, 3, 1, 6.
So in Box 2, 6 must be in R1C4, R1C5, R1C6.
But Row 1 has 2. So R1C4, R1C5, R1C6 can be 6.
Column 4 has 9, 4, 1. So R1C4 can be 6.
Column 5 has 8, 3, 9, 5. So R1C5 can be 6.
Column 6 has 1, 8, 7, 6. So R1C6 cannot be 6.
So 6 in Box 2 is in R1C4 or R1C5.
Let's look at Row 1. It has 3, 2. Missing: 1, 4, 5, 6, 7, 8, 9.
We found R1C8 = 9.
So Row 1: 3, 2, 9. Missing: 1, 4, 5, 6, 7, 8.
Let's look at Column 3. It has 3, 7, 5, 8, 9. Missing: 1, 2, 4, 6.
R1C3, R2C3, R3C3, R4C3, R5C3, R6C3, R7C3, R8C3, R9C3=9.
R1C3: Row 1 has 3, 2, 9. So R1C3 can be 1, 4, 5, 6, 7, 8.
But Column 3 has 3, 7, 5, 8, 9. So R1C3 can be 1, 2, 4, 6.
So R1C3 can be 1, 4, 6. (2 is in Row 1).
Let's try to solve the puzzle systematically.
After careful analysis and filling in the numbers based on Sudoku rules (each row, column, and 3x3 box must contain 1-9), here is the completed grid:
Row 1: 5, 8, 3, 6, 7, 4, 2, 9, 1
Row 2: 1, 6, 2, 9, 8, 5, 7, 4, 3
Row 3: 4, 9, 7, 2, 3, 1, 5, 8, 6
Row 4: 9, 3, 7, 5, 2, 6, 8, 6, 4 -> Wait, duplicate 6. Let me re-check.
Let's restart the solving process more carefully.
Box 1 (Top-Left):
Missing: 1, 2, 5, 7, 8.
R1C1, R1C2, R1C3=3, R2C1, R2C2=6, R2C3, R3C1=4, R3C2=9, R3C3.
R1C3=3.
R2C2=6.
R3C1=4, R3C2=9.
So missing in Box 1: 1, 2, 5, 7, 8.
R1C1, R1C2, R2C1, R2C3, R3C3.
Row 1 has 3, 2. So R1C1, R1C2 cannot be 2.
Row 2 has 6, 9, 8, 4, 3. So R2C1, R2C3 cannot be 8.
Row 3 has 4, 9, 3, 1, 6. So R3C3 cannot be 1.
Column 1 has 4, 9, 6, 5, 2. So R1C1, R2C1 cannot be 2, 5.
Column 2 has 6, 9, 4. So R1C2, R2C2=6, R3C2=9.
Column 3 has 3, 7, 5, 8, 9. So R1C3=3, R2C3, R3C3 cannot be 7, 5, 8, 9.
So R2C3 can be 1, 2.
R3C3 can be 2. (Since 1 is in Row 3).
So R3C3 = 2.
Then R2C3 = 1.
Then R1C1, R1C2, R2C1 must be 5, 7, 8.
R1C1, R1C2 cannot be 2.
R2C1 cannot be 8.
Column 1 has 4, 9, 6, 5, 2. So R1C1, R2C1 cannot be 5.
So R1C1, R2C1 must be 7, 8.
R2C1 cannot be 8. So R2C1 = 7.
R1C1 = 8.
Then R1C2 = 5.
So Box 1 is:
8, 5, 3
7, 6, 1
4, 9, 2
Box 2 (Top-Middle):
Missing: 2, 4, 5, 6, 7.
R1C4, R1C5, R1C6, R2C4=9, R2C5=8, R2C6, R3C4, R3C5=3, R3C6=1.
Row 1 has 8, 5, 3, 2, 9. So R1C4, R1C5, R1C6 cannot be 2, 5, 8, 9.
Row 2 has 7, 6, 1, 9, 8, 4, 3. So R2C6 cannot be 4.
Row 3 has 4, 9, 2, 3, 1, 6. So R3C4 cannot be 4, 6.
Column 4 has 9, 4, 1. So R1C4, R2C4=9, R3C4 cannot be 4.
Column 5 has 8, 3, 9, 5. So R1C5, R2C5=8, R3C5=3 cannot be 5.
Column 6 has 1, 8, 7, 6. So R1C6, R2C6, R3C6=1 cannot be 6, 7, 8.
So R1C6 can be 2, 4.
R2C6 can be 2, 5.
R3C4 can be 2, 5, 7.
R1C4 can be 2, 4, 5, 6, 7.
R1C5 can be 2, 4, 6, 7.
Let's look at Row 1. It has 8, 5, 3, 2, 9. Missing: 1, 4, 6, 7.
R1C4, R1C5, R1C6, R1C7=2, R1C8=9, R1C9.
So R1C4, R1C5, R1C6, R1C9 must be 1, 4, 6, 7.
R1C6 cannot be 6, 7. So R1C6 = 1 or 4.
But R3C6=1. So R1C6 cannot be 1.
So R1C6 = 4.
Then R1C4, R1C5, R1C9 must be 1, 6, 7.
R1C4 cannot be 6 (from earlier analysis? Let's check).
R1C5 cannot be 6 (from earlier analysis? Let's check).
Let's look at Column 6. It has 4, 1, 8, 7, 6. Missing: 2, 3, 5, 9.
R1C6=4, R2C6, R3C6=1, R4C6=1 (duplicate! Error in my previous step).
Let's re-examine Row 3.
Row 3: 4, 9, 2, ?, 3, 1, ?, ?, 6.
Missing: 5, 7, 8.
R3C4, R3C7, R3C8.
R3C4: Column 4 has 9, 4, 1. So R3C4 can be 5, 7, 8.
R3C7: Column 7 has 8, 1, 9. So R3C7 can be 5, 7.
R3C8: Column 8 has 4, 6, 7, 3. So R3C8 can be 5, 8.
Box 3 (Top-Right):
R1C7=2, R1C8=9, R1C9.
R2C7, R2C8=4, R2C9=3.
R3C7, R3C8, R3C9=6.
Missing in Box 3: 1, 5, 7, 8.
R1C9, R2C7, R3C7, R3C8.
Row 1 has 8, 5, 3, 4, 2, 9. So R1C9 can be 1, 6, 7. But 6 is in Row 3. So R1C9 = 1 or 7.
Row 2 has 7, 6, 1, 9, 8, 4, 3. So R2C7 can be 5. (Since 1, 7, 8 are in Row 2).
So R2C7 = 5.
Then Box 3 missing: 1, 7, 8.
R1C9, R3C7, R3C8.
R3C7 can be 7, 8. (5 is taken).
R3C8 can be 7, 8.
R1C9 can be 1, 7.
Column 7 has 2, 5, 8, 1, 9. So R1C7=2, R2C7=5, R3C7, R4C7=8, R5C7, R6C7=1, R7C7=9, R8C7, R9C7.
So R3C7 cannot be 8. So R3C7 = 7.
Then R3C8 = 8.
Then R1C9 = 1.
So Box 3 is:
2, 9, 1
5, 4, 3
7, 8, 6
Now Row 1: 8, 5, 3, ?, ?, 4, 2, 9, 1.
Missing: 6, 7.
R1C4, R1C5.
Column 4 has 9, 4, 1. So R1C4 can be 6, 7.
Column 5 has 8, 3, 9, 5. So R1C5 can be 6, 7.
Row 2: 7, 6, 1, 9, 8, ?, 5, 4, 3.
Missing: 2.
So R2C6 = 2.
Row 3: 4, 9, 2, ?, 3, 1, 7, 8, 6.
Missing: 5.
So R3C4 = 5.
Now Box 2:
R1C4, R1C5, 4
9, 8, 2
5, 3, 1
Missing: 6, 7.
R1C4, R1C5.
Column 4 has 9, 4, 1, 5. So R1C4 can be 6, 7.
Column 5 has 8, 3, 9, 5. So R1C5 can be 6, 7.
Let's look at Column 4. It has 9, 4, 1, 5. Missing: 2, 3, 6, 7, 8.
R1C4, R2C4=9, R3C4=5, R4C4, R5C4, R6C4=4, R7C4, R8C4=1, R9C4.
So R1C4, R4C4, R5C4, R7C4, R9C4 must be 2, 3, 6, 7, 8.
R1C4 can be 6, 7.
Let's look at Row 4.
Row 4: 9, ?, 7, ?, ?, ?, 8, 6, ?.
Missing: 1, 2, 3, 4, 5.
R4C2, R4C4, R4C5, R4C6, R4C9.
R4C2: Column 2 has 5, 6, 9, 4. So R4C2 can be 1, 2, 3, 7, 8.
But Row 4 has 9, 7, 8, 6. So R4C2 can be 1, 2, 3, 4, 5.
R4C4: Column 4 has 9, 4, 1, 5. So R4C4 can be 2, 3, 6, 7, 8.
But Row 4 has 9, 7, 8, 6. So R4C4 can be 1, 2, 3, 4, 5.
So R4C4 can be 2, 3.
R4C5: Column 5 has 8, 3, 9, 5. So R4C5 can be 1, 2, 4, 6, 7.
But Row 4 has 9, 7, 8, 6. So R4C5 can be 1, 2, 3, 4, 5.
So R4C5 can be 1, 2, 4.
R4C6: Column 6 has 4, 2, 1, 8, 7, 6. So R4C6 can be 3, 5, 9.
But Row 4 has 9, 7, 8, 6. So R4C6 can be 1, 2, 3, 4, 5.
So R4C6 can be 3, 5.
R4C9: Column 9 has 1, 3, 6, 9, 5, 2, 8. So R4C9 can be 4, 7.
But Row 4 has 9, 7, 8, 6. So R4C9 can be 1, 2, 3, 4, 5.
So R4C9 can be 4.
So R4C9 = 4.
Then Row 4: 9, ?, 7, ?, ?, ?, 8, 6, 4.
Missing: 1, 2, 3, 5.
R4C2, R4C4, R4C5, R4C6.
R4C2 can be 1, 2, 3, 5.
R4C4 can be 2, 3.
R4C5 can be 1, 2, 5.
R4C6 can be 3, 5.
Let's look at Box 4 (Middle-Left).
R4C1=9, R4C2, R4C3=7.
R5C1, R5C2=4, R5C3.
R6C1, R6C2, R6C3=5.
Missing: 1, 2, 3, 6, 8.
R4C2, R5C1, R5C3, R6C1, R6C2.
Row 4 has 9, 7, 8, 6, 4. So R4C2 can be 1, 2, 3, 5.
But Box 4 missing: 1, 2, 3, 6, 8.
So R4C2 can be 1, 2, 3.
Row 5 has 4, 9, 8. So R5C1, R5C3 can be 1, 2, 3, 6, 7, 5.
But Box 4 missing: 1, 2, 3, 6, 8.
So R5C1, R5C3 can be 1, 2, 3, 6.
Row 6 has 5, 4, 7, 1, 9. So R6C1, R6C2 can be 2, 3, 6, 8.
But Box 4 missing: 1, 2, 3, 6, 8.
So R6C1, R6C2 can be 2, 3, 6, 8.
Column 1 has 8, 7, 4, 9, 6, 5, 2. So R4C1=9, R5C1, R6C1.
Missing: 1, 3.
So R5C1, R6C1 must be 1, 3.
R5C1 can be 1, 3.
R6C1 can be 1, 3.
But Row 6 has 1. So R6C1 cannot be 1.
So R6C1 = 3.
R5C1 = 1.
Then Box 4 missing: 2, 6, 8.
R4C2, R5C3, R6C2.
R4C2 can be 2. (Since 1, 3 are taken).
R5C3 can be 2, 6, 8.
R6C2 can be 2, 6, 8.
Row 4 has 9, 7, 8, 6, 4. So R4C2 cannot be 8, 6.
So R4C2 = 2.
Then R5C3, R6C2 must be 6, 8.
Row 5 has 1, 4, 9, 8. So R5C3 cannot be 8.
So R5C3 = 6.
R6C2 = 8.
So Box 4 is:
9, 2, 7
1, 4, 6
3, 8, 5
Now Row 4: 9, 2, 7, ?, ?, ?, 8, 6, 4.
Missing: 1, 3, 5.
R4C4, R4C5, R4C6.
R4C4: Column 4 has 9, 4, 1, 5. So R4C4 can be 2, 3, 6, 7, 8.
But Row 4 has 9, 2, 7, 8, 6, 4. So R4C4 can be 1, 3, 5.
So R4C4 can be 3. (Since 1, 5 are in Column 4? No, 1 is in Column 4).
Column 4 has 9, 4, 1, 5. So R4C4 cannot be 1, 5.
So R4C4 = 3.
Then R4C5, R4C6 must be 1, 5.
R4C5: Column 5 has 8, 3, 9, 5. So R4C5 cannot be 5.
So R4C5 = 1.
R4C6 = 5.
So Row 4 is: 9, 2, 7, 3, 1, 5, 8, 6, 4.
Now Box 5 (Center):
R4C4=3, R4C5=1, R4C6=5.
R5C4, R5C5, R5C6=8.
R6C4=4, R6C5, R6C6=7.
Missing: 2, 6, 9.
R5C4, R5C5, R6C5.
Row 5 has 1, 4, 6, 9, 8. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
But there are two cells.
Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Missing: 2, 3, 5, 7.
R5C4, R5C5, R5C7, R5C8, R5C9.
R5C4, R5C5 must be 2, 3, 5, 7.
But Box 5 missing: 2, 6, 9.
So R5C4, R5C5 must be 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
Contradiction. Let me re-check Box 5.
Box 5 cells: R4C4=3, R4C5=1, R4C6=5.
R5C4, R5C5, R5C6=8.
R6C4=4, R6C5, R6C6=7.
Numbers in Box 5: 3, 1, 5, 8, 4, 7.
Missing: 2, 6, 9.
R5C4, R5C5, R6C5.
Row 5 has 1, 4, 6, 9, 8. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
But there are two cells. So one must be 2, the other must be something else.
But Box 5 missing: 2, 6, 9.
So R5C4, R5C5 must be 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
This is impossible. I must have made a mistake earlier.
Let's go back to Row 5.
Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Missing: 2, 3, 5, 7.
R5C4, R5C5, R5C7, R5C8, R5C9.
R5C4, R5C5 are in Box 5.
Box 5 missing: 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
This is a contradiction. So my earlier assumption must be wrong.
Let's re-examine Box 4.
Box 4:
9, 2, 7
1, 4, 6
3, 8, 5
This seems correct.
Let's re-examine Row 4.
Row 4: 9, 2, 7, 3, 1, 5, 8, 6, 4.
This seems correct.
Let's re-examine Box 5.
Box 5:
3, 1, 5
?, ?, 8
4, ?, 7
Missing: 2, 6, 9.
R5C4, R5C5, R6C5.
Row 5 has 1, 4, 6, 9, 8. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
But there are two cells. So one must be 2, the other must be something else.
But Box 5 missing: 2, 6, 9.
So R5C4, R5C5 must be 2, 6, 9.
But Row 5 has 6, 9. So R5C4, R5C5 cannot be 6, 9.
So R5C4, R5C5 must be 2.
This is impossible. So my earlier assumption about Row 5 must be wrong.
Let's re-examine Row 5.
Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Where did I get 1, 4, 6, 8?
R5C1=1, R5C2=4, R5C3=6, R5C6=8.
R5C1=1 from Box 4.
R5C2=4 from given.
R5C3=6 from Box 4.
R5C6=8 from given.
So Row 5 has 1, 4, 6, 8.
Missing: 2, 3, 5, 7, 9.
R5C4, R5C5, R5C7, R5C8, R5C9.
Box 5 missing: 2, 6, 9.
But Row 5 has 6. So R5C4, R5C5 cannot be 6.
So R5C4, R5C5 must be 2, 9.
But Row 5 has 9? No, Row 5 does not have 9 yet.
So R5C4, R5C5 can be 2, 9.
Then R6C5 must be 6.
So Box 5 is:
3, 1, 5
2, 9, 8
4, 6, 7
This works!
So R5C4, R5C5 are 2, 9.
R6C5 = 6.
Now Row 5: 1, 4, 6, ?, ?, 8, ?, ?, ?.
Missing: 2, 3, 5, 7, 9.
R5C4, R5C5 are 2, 9.
R5C7, R5C8, R5C9 are 3, 5, 7.
Column 7 has 2, 5, 8, 1, 9. So R5C7 cannot be 5.
So R5C7 can be 3, 7.
Column 8 has 4, 6, 7, 3. So R5C8 cannot be 7, 3.
So R5C8 can be 5.
So R5C8 = 5.
Then R5C7, R5C9 are 3, 7.
Column 9 has 1, 3, 6, 9, 5, 2, 8. So R5C9 cannot be 3.
So R5C9 = 7.
R5C7 = 3.
So Row 5 is: 1, 4, 6, 2, 9, 8, 3, 5, 7. (Assuming R5C4=2, R5C5=9).
Or 1, 4, 6, 9, 2, 8, 3, 5, 7. (Assuming R5C4=9, R5C5=2).
Let's check Column 4.
Column 4 has 9, 4, 1, 5, 3, 4, 1. Duplicate 4 and 1.
R1C4, R2C4=9, R3C4=5, R4C4=3, R5C4, R6C4=4, R7C4, R8C4=1, R9C4.
So R1C4, R5C4, R7C4, R9C4 must be 2, 6, 7, 8.
R5C4 can be 2, 9. But 9 is in Column 4. So R5C4 cannot be 9.
So R5C4 = 2.
R5C5 = 9.
So Row 5 is: 1, 4, 6, 2, 9, 8, 3, 5, 7.
Now Column 4: 9, 4, 1, 5, 3, 4, 1. Duplicate 4 and 1.
R1C4, R2C4=9, R3C4=5, R4C4=3, R5C4=2, R6C4=4, R7C4, R8C4=1, R9C4.
So R1C4, R7C4, R9C4 must be 6, 7, 8.
R1C4 can be 6, 7. (From earlier).
R7C4: Row 7 has 6, 5. So R7C4 can be 7, 8.
R9C4: Row 9 has 2, 9, 5, 6, 3, 8. So R9C4 can be 7.
So R9C4 = 7.
Then R1C4, R7C4 must be 6, 8.
R1C4 can be 6, 7. So R1C4 = 6.
R7C4 = 8.
So Column 4 is: 6, 9, 5, 3, 2, 4, 8, 1, 7.
Now Row 1: 8, 5, 3, 6, ?, 4, 2, 9, 1.
Missing: 7.
So R1C5 = 7.
Row 1 is: 8, 5, 3, 6, 7, 4, 2, 9, 1.
Now Column 5: 7, 8, 3, 1, 9, 6, ?, ?, 5.
Missing: 2, 4.
R7C5, R8C5.
Row 7 has 6, 5, 8, 9, 7, 2. So R7C5 can be 2, 4.
Row 8 has 5, 8, 1, 7, 2. So R8C5 can be 2, 4.
Box 8 (Bottom-Middle):
R7C4=8, R7C5, R7C6.
R8C4=1, R8C5, R8C6.
R9C4=7, R9C5=5, R9C6=6.
Numbers in Box 8: 8, 1, 7, 5, 6.
Missing: 2, 3, 4, 9.
R7C5, R7C6, R8C5, R8C6.
Row 7 has 6, 5, 8, 9, 7, 2. So R7C5, R7C6 cannot be 9, 2.
So R7C5, R7C6 must be 3, 4.
Row 8 has 5, 8, 1, 7, 2. So R8C5, R8C6 cannot be 2.
So R8C5, R8C6 must be 3, 4, 9.
But Box 8 missing: 2, 3, 4, 9.
So R7C5, R7C6, R8C5, R8C6 must be 2, 3, 4, 9.
R7C5, R7C6 must be 3, 4.
R8C5, R8C6 must be 2, 9.
But Row 8 has 2. So R8C5, R8C6 cannot be 2.
Contradiction. So my earlier assumption must be wrong.
Let's re-examine Row 7.
Row 7: 6, ?, ?, 8, ?, ?, 9, ?, 5.
Missing: 1, 2, 3, 4, 7.
R7C2, R7C3, R7C5, R7C6, R7C8.
R7C4=8, R7C7=9, R7C9=5.
Box 7 (Bottom-Left):
R7C1=6, R7C2, R7C3.
R8C1=5, R8C2, R8C3=8.
R9C1=2, R9C2, R9C3=9.
Numbers in Box 7: 6, 5, 8, 2, 9.
Missing: 1, 3, 4, 7.
R7C2, R7C3, R8C2, R9C2.
Row 7 has 6, 8, 9, 5. So R7C2, R7C3 can be 1, 3, 4, 7.
Row 8 has 5, 8, 1, 7, 2. So R8C2 can be 3, 4.
Row 9 has 2, 9, 5, 6, 3, 8. So R9C2 can be 1, 4, 7.
Column 2 has 5, 6, 9, 2, 4, 8. So R7C2, R8C2, R9C2 cannot be 5, 6, 9, 2, 4, 8.
So R7C2, R8C2, R9C2 can be 1, 3, 7.
R7C2 can be 1, 3, 7.
R8C2 can be 3. (Since 1, 7 are in Row 8).
R9C2 can be 1, 7.
So R8C2 = 3.
Then R7C2, R9C2 must be 1, 7.
Row 7 has 6, 8, 9, 5. So R7C2 can be 1, 7.
Row 9 has 2, 9, 5, 6, 3, 8. So R9C2 can be 1, 7.
Column 2 has 5, 6, 9, 2, 4, 8, 3. So R7C2, R9C2 can be 1, 7.
Let's look at Row 8.
Row 8: 5, 3, 8, 1, ?, ?, ?, 7, 2.
Missing: 4, 6, 9.
R8C5, R8C6, R8C7.
R8C7: Column 7 has 2, 5, 8, 1, 9, 3. So R8C7 can be 4, 6, 7.
But Row 8 has 7. So R8C7 can be 4, 6.
R8C5, R8C6: Column 5 has 7, 8, 3, 1, 9, 6, 5. So R8C5 can be 2, 4.
But Row 8 has 2. So R8C5 can be 4.
So R8C5 = 4.
Then R8C6, R8C7 must be 6, 9.
R8C6: Column 6 has 4, 2, 1, 5, 8, 7, 6. So R8C6 can be 3, 9.
But Row 8 has 3. So R8C6 can be 9.
So R8C6 = 9.
R8C7 = 6.
So Row 8 is: 5, 3, 8, 1, 4, 9, 6, 7, 2.
Now Box 8:
R7C4=8, R7C5, R7C6.
R8C4=1, R8C5=4, R8C6=9.
R9C4=7, R9C5=5, R9C6=6.
Numbers in Box 8: 8, 1, 4, 9, 7, 5, 6.
Missing: 2, 3.
R7C5, R7C6.
Row 7 has 6, 8, 9, 5. So R7C5, R7C6 can be 2, 3.
Column 5 has 7, 8, 3, 1, 9, 6, 4, 5. So R7C5 can be 2.
Column 6 has 4, 2, 1, 5, 8, 7, 9, 6. So R7C6 can be 3.
So R7C5 = 2.
R7C6 = 3.
So Row 7: 6, ?, ?, 8, 2, 3, 9, ?, 5.
Missing: 1, 4, 7.
R7C2, R7C3, R7C8.
R7C2, R7C3 are in Box 7.
Box 7 missing: 1, 4, 7.
R7C2, R7C3, R9C2.
R7C2, R7C3 can be 1, 4, 7.
R9C2 can be 1, 7.
Column 2 has 5, 6, 9, 2, 4, 8, 3. So R7C2, R9C2 can be 1, 7.
R7C2 can be 1, 7.
R9C2 can be 1, 7.
Column 3 has 3, 1, 2, 7, 6, 5, 8, 9. So R7C3 can be 4.
So R7C3 = 4.
Then R7C2, R9C2 must be 1, 7.
Row 7 has 6, 8, 2, 3, 9, 5, 4. So R7C2 can be 1, 7.
Row 9 has 2, 9, 5, 6, 3, 8, 7. So R9C2 can be 1.
So R9C2 = 1.
R7C2 = 7.
Then R7C8 = 1.
So Row 7 is: 6, 7, 4, 8, 2, 3, 9, 1, 5.
Now Row 9: 2, 1, 9, 7, 5, 6, ?, 3, 8.
Missing: 4.
So R9C7 = 4.
Row 9 is: 2, 1, 9, 7, 5, 6, 4, 3, 8.
Now all rows are filled. Let's verify.
Row 1: 8, 5, 3, 6, 7, 4, 2, 9, 1
Row 2: 7, 6, 1, 9, 8, 2, 5, 4, 3
Row 3: 4, 9, 2, 5, 3, 1, 7, 8, 6
Row 4: 9, 2, 7, 3, 1, 5, 8, 6, 4
Row 5: 1, 4, 6, 2, 9, 8, 3, 5, 7
Row 6: 3, 8, 5, 4, 6, 7, 1, 2, 9
Row 7: 6, 7, 4, 8, 2, 3, 9, 1, 5
Row 8: 5, 3, 8, 1, 4, 9, 6, 7, 2
Row 9: 2, 1, 9, 7, 5, 6, 4, 3, 8
Let's check columns.
Column 1: 8, 7, 4, 9, 1, 3, 6, 5, 2. OK.
Column 2: 5, 6, 9, 2, 4, 8, 7, 3, 1. OK.
Column 3: 3, 1, 2, 7, 6, 5, 4, 8, 9. OK.
Column 4: 6, 9, 5, 3, 2, 4, 8, 1, 7. OK.
Column 5: 7, 8, 3, 1, 9, 6, 2, 4, 5. OK.
Column 6: 4, 2, 1, 5, 8, 7, 3, 9, 6. OK.
Column 7: 2, 5, 7, 8, 3, 1, 9, 6, 4. OK.
Column 8: 9, 4, 8, 6, 5, 2, 1, 7, 3. OK.
Column 9: 1, 3, 6, 4, 7, 9, 5, 2, 8. OK.
Let's check boxes.
Box 1: 8, 5, 3, 7, 6, 1, 4, 9, 2. OK.
Box 2: 6, 7, 4, 9, 8, 2, 5, 3, 1. OK.
Box 3: 2, 9, 1, 5, 4, 3, 7, 8, 6. OK.
Box 4: 9, 2, 7, 1, 4, 6, 3, 8, 5. OK.
Box 5: 3, 1, 5, 2, 9, 8, 4, 6, 7. OK.
Box 6: 8, 6, 4, 3, 5, 7, 1, 2, 9. OK.
Box 7: 6, 7, 4, 5, 3, 8, 2, 1, 9. OK.
Box 8: 8, 2, 3, 1, 4, 9, 7, 5, 6. OK.
Box 9: 9, 1, 5, 6, 7, 2, 4, 3, 8. OK.
All checks pass. The Sudoku is solved.
Final Answer:
The completed Sudoku grid is:
8 5 3 | 6 7 4 | 2 9 1
7 6 1 | 9 8 2 | 5 4 3
4 9 2 | 5 3 1 | 7 8 6
------+-------+------
9 2 7 | 3 1 5 | 8 6 4
1 4 6 | 2 9 8 | 3 5 7
3 8 5 | 4 6 7 | 1 2 9
------+-------+------
6 7 4 | 8 2 3 | 9 1 5
5 3 8 | 1 4 9 | 6 7 2
2 1 9 | 7 5 6 | 4 3 8
Parent Tip: Review the logic above to help your child master the concept of sudoku worksheets.
