010084060 280000419 046510000 023605080 000042153 ...

Analyzing the puzzle

I'm working through a 9x9 sudoku grid with some initial numbers filled in, aiming to solve it methodically through logical reasoning or identifying patterns.

Identifying missing values

I’m piecing together the missing digits for row 1, noting that 2, 3, 5, 7, and 9 are absent. Now, I’ll compare against columns and boxes to see if these digits are present.

Verifying intersection

I’m checking the missing digits in row 1 against column 1. The potential values are 3, 5, and 7. Now, I’ll compare these with the top left 3x3 box.

Analyzing columns

I’m examining column 1 and 2, noting the missing digits. Column 1 lacks 1, 3, 5, 6, 7, and column 2 is missing 3, 6, 7, 9.

Tracking the progress

I’ve identified the missing digits in columns 3 through 9, and I’m working through the missing digits in boxes 1 through 4. No single missing digit solution yet.

Revisiting possibilities

I’m analyzing missing digits in boxes 5 through 9 and row 5. No single missing digit solution is evident yet, indicating further reasoning or pattern recognition is needed.

Evaluating intersections

I’m focusing on intersections of missing digits in row 5, column 2, and box 4, and intersections of missing digits in row 6, column 4, and box 5. No single missing digit found yet.

Mapping out intersections

I’m working through intersections in row 7, column 2, and box 7, and row 6, column 8, and box 6, to pinpoint any single missing digits.

Cross-checking regions

I’m piecing together the intersections of missing digits in row 7, column 2, and box 7, ensuring not to mistakenly include 9, which is already present in box 7.

Clarifying intersections

I’m thinking about the overlapping missing digits in row 7, column 2, and box 7, concluding that the possible values for R7C2 are 3 or 7.

Identifying possibilities

I’m noting the potential values for R7C3 as 4, 5, and 7, after examining intersections of missing digits in row 7, column 3, and box 7.

Pinpointing positions

I’m identifying the unique placement of missing digits in row 2, examining columns 3, 4, 5, and 6 for the digit 5's exact spot. Progress is being made with each step.

Figuring out placements

OK, let me see. The digit 7 fits uniquely into row 2, columns 3 and 4, as those columns are missing 7 and have no restrictions.

Mapping out possibilities

I’m noting that digit 7 can go in rows 2 and 1, columns 3, 4, and 6, without any single solution emerging. Progress is being made in identifying potential alignments.

Identifying potential spots

I'm evaluating digit 3's and digit 5's placements in row 1, considering their column restrictions. These digits have multiple possible positions, suggesting more iterative refinements are necessary.

Evaluating placements

I’m checking the possibilities for digits 7 and 9 in row 1, considering their column constraints. Both digits have multiple potential placements, indicating a need for more detailed analysis.

Pinpointing the digit

Interestingly enough, I’ve determined that the digit 2 is uniquely placed in row 8, column 3. This conclusion is drawn from intersecting row, column, and box constraints.

Adjusting and narrowing down

I’ve updated the sudoku grid with the digit 2 in row 8, column 3. The missing digits in rows, columns, and boxes are now more precise, aiding in further solving.

Spotting unique digits

I’m piecing together intersections of missing digits in row 1, column 3, and box 1, and similarly in row 2 and box 1, for unique digit placements. Progress is steady!

Assessing potential spots

I'm thinking about unique placements for digit 1 in row 8, columns 1 and 9. Checking column constraints and intersections indicates multiple possibilities, showing clear need for further analysis.

Assessing cell options

Mapping out the possibilities for digits 4 and 5 in row 8, columns 8 and 9, noting constraints in columns and boxes. Progress is being made towards pinpointing exact spots.

Piecing together options

I’m piecing together digit 6's and digit 8's placements in row 8, considering column and box constraints. Both digits have multiple possible spots, indicating a need for more detailed analysis.

Figuring out the options

OK, let's see: digit 1 in column 1 is feasible in rows 4 and 8. Crucially, column 1 is missing 3, 5, 6, and 7.

Placing options

I'm carefully considering digit 3 in column 1. It can go in R1C1 or R3C1. Digit 5 also fits in R1C1, so further checks are needed to finalize.

Mapping out spots

I'm piecing together the possible spots for digits 5 and 6 in column 1: digit 5 can go in R1C1 or R8C1, while digit 6 can go in R5C1 or R8C1.

Assessing digit positions

I'm working through digit 7's spots in column 1: R1C1, R3C1, R4C1, or R5C1. Considering box-line reduction and row 3's unique cell for digit 2.

Figuring out cell possibilities

OK, let's finalize digit 3 in row 7, columns 3, 4, and 6. Then, digit 4 goes in row 8, columns 8 and 9, removing the options for digit 9.

Weighing options

I’m evaluating digit 3's and digit 7's potential placements in row 3. They have multiple possible spots, so further strategy refinement is needed.

Assessing digit locations

I'm identifying the possible spots for digits 8 and 9 in row 3, considering their intersections in columns and boxes. Both digits have multiple potential placements.

Pinpointing options

I’m concluding that digit 8 can be placed in R3C9 and digit 9 in R3C6, with digit 7 possibly in R3C6 as well, resulting in unique spot assignments.

Mapping out digit options

Starting with box3, digit 7 can’t go in R3C9. I’m concluding that digit 8 is viable in R3C9, and noting inconsistencies in rows and columns, adjusting my approach accordingly.

Adjusting cell choices

I’m thinking through the empty cells in column 7, eliminating digit 9 from R1C7 and narrowing down possibilities to 5 and 7. This helps refine the sudoku grid.

Confirming digit spots

I'm ensuring digits 5 and 7 are consistently placed in rows 1 and 3. Clarity is forming as I verify these positions against box constraints.

Revisiting digit choices

OK, let's recheck the placements for rows 1 and 3. I’ve noticed digit overlaps in R1C7 and R1C9, suggesting the need for a fresh look at their intersections.

Pinpointing specific values

Hmm, I’m thinking about the intersections of missing digits in row 1 and column 1, which lead me to R1C1 = 3. This logic forms the basis for further constraints.

Finalizing digit assignments

I’ve noticed R1C1=3, marking significant progress. It’s interesting to see that backed up by further intersections in column and box 1, digit placements are becoming more precise.

Isolating digits

I’ve identified R1C3 as 9 and R1C4 as 2, ensuring digit assignments are unique within their respective rows and columns. This approach helps refine the grid systematically.

Finalizing assignments

I’ve determined R1C4 as 2, updated col4 to {1,3,4,7,8}, and bumped R1C7 and R1C9 down to {5,7}. The grid is almost fully filled.

Evaluating row possibilities

I’m piecing together that intersections in row2C3 with col3 révèle {5,7}, while intersections in row2C4, row2C5, and row2C6 yield no clear single digit, so progress is steady.

Establishing the single

OK, let me see. By intersecting row4C5 with col5, I determined row4C5=9. This insight corrected row4 and updated col5 to refine the grid further.

Placing 7

I determined row4C7=7 by overlapping col7 and row4, leading to {5,6,9} in col7 and {1,4} in row4.

Finalizing row4

I ensured R4C1=1 by confirming 4 isn't in col1's missing set. With R4C1=1 established, I determined R4C9=4, completing row4.

Concluding row5 and col3 updates

I confirmed R5C3=7 from intersecting row5 and col3 missing sets. This finalizes row5 as [0,0,7,0,4,2,1,5,3] and updates col3, removing 7 from its missing set.

Confirming the digit

I determined R5C1=6 by intersecting missing sets from row5 and col1. This update finalized row5 as [6,0,7,0,4,2,1,5,3] and adjusted col1's missing digits to {5,7}.

Pinpointing digits

I identified that R5C2=9 and R5C4=8, completing row5 as [6,9,7,8,4,2,1,5,3]. These placements were motivated by intersecting missing sets from rows and columns.

Pinpointing a digit

I noted that R6C5=3, as 6 isn't absent from box5. This narrowed down row6 to [4,5,8,?,3,1,?,?,2], with col5 missing {5,6}.

Mapping intersections

I’m trying to figure out row6 and column intersections. Placing 3 helped pinpoint likely digits but left multiple possibilities for some cells.

Identifying missing pieces

Placing 9 in R6C8 and 6 in R6C7 completes row6. Looking at row7, it’s missing {1,3,4,5,7}, addressing the intersection with col2.

Assessing intersections

I’m curious about the intersections of col2 and box8 with row7's missing digits. None of them yield a single missing number for R7C2.

Decrypting intersections

OK, let me see: the intersections in col5 and row7 reveal that R7C5=5 is locked in. This adjustment affects the remaining empty spots in both col5 and row7.

Confirming the goal

I’m piecing together the final number placements in col5 and row9. All spots are now accounted for, completing these sections of the puzzle.

Ensuring completion

Filling in the gaps now: R4C5, R6C5, and R2C5. These are the only missing 5 spots left, completing col5's 9 digits efficiently.

Pinpointing missing numbers

Checking row2, I spot that R2C3 is blank and intersects with col3 and box1 to reveal the digit 5. This fills R2C3 definitively.

Pinpointing the gap

OK, let me see. By placing 6 in R2C5, the blanks R2C3, R2C4, and R2C6 now only need 3, 5, or 7, respectively.

Wrapping up row 7

By filling in 4 for R7C3, the final missing digit for row 7, the row becomes [9,0,4,0,5,0,8,2,6].

Checking for errors

I'm mapping out row 7, identifying digits {9,4,5,8,2,6} used so far, and confirming {1,3,7} as missing. Any inconsistencies or errors are flagged for review.

Analyzing gaps

Taking a closer look at col6, it holds digits {4,5,2,1} and is missing {3,6,7,8,9}. Examining intersections, I'm working through row 3 and row 7 to locate any missing values.

Piecing together the grid

Mapping out a systematic approach to fill in the grid, I've worked through R1C7 and R1C9, earlier establishing they hold values {5,7}. Now, I'm focusing on determining their exact order.

Clarifying digit placement

I resolved the R1C7 conflict, assigning R1C7=5 and R1C9=7. This adjustment completed row1 and refined col9, leaving {1,5,8} as the remaining digits.

Confirming digit assignments

I’m finalizing row3 by confirming R3C1 and R3C9, ensuring the remaining digits are correctly placed. This approach clarifies the digit assignments and enhances comprehension.

Confirming placements

I’m verifying digit placements in row3, particularly R3C1 and R3C7, ensuring there are no conflicts and all other squares are correctly filled.

Identifying missing digits

I’m noting that row3 is missing 3, 7, 8, and 9, while col1 is missing 5 and 7. This indicates R3C1 and R8C1 must be 5 and 7.

Identifying placement

I’m piecing together the digits for row3 and col1. By intersecting their missing numbers, it appears R3C1 is 7, leaving row3 with missing digits 3, 8, and 9.

Mapping digit placements

I’m figuring out digits for R3C6 by cross-referencing with col6 and box2. This method leaves R3C6 containing {3,8,9}, which are the missing numbers for row3.

Figuring out digits

I'm curious about row2 and row3, narrowing down R3C6 to {3,9}. This aligns with row3 and box2's missing digits.

Piecing together options

I’m mapping out row3C8 as {3,9}, leading to row3C9 being 8. This narrows down row3C6 and row3C8 to {3,9}.

Updating column details

I'm verifying if placing 8 in R3C9 affects col9 constraints for R8C9 and R9C9, ensuring compliance with missing digits {1,5}.

Placing the digit

I’m working through column constraints, identifying that R8C1 should be 5. This ensures all constraints for col1 and row8 are aligned, filling the digit in the matrix.

Filling the grid

OK, let me see. Column 1 is done with digits 1-9. Now, it's about placing 1, 4, 6, and 8 in row 8.

Laying out the digits

OK, let's see. I placed 6 in R8C2. Now, we're focusing on col2 and row8, noting missing digits 1, 4, and 8 for empty squares.

Nailing down col2

I confirmed row9C2=3. Now, focusing on col2 and row8, both missing 7. Next, we'll tackle col6, aiming to finalize the remaining empty squares.

Whittling down possibilities

I determined row8C6=8. Now, with col6 missing {3,6,7,9} and row8C8 already having 8, the only empty squares in row8 are R8C8=1 and R8C9=4.

Wrapping it up

I determined that row8C8=4, leaving R8C9=1. Interestingly enough, row9 remains partially filled with digits.

Piecing together the solution

I identified that col9 is missing 5 and R9C9=5. This placement updated row9, which now has R9C9=5, leaving missing numbers {3,4,6,7,9}.

Mapping row9's progress

I identified that row9 is missing {3,7,9} after eliminating 9, reducing it to {3,7}. This update adjusts the final intersections for R9C2 and R9C7.

Mapping out intersections

I’m gathering info on row9C8 by analyzing col8's usage. This will clarify row3C8 as part of the pair {3,9}, before concluding with row9C9 as 5.

Clarifying intersections

OK, let me see. By cross-referencing col8 and row3C8, I determine that row3C8 is 3. This is accomplished by noting the digits used and missing.

Adjusting my approach

I realize that placing 3 in row3C8 conflicts with existing 3s. To avoid duplicates, I opt to assign 9 to row3C6 and 3 to row3C8, ensuring consistency.

Mixing things up

I assign 9 to row3C6 and 3 to row3C8, sidestepping repeated digits. Next, I ensure all numbers 1 to 9 fit seamlessly into row3.

Aligning numbers

OK, let's verify that all numbers from 1 to 9 fit into row9 without overlaps. Confirming intersecting values and updating missing digits to ensure a complete row.

Filling in the gaps

OK, I’m digging into row9 by earlier calculated missing digits. By aligning column and row intersections, I’m sure each square holds a unique number from 1 to 9.

Aligning intersections

I'm pairing the missing digits from rows, columns, and boxes, ensuring each square in the Sudoku grid carries a distinct number from 1 to 9.

Pinpointing intersections

Let's match missing digits from rows, columns, and boxes, ensuring each Sudoku square has a unique number from 1 to 9, while cross-checking for any contradictions.

Pinpointing intersections

Matching missing digits from rows, columns, and boxes ensures each Sudoku square has a unique number from 1 to 9, while avoiding contradictions in the logical setup.

Ensuring unique numbers

I completed each row and column by assigning the missing numbers from Sudoku rows, columns, and boxes, making sure each number 1 to 9 is unique in every row and column.

Ensuring accuracy

I confirmed that each row from 1 to 9 in the Sudoku grid contains all digits from 1 to 9, with no repetitions, ensuring the puzzle's accuracy.

Verifying distinct digits

I checked that each Sudoku row and column contains all digits from 1 to 9 without repetition, confirming the solution is comprehensive and accurate.

思考，持续 11m 33s

Here is a fully worked‐out solution. I'll rewrite the puzzle in a 9×9 grid, where “0” indicates an empty cell:

Row 1: 0 1 0 | 0 8 4 | 0 6 0
Row 2: 2 8 0 | 0 0 0 | 4 1 9
Row 3: 0 4 6 | 5 1 0 | 0 0 0
       ------+-------+------
Row 4: 0 2 3 | 6 0 5 | 0 8 0
Row 5: 0 0 0 | 0 4 2 | 1 5 3
Row 6: 4 5 8 | 0 0 1 | 0 0 2
       ------+-------+------
Row 7: 9 0 0 | 0 0 0 | 8 2 6
Row 8: 0 0 0 | 9 7 0 | 3 0 0
Row 9: 8 0 1 | 0 2 0 | 0 0 0

A concise route to the final answer (skipping many of the intermediate pencil‐mark steps) leads to this unique solved grid:

   R1: 3 1 9 | 2 8 4 | 5 6 7
   R2: 2 8 5 | 3 6 7 | 4 1 9
   R3: 7 4 6 | 5 1 9 | 2 3 8
             +-------+------
   R4: 1 2 3 | 6 9 5 | 7 8 4
   R5: 6 9 7 | 8 4 2 | 1 5 3
   R6: 4 5 8 | 7 3 1 | 6 9 2
             +-------+------
   R7: 9 7 4 | 1 5 3 | 8 2 6
   R8: 5 6 2 | 9 7 8 | 3 4 1
   R9: 8 3 1 | 4 2 6 | 9 7 5

Verifying each row, column, and 3×3 box will confirm all digits 1–9 appear exactly once.