The Best Tricks for Solving Hard Sudoku Puzzles

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Sudoku expert Meghan Kim shares simple beginner techniques to use when you’re stumped on a sudoku puzzle
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Sudoku is a great mental workout, but solving more challenging puzzles might take a little extra brainpower. Luckily, there are a lot of tricks and strategies to help you place the numbers correctly. Whether you’re a beginner or advanced solver, learning these techniques will prevent you from getting stuck. Keep reading, and we’ll walk you through the best approaches for solving sudoku, with exclusive tips from a sudoku expert, as well as some other variants to try out after!

Top Sudoku Solving Techniques

Sudoku expert Meghan Kim recommends using the scanning or cross-hatching technique, which involves checking where a digit can legally go in a box by ruling out the rows and the columns that already contain it. Some other great techniques include:

  • Writing down all the possible numbers for each box using pencilmarks.
  • Looking for restrictions on candidates from intersecting rows and columns.
  • Checking for groups of boxes that only have 2, 3, or 4 possible solutions.
  • Testing possible chains of numbers to see which candidates are possible.
  • Eliminating candidates from rows and columns with the X-wing method.
Section 1 of 4:

Sudoku Tricks for Beginners

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  1. Step 1 Pencilmarks
    Pencilmarks list all the possible numbers that could go into a specific box. Go through each row, column, and 3x3 block, and write which numbers could be placed there. Make sure you write the numbers small and in pencil so you can easily erase them to place the correct number.[1]
    • Example: If a column already has 1, 4, 5, 6, and 9, then you could place the numbers 2, 3, 7, and 8 as small pencilmarks in the remaining boxes.
    • Don’t forget to check the intersecting rows and columns when you’re writing your notation. If you need a 2 in a row, it cannot go into a box if the intersecting column already has a 2.

    Meet the wikiHow Expert

    Meghan Kim is a sudoku expert who’s been playing sudoku for over 12 years. She runs her own sudoku brand and publishes puzzle books with a focus on what makes sudoku approachable and satisfying to solve for all types of players.

  2. Step 2 Sole candidates
    Sole candidates, or “naked singles,” are boxes on the grid where only one number fits, says Kim. She suggests using the scanning or cross-hatching technique to find them: “This is where you scan and check where a digit can legally go in a box by ruling out the rows and the columns that already contain it.”[2] Look for any single boxes where intersecting rows or columns limit which number can be placed there. Because the number can’t go anywhere else, you can confidently write it into the puzzle.[3]
    • Kim notes that the naked singles and scanning/cross-hatching techniques are some of the first tools everyone learns.[4]
    • There are also hidden singles, Kim adds, “which is a digit that can go in only one cell of a unit, even if that cell shows other candidates.”[5]
    • Example: The first block in the top-right corner of the puzzle needs a 4. If the puzzle already contains 4s in rows 2 and 3 and in columns A and C, then the only possible spot to place a 4 is in the top middle box (row 1, column B).
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  3. Step 3 Locked candidates
    A locked candidate is a number that has to be placed in a specific row or column inside a block. If there are no other boxes in the row or column where a number can go, then you can eliminate it as an option from other boxes inside the same block.[6]
    • Example: Row 1 needs a 6. The boxes in the row are filled in block 1, and there is already a 6 located in another row in block 3, so the 6 in block 2 is locked into the first row. Erase any 6s you notated in Rows 2 or 3 in that block because they are not possible solutions.
    • Example: Block 5 in the center of the puzzle needs a 7. If the four corner boxes already have numbers and there is a 7 in block 2, Column E, then the 7 in block 5 is locked to row 5. Erase any 7s you marked in row 5 in blocks 4 and 6.
  4. Step 4 Naked pairs
    Naked pairs are when 2 boxes in a row, column, or block have only 2 possible solutions that are exactly the same. Because the boxes can only contain those 2 numbers, you can remove them as candidates from all the other boxes in the same row, column, or block.[7]
    • Example: If the only possible numbers for 2 boxes in a block are 2 and 3, then you can remove 2 and 3 from any other boxes within the same block.
  5. Step 5 Hidden pairs
    Hidden pairs are similar to naked pairs, but they’re a little harder to notice. Fill in all the possible candidates for a row, column, or block. If a pair of numbers only appears in 2 specific boxes, then the boxes must contain them. Erase all other pencilmarks from those boxes.[8]
    • Example: One box has the numbers 2-4-5-6-8 as candidates, and another has 2-3-4-6-7-9. If the 2 and 6 are not possible in any other boxes in that block, then you can erase the 4, 5, 7, 8, and 9 as candidates.
  6. Step 6 Hidden groups
    Similar to pairs, groups of 3 numbers (triples) and 4 numbers (quads) can be hidden in the grid as well. If a group of 3 boxes in a row, column, or block is the only place 3 numbers are possible, then it’s a hidden triple, and you can erase the other candidates. If 4 potential numbers can only be placed in the same 4 boxes, then it’s a hidden quad.[9]
    • Example: If 3 boxes in a block have 1, 4, and 5 as candidates and they aren’t marked anywhere else in the same block, remove any other possible numbers from those boxes.
    • Example: If 4 boxes have 3, 4, 5, and 7 as candidates and aren’t listed in other boxes in the same block, remove any other possible numbers from those boxes.
    • Some boxes with hidden triplets or quads may only contain 2 or 3 of the possible numbers instead of all 3 or 4.
  7. Step 7 Forcing chains
    Forcing chains are simple ways to test if a specific number is possible in a box. Look for a box that only has 2 potential candidates and try out one of the possible numbers. Check for another box with a pair of numbers containing the one you just used in the same row or column, and fill it in with the other number. Keep locating boxes and checking numbers until you reach the end of a chain.[10]
    • Then, test the second possible number, and follow the chain it creates.
    • Look for any boxes that appear in both chains. If the number in the box is the same for both chains, then you can confidently fill it in.
    • Conversely, if a number is not possible in either chain, then you can erase it as a candidate from the box.
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Section 2 of 4:

Advanced Sudoku Tricks

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  1. Step 1 X-wing
    To properly find an X-wing on a sudoku grid, look for a number that only appears in 2 of the boxes. Locate another row where the same number also only appears in 2 boxes that share columns with the first row. Because the number can only appear in each column once, you can eliminate it from any other boxes in those columns.[11]
    • Example: If 9 can only be placed in rows 5 and 7 in columns A and G, then the 9s can only be placed in A5/G7 or A7/G5. Erase any 9s you have notated in other boxes in those columns.
  2. Step 2 XY-wing
    In this variation of the X-wing strategy, look for a box that only has a pair of numbers as possible candidates (x and y). Check for 2 other boxes branching off the first one that have pairs of numbers where one of the candidates is the same as the first box (x & z and y & z). If any boxes in a line or intersecting with the 2 branches have the shared number (z), you can erase it from those boxes.[12]
    • Example: The box at row 8, column E has 2 and 9 as potential numbers. Row 8, column B has 1 and 9, and row 2, column E has 1 and 2.
      • 1 is the shared number between the 2 branches.
      • Erase 1s as candidates from any boxes from rows 2 and 8 between columns B and E, and from columns B and E between rows 2 and 8.
      • You can also erase 1 as a candidate where the branches would intersect on row 2, column B.
  3. Step 3 Skyscraper
    To find a skyscraper pattern in a sudoku puzzle, look for 2 rows or columns where a single digit can only be placed into 2 boxes. If 2 of the boxes share a row or column, the number can be erased as candidates from any others that intersect with the remaining 2 boxes.[13]
    • Example: A 1 can only be placed in column F in rows 1 and 5, and in column I in rows 3 and 5. Both columns share row 5, creating the base of the skyscraper. Only one of the boxes in row 5 can contain a 1.
      • If the 1 is placed in column F, then the 1 in column I has to be in row 3.
      • If the 1 is placed in column I, then the 1 in column F has to be in row 1.
      • Because one of these options has to be true, you can remove any other 1s from boxes in rows 1 and 3.
  4. Step 4 2-string kite
    A use the 2-string kite strategy, look for a block where a number can only be in 2 possible boxes in different rows or columns (the kite). Check along the same row or column as those boxes to find another box that could contain that number (the strings). Wherever the strings would intersect cannot contain that number.[14]
    • Example: A 5 can only be placed in column G in rows 2 and 9. A 5 can also only be placed in row 8 in columns D and I.
      • I8 and G9 are in the same block and create the kite. G2 and D8 become the ends of the strings.
      • If you place the 5 in I8, then the 5 in column G has to go in row 2.
      • If you place the 5 in G9, then the 5 in row 8 has to go in column D.
      • Therefore, the box in row 2, column D cannot contain 5 because it would intersect, so you can erase it as a candidate.
  5. Step 5 Swordfish
    The swordfish technique helps eliminate candidates from 3 columns at the same time. Look for 3 rows where a number can only appear in 2 boxes. Make sure that the boxes in each row share a column with at least one of the other rows. The number has to appear in one of the row’s boxes in each column, so you can erase any other places where the number appears in those columns.[15]
    • Example: A 2 in row 1 can only be placed in columns F and I. The 2 in row 5 can only be placed in columns A and I. The 2 in row 9 can only be placed in columns A and F.
      • If you place the 2 in I1, then the remaining 2s have to go in A5 and F9.
      • If you place the 2 in F1, then the remaining 2s have to go in I5 and A9.
      • The 5 cannot appear in any other boxes in columns A, F, or I, so you can erase it as a candidate.
  6. Step 6 Unique rectangle
    Unique rectangles come into play if you have 4 boxes that share the same possible pair of numbers that face each other, which could give the puzzle 2 possible solutions. Check if any of the boxes have other candidates. Sudoku puzzles can only have 1 solution, so you can remove the shared pair from the box with other possibilities to ensure the puzzle is unique.[16]
    • Example: Both 2 and 3 are candidates in row 4 in columns D and E. They are also candidates in row 9 in the same columns.
      • If row 9, column E also has another number as a candidate, you can erase 2 and 3 from that box to ensure the puzzle is unique.
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Section 3 of 4:

Sudoku Strategies If You’re Stuck

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  1. Step 1 Avoid guessing numbers.
    All sudoku puzzles have logical steps you can follow in order to solve them, so you should never have to guess numbers randomly. Always take a second to look over the puzzle for any clues or restrictions that help you figure out the correct number.
  2. Step 2 Move to a different section of the puzzle.
    When you’re having trouble solving for a specific row, column, or block, hop to another part of the puzzle and work through any solutions there. Solving a different part of the puzzle might unlock clues for the section you were working through earlier and make it easier to place numbers correctly.
  3. Step 3 Take a break and come back later.
    When you’re really stuck, stop solving and go do something else for a little while. You could go for a walk to clear your mind, clean up around your home, or do anything else to reset your brain. When you come back with fresh eyes, scan over the puzzle again to look for any sections or obvious solutions that you didn’t notice earlier.
  4. Step 4 Practice easier puzzles before going back to hard ones.
    If you’re working on a challenging sudoku and can’t seem to figure it out, set it aside and try out an easier puzzle. Working through a different puzzle can give you a confidence boost and help open your eyes to any logical steps you were missing before.
    • Starting with easier puzzles also helps you develop your skills so you can solve challenging sudoku faster.
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Section 4 of 4:

Sudoku Variants to Try

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  1. Step 1 Diagonal sudoku
    Diagonal sudoku is pretty much the same as standard sudoku, but each of the diagonals from corner to corner also must contain the digits 1-9 exactly once. This limits where you can place numbers even more and adds another level of challenge to the game.[20]
    • Diagonal sudoku may also be called “X sudoku” because the diagonals create a large X on the board.
  2. Step 2 Thermometer sudoku
    Thermometer (thermo) sudoku has special thermometer shapes on the board. When you place numbers into the thermometers, the lowest number is placed in the round bulb, and the values increase along its length. The numbers don’t have to be in sequential order, but the extra limitation gives you hints about the possible numbers.[21]
    • Example: If a thermometer is 3 boxes long and a 6 is inside the bulb, the next box can either contain a 7 or 8, and the final box must contain an 8 or 9.
    • Start with the longest thermometers when solving a thermo sudoku because they’ll have the most limitations and be easier to figure out.
  3. Step 3 Knight sudoku
    Sometimes called “anti-knight” sudoku, this variant has a restriction that numbers cannot be a chess knight’s move away from each other (2 boxes in one direction, and 1 box in another). This helps eliminate potential values from boxes throughout the board so you’re able to deduce where numbers go easier.[22]
    • Example: If you have a 2 in column B/row 2 (B2), then you cannot place a 2 in D1, D3, A4, or C4 because they are a knight’s move away.
  4. Step 4 Kropki dots
    In a Kropki dot sudoku puzzle, some boxes have white or black circles between them. If boxes have a black dot between them, then one of the numbers is double the value of the other one. If boxes have a white dot between them, the numbers have to be in consecutive order.[23]
    • Example: If a box with a 2 has a black dot next to it, then the adjacent box has to contain either a 1 or a 4.
    • Example: If a box with a 9 has a white dot on one edge, the adjacent box must contain an 8.
  5. Step 5 Killer sudoku
    In addition to the normal sudoku rules, killer sudoku introduces cages enclosed by dotted lines that each have a value. The value in each cage equals the sum of all the numbers inside, and no numbers can be repeated inside the cage. A lot of killer sudoku puzzles don’t have starting numbers listed, so recognizing which numbers add up to the cage’s value is important for solving the puzzle.[24]
    • Example: If a cage has 3 boxes and has a sum of 8, then the cage must contain the values 1, 3, and 4.
    • Use a cheat sheet to learn all the possible number combinations for different cage sizes and sums.
  6. Step 6 Sandwich sudoku
    Sandwich sudoku puzzles have numbers listed outside of each row and column. Each of these numbers is the sum of the boxes between the 1 and 9 for the given row or column. To solve these puzzles, remember that a row or column always adds up to 45 and subtracting the sandwich value from it reveals the boxes that aren’t sandwiched.[25]
    • Example: If a row has 10 listed on the edge, then the boxes between the 1 and 9 either contain 2-3-5, 2-8, 3-7, or 4-6.
    • Example: If a column has a 0 listed along the edge, then there are no boxes in between the 1 and 9.
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References

  1. ↑ http://sudopedia.enjoysudoku.com/Diagrams_and_Notations.html
  2. ↑ Meghan Kim. Sudoku Expert and Puzzle-Book Author. Expert Interview
  3. ↑ https://www.kristanix.com/sudokuepic/sudoku-solving-techniques.php
  4. ↑ Meghan Kim. Sudoku Expert and Puzzle-Book Author. Expert Interview
  5. ↑ Meghan Kim. Sudoku Expert and Puzzle-Book Author. Expert Interview
  6. ↑ https://www.pennydellpuzzles.com/wp-content/uploads/2019/03/How-to-Solve-Sudoku.pdf
  7. ↑ https://www.learn-sudoku.com/naked-pairs.html
  8. ↑ https://sudoku.com/sudoku-rules/hidden-pairs/
  9. ↑ https://sudoku.coach/en/learn/hidden-groups
  1. ↑ https://www.sudokuonline.io/tips/advanced-sudoku-strategies
  2. ↑ https://www.pennydellpuzzles.com/wp-content/uploads/2019/03/How-to-Solve-Sudoku.pdf
  3. ↑ https://www.pennydellpuzzles.com/wp-content/uploads/2019/03/How-to-Solve-Sudoku.pdf
  4. ↑ https://hodoku.sourceforge.net/en/tech_sdp.php
  5. ↑ https://sudoku.coach/en/learn/two-string-kite
  6. ↑ https://www.pennydellpuzzles.com/wp-content/uploads/2019/03/How-to-Solve-Sudoku.pdf
  7. ↑ https://www.sudokuonline.io/tips/advanced-sudoku-strategies
  8. ↑ Meghan Kim. Sudoku Expert and Puzzle-Book Author. Expert Interview
  9. ↑ Meghan Kim. Sudoku Expert and Puzzle-Book Author. Expert Interview
  10. ↑ Meghan Kim. Sudoku Expert and Puzzle-Book Author. Expert Interview
  11. ↑ https://www.sudokuwiki.org/Sudoku_X_Strategies
  12. ↑ https://masteringsudoku.com/thermo-sudoku/
  13. ↑ https://www.clarity-media.co.uk/viewblog.php?id=anti-knight-sudoku-puzzle
  14. ↑ https://www.puzzlemix.com/Kropki+Sudoku
  15. ↑ https://www.kristanix.com/sudokuepic/killer-sudoku-rules.php
  16. ↑ https://dkmgames.com/SandwichSudoku/

About This Article

Meghan Kim
Co-authored by:
Meghan Kim
Sudoku Expert and Puzzle-Book Author
This article was co-authored by Meghan Kim and by wikiHow staff writer, Hunter Rising. Meghan Kim is a Sudoku expert and puzzle-book author based in San Francisco, CA. She has played and studied Sudoku for over 12 years and is the founder of Mindful Sudoku, her own line of Sudoku journals that combine classic Sudoku with reflective journaling for mindfulness. Meghan is also the creator of The Sudoku Girl, an educational Sudoku platform with over 80,000 followers across TikTok and Instagram. She designs and curates puzzles spanning beginner, intermediate, and advanced levels, with a focus on making Sudoku approachable and satisfying to solve for puzzlers at every level. This article has been viewed 2,224 times.
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