Sudoku

Sudoku is the 9×9 number-placement puzzle that’s spread from a niche American puzzle magazine in the 1980s to a global phenomenon since 2004. The rules are minimal: place digits 1-9 in each row, column, and 3×3 sub-grid so that no digit repeats. The puzzle starts partially filled; you complete it through pure logical deduction (no guessing required for a properly-constructed puzzle).

How to play

• The 9×9 grid is divided into nine 3×3 sub-grids (“boxes” or “regions”).
• Each row, column, and box must contain each digit 1-9 exactly once.
• The puzzle gives you some starting digits (“clues”). Fill in the rest.
• A well-constructed Sudoku has exactly one solution.
• Click a cell to select. Type 1-9 to enter a digit. Use pencil marks (small numbers) to track possibilities.

Beginner techniques

• Naked singles. A cell that can only be one digit (because every other digit is excluded by row, column, or box) is a “naked single”. Fill it in immediately.
• Hidden singles. A digit that can only go in one cell of a row/column/box, even if that cell could theoretically hold other digits, is a “hidden single”. Look for them in each of the 27 units (9 rows + 9 columns + 9 boxes).
• Scan, don’t think. Most easy and medium puzzles solve entirely through repeated scanning for naked and hidden singles. Pick a digit (say “5”), look at where 5 already is, find rows/columns/boxes where 5 must go in one specific cell.
• Cross-hatching. Pick a digit. Look at one row of three boxes. Where can the digit go in each box? Often constraints from rows/columns force it to one cell.
• Don’t pencil-mark every cell. Heavy pencil-marking is for hard puzzles. Easy puzzles solve by direct scanning — pencil marks just slow you down.

Intermediate techniques

• Naked pairs. Two cells in a unit (row/column/box) where the only candidate digits are exactly the same two (e.g. both can only be 4 or 7). One of them is 4, the other is 7 — you don’t know which, but you know NO OTHER cell in that unit can be 4 or 7.
• Hidden pairs. Two digits in a unit that can only appear in two specific cells. Those cells must contain those two digits, even if they have other candidates.
• Pointing pairs / triples. If a digit’s candidates within a box are confined to one row (or column) of that box, that digit can be eliminated from the rest of that row (or column) outside the box.
• Box-line reduction. The reverse: if a digit’s candidates in a row/column are confined to one box, you can eliminate it from the rest of that box.
• X-Wing. A digit appears as a candidate in exactly two cells in two different rows, AND those cells are in the same two columns. Then those two columns can’t have that digit anywhere else.

Advanced techniques

• Swordfish. Like X-Wing but with three rows × three columns instead of two × two. Rare but extremely satisfying when found.
• XY-Wing. Three cells with candidate sets {x,y}, {y,z}, {x,z} that share a unit. Any cell that sees both endpoints can eliminate digit z.
• Coloring / chains. Mark candidates with colors based on logical implications. When two same-colored candidates appear in the same unit, that color must be wrong — eliminate all of that color.
• Forcing chains. Hypothetically assume a candidate is true, follow the consequences, see if they lead to contradiction. If so, the candidate must be false.
• Nishio (single-cell trial). Pick a cell, try one candidate, see if the puzzle solves or hits a contradiction. Technically “guessing” but well-defined as a manual technique.

Sudoku math facts

• Number of valid completed grids: 6,670,903,752,021,072,936,960 (about 6.67 × 10^21). Calculated by Bertram Felgenhauer and Frazer Jarvis (2005).
• Minimum clues for unique solution: 17. Proved by McGuire, Tugemann, and Civario (2012) using exhaustive computer search. No Sudoku with 16 or fewer clues has a unique solution.
• NP-completeness: generalized Sudoku (n²×n²) is NP-complete. Standard 9×9 Sudoku is technically solvable in constant time (just bounded), but for humans it’s very much in the “feels hard” zone.
• Construction: creating a hard Sudoku is harder than solving one. Generators typically construct a complete grid, then remove cells while testing uniqueness with a solver.

Sudoku history

• 1979 — Number Place: American architect Howard Garns publishes the first modern Sudoku as “Number Place” in Dell Magazines. Anonymous at the time.
• 1984 — Sudoku name: Nikoli (Japanese puzzle magazine) re-publishes Garns’ format under the name “Sūji wa dokushin ni kagiru” (later shortened to “Sudoku” — “single number”).
• 2004 — UK breakout: Wayne Gould pitches Sudoku to The Times of London. Within a year, every UK national newspaper has a daily Sudoku.
• 2005-2006 — Worldwide explosion: Sudoku books dominate bestseller lists. Estimated 100+ million people playing daily by 2006.
• 2012 — Minimum-17 proven: the mathematical lower bound on clues for unique solvability is established.

Common mistakes

• Guessing too early. Easy and medium Sudoku NEVER require guesses. If you’re stuck, you’ve missed a technique — re-scan, don’t gamble.
• Pencil-marking without erasing. When you place a digit, immediately update pencil marks in the affected row/column/box. Stale candidates lead to wrong placements.
• Tunnel vision on one box. Sudoku constraints flow through rows AND columns AND boxes. When stuck in one box, switch perspectives — look at the same digit in a different unit type.
• Working in pen. Until you’re consistently solving hard puzzles, pencil-and-eraser is essential.

Variants

• Killer Sudoku. Adds “cages” — groups of cells whose digits must sum to a given number, with no repeats within the cage.
• Diagonal Sudoku (“X Sudoku”). Both diagonals must also contain 1-9 exactly once.
• Hyper Sudoku. Four additional 3×3 regions overlap the main grid.
• Mini Sudoku (6×6). 6×6 grid with 2×3 blocks. Easier for beginners.
• Samurai Sudoku. Five 9×9 grids share corners. Solve all five interlocked.

FAQ

• Q: Do I need to know math? A: No. Sudoku uses the digits 1-9 as symbols only — no addition, no subtraction. Killer Sudoku adds basic addition.
• Q: Is every Sudoku solvable without guessing? A: Well-constructed published Sudoku, yes. Random or poorly-generated grids may require trial and error.
• Q: What’s the world record? A: Top human solvers complete Expert puzzles in under 2 minutes. The world record for solving a Sudoku (any difficulty) is held by Yuhei Kusui at about 1:23 on an Easy puzzle.
• Q: How many starting clues do hard puzzles use? A: Typically 22-28. The minimum for any unique-solution Sudoku is 17, but those are rare and very difficult.

For other logic puzzles, try Minesweeper, Picross, Lights Out, or Crossword.