Fill the grid
Difficulty: gentle to moderate. Seven givens, one deliberate starting point, no guessing required.
- Place the digits 1, 2, 3 and 4 in every row.
- Place 1, 2, 3 and 4 in every column.
- Place 1, 2, 3 and 4 in each of the four 2 × 2 boxes (marked by the thicker lines).
| 1 | · | · | 4 |
| 3 | 4 | · | · |
| · | · | 4 | · |
| 4 | · | · | 1 |
Reveal the walkthrough
- Top-left box holds 1, 3 and 4. Row 1, column 2 must be 2.
- Row 1 now has 1, 2, 4. Row 1, column 3 is 3.
- Column 1 has 1, 3, 4. Row 3, column 1 is 2.
- Bottom-left box holds 2 and 4, so its two empty cells need 1 and 3. Row 4 already contains 1, so row 4, column 2 is 3 and row 3, column 2 is 1.
- Row 4 has 4, 3, 1. Row 4, column 3 is 2.
- Row 3 has 2, 1, 4. Row 3, column 4 is 3.
- Row 2 needs 1 and 2. Column 3 already contains 2 (row 4), so row 2, column 3 is 1 and row 2, column 4 is 2.
| 1 | 2 | 3 | 4 |
| 3 | 4 | 1 | 2 |
| 2 | 1 | 4 | 3 |
| 4 | 3 | 2 | 1 |
If you got stuck early, it was probably because the 2s and 3s look interchangeable at first. The given 3 in row 2 is the single clue that breaks that symmetry; without it, the grid would have two solutions.
Where these grids come from
The modern number-placement grid has a surprisingly traceable history. Howard Garns, an architect from Indiana, is generally credited with the format that appeared as "Number Place" in Dell Pencil Puzzles & Word Games in 1979. The Japanese publisher Nikoli took the idea up in 1984, and its president Maki Kaji gave it the name Sudoku. Long before either, the Lo Shu magic square — the 3 × 3 arrangement on our home page — showed that numbers placed under strict rules can be satisfying in themselves.
Related formats include Kakuro, where digits must add to given totals, and KenKen, created by the Japanese teacher Tetsuya Miyamoto. They differ in detail but share one principle: the rules leave exactly one consistent arrangement.
Three techniques that do most of the work
1 · Candidates
For any empty cell, list the digits not already used in its row, column and box. Write them small in the corner. On a 4 × 4 grid you can often keep candidates in your head; on a 9 × 9 grid, pencil marks save you from rechecking the same lines again and again.
2 · Single candidate
When a cell has only one candidate left, place it. Step 1 of the walkthrough is exactly this: the top-left box already contained 1, 3 and 4, so the remaining cell could hold only 2.
3 · Single position
Turn the question around. Instead of asking "what goes in this cell?", ask "where can this digit go in this row, column or box?" If a digit has only one legal home in a unit, it goes there. Step 4 used a version of this: of the two empty cells in the bottom-left box, only one could take a 3.
A grid is a conversation between rows, columns and boxes. Listen to all three before you write anything in ink.
Why guessing breaks a solve
Guessing feels faster, and sometimes it lands. The trouble is what happens when it does not. A wrong guess can sit unnoticed for many moves, producing placements that look legal until a contradiction appears far from its cause. At that point you cannot tell which of your entries are reasoned and which inherited the guess. Working from candidates keeps every placement traceable, which makes mistakes easy to locate and fix.
Published grids are designed so that logic alone suffices. If you find yourself needing to guess, it usually means a technique has been missed rather than that the puzzle requires luck. Step back, recount candidates in the most crowded box, and look for a digit with a single position.
A note on magic squares
For 3 × 3 squares using 1 to 9, the total of all nine numbers is 45. Three rows share that total equally, so every line must add to 15. Knowing the target before you begin turns a search into a short calculation — a small example of reasoning about structure before reasoning about cells.
Continue practising
Deduction grids
The same elimination habit, applied to people, places and orders.
Cross out what cannot be true