How to Solve a 6x6 Sudoku, With an Original Practice Grid

A 6x6 sudoku is the friendliest way into sudoku. It uses only the digits 1 to 6, the grid fits on a sticky note, and almost every beginner grid can be finished with two simple techniques. NRICH, the mathematics enrichment project at the University of Cambridge, keeps a whole collection of sudoku-style puzzles, from 3 by 3 and 4 by 4 "Mathdokus" up to 6 by 6 versions, because filling a grid is a clean way to practice careful reasoning.

Below you will find the rules, the two techniques, a short worked start, and an original practice grid. The grid was checked against every possible way of filling it, and exactly one completed grid fits the clues, so you never need to guess.

The rules of a 6x6 sudoku

• The grid has 6 rows and 6 columns.
• It is split into six boxes, each 2 rows tall and 3 columns wide.
• Every row, every column, and every box must contain the digits 1 to 6 exactly once.

That is all. There is no arithmetic involved; the digits could just as well be six different symbols.

Technique 1: the only digit that fits a cell

Pick an empty cell and list the digits already used in its row, its column, and its box. If five of the six digits are already used, the missing one must go in that cell. Sudoku solvers often call this a "naked single."

Technique 2: the only cell that fits a digit

Flip the question around. Pick a row, column, or box and a digit it still needs, say 3. Check each empty cell in that unit and cross off the ones where a 3 is blocked by the cell's other row, column, or box. If only one cell is left, the 3 goes there. This is usually called a "hidden single."

NRICH's guidance on working systematically describes this kind of habit in general terms: work in a structured order and keep a record, so you can see underlying patterns instead of a mess. For sudoku, that means sweeping the grid in a fixed order (row 1 to row 6, then columns, then boxes) rather than jumping around.

The practice grid

Rows are numbered 1 to 6 from the top and columns 1 to 6 from the left. A dot is an empty cell. Boxes are rows 1 and 2, rows 3 and 4, and rows 5 and 6, each split into columns 1 to 3 and columns 4 to 6.

Row	C1	C2	C3	C4	C5	C6
1	4	.	.	.	.	.
2	.	.	2	.	.	.
3	.	4	1	.	5	.
4	.	.	5	1	.	.
5	5	.	.	3	6	.
6	.	2	.	.	.	4

There are 12 clues. Try it before reading the walkthrough.

A worked start

Step 1, row 5, column 2. Row 5 already has 5, 3, and 6, so it still needs 1, 2, and 4. Column 2 contains a 4 (row 3) and a 2 (row 6). The only digit left for this cell is 1.

Step 2, row 5, column 3. Row 5 now needs 2 and 4. Column 3 already contains a 2 (row 2), so this cell is 4, and the last gap in row 5, column 6, must be 2.

Step 3, row 6, column 4. Row 6 needs 1, 3, 5, and 6. Column 4 already holds 1 and 3. The bottom-right box (rows 5 and 6, columns 4 to 6) already holds 3, 6, 2, and 4. The only digit that survives all three checks is 5.

Step 4, row 6, column 5. Row 6 now needs 1, 3, and 6. Column 5 holds 5 and 6, and the bottom-right box holds 3. That leaves 1.

Step 5, the 1 in row 1. Now switch to Technique 2. Row 1 needs a 1 somewhere. Columns 2, 3, 4, and 5 each already contain a 1 (in rows 5, 3, 4, and 6). Row 1, column 1 is taken by the 4. So the 1 in row 1 can only go in column 6.

Keep alternating between the two techniques. Every remaining cell in this grid can be reached that way, without trial and error.

Common mistakes on small grids

• Forgetting the box. On a 6x6 grid the boxes are rectangles, not squares, so it is easy to check a 3 by 2 area by accident. Lightly outline the six boxes before you start.
• Writing in a digit "because it looks right." If you cannot name the row, column, or box that forces it, leave it blank for now.
• Stopping when one technique runs dry. When no cell has a single candidate, switch to asking where each missing digit can go. The second question often unlocks the first again.

Answer

The completed grid, which is the only one that satisfies all 12 clues:

Row	C1	C2	C3	C4	C5	C6
1	4	5	3	6	2	1
2	1	6	2	4	3	5
3	6	4	1	2	5	3
4	2	3	5	1	4	6
5	5	1	4	3	6	2
6	3	2	6	5	1	4

To check your own grid, add up any row, column, or box. Each should total 21, because 1 + 2 + 3 + 4 + 5 + 6 = 21. A total of 21 does not prove a unit is correct, but any other total proves it is wrong.

Key takeaways

• A 6x6 sudoku uses the digits 1 to 6 once in every row, column, and 2 by 3 box.
• Technique 1 asks which digit is the only one left for a cell; Technique 2 asks which cell is the only one left for a digit.
• Sweep the grid in a fixed order and switch techniques when one stops producing placements.
• A well-made beginner grid, like the one above, never requires guessing.
• Every finished row, column, and box sums to 21, which makes a quick error check.