Sudoku is a logic-based number-placement puzzle. The objective is to fill a 9x9 grid with digits so that each column, each row, and each of the nine boxes in the grid contain all the digits from 1 to 9.

I love Sudoku. It fits my interest in problem-solving and logical thinking. Sudoku is an exercise in reasoning, strategising, and pattern recognition, providing a satisfying mental challenge that keeps the mind active. I use a nice sudoku Android app, “the clean one,” that supports notation, nice colour themes, and different difficulty levels. You can find it here: https://play.google.com/store/apps/details?id=ee.dustland.android.dustlandsudoku&hl=en&pli=1

## Terminology

Sudoku is relatively simple to describe. There are only a few terms to be aware of.

**Board**: A 9x9 grid

**Box**: Within the board are nine boxes

**Cell**: Each box is divided into nine cells. Each cell should contain a number between 1 and 9. Each number can only be used once within a box.

**Column**: A vertical set of cells. This could be three cells within a single box or nine cells across the entire board. For the latter, each cell in the column must contain a number between 1 and 9, and no number can be repeated in the column.

**Row**: a horizontal set of cells. This could be three cells within a single box or nine cells across the entire board. For the latter, each cell in the row must contain a number between 1 and 9, and no number can be repeated in the row.

## A bit of history

Given the name, it may be a surprise that Sudoku's origins can be traced back to Switzerland in the late 18th century. It became popular in Japan in the 1980s, where its name derives from *"Sūji wa dokushin ni kagiru"*, meaning *"the digits are limited to one occurrence"*.

Sudoku puzzles are created with mathematical algorithms and logic rules. The goal is to create a puzzle that has one and only one solution, providing an engaging and challenging experience for the player.

The process usually starts by filling a complete 9x9 grid, following the standard Sudoku rules. Every row, column, and 3x3 box contains all the numbers from 1 to 9 without any repetition. Here's an example:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 |

7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 |

5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 |

8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 |

6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 |

9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

The algorithm then proceeds to remove numbers from this grid. This is done systematically and strategically to ensure the resulting puzzle maintains a unique solution.

As numbers are removed, the puzzle's uniqueness must be preserved. It involves complex algorithmic checks to ensure no other solutions can be derived from the given clues. This is crucial; if a puzzle has more than one solution, it can't be solved using logic alone, which goes against the nature of Sudoku.

Here is an example of a puzzle created from the grid above.

1 | 3 | 4 | 7 | 8 | ||||

4 | 8 | 1 | 3 | |||||

8 | 1 | 3 | 5 | |||||

2 | 3 | 6 | 7 | 9 | 1 | |||

7 | 9 | 2 | 3 | 4 | ||||

8 | 9 | 1 | 3 | 5 | 6 | 7 | ||

5 | 6 | 7 | 8 | 1 | 2 | |||

6 | 9 | 1 | 3 | 4 | 5 | |||

1 | 5 | 6 | 8 |

Creating a valid Sudoku puzzle is a complex interplay of mathematics and logic, designed to deliver a puzzle that presents an enjoyable challenge without frustrating ambiguity or multiple solutions. Typically, the more numbers removed, the higher the puzzle's difficulty level.

## How to play

To achieve a winning state, the player places numbers in the cells following these simple rules:

- Each 9-cell row, column, and box on the board must contain all numbers from 1 to 9.
- A number must not repeat within a row, column, or box.

While playing, players may use commented numbers, also known as pencil marks, to fill out the potential numbers that could fit each cell. These are typically added with a smaller or different colour text.

Through various strategies, pencil marks are removed until only one viable number for each cell remains, which is then noted in the standard text. Apps excel at this approach as it's easier and cleaner to add/remove text. When doing it on paper, a pencil is typically used with an eraser to remove them once the correct number has been identified.

### Pencil marks

1 | 265 | 3 | 4 | 25 | 2569 | 7 | 8 | 69 |

4 | 2675 | 269 | 2579 | 8 | 2569 | 1 | 2 | 3 |

79 | 8 | 26 | 1 | 2 | 3 | 469 | 5 | 9 |

2 | 3 | 4 | 58 | 6 | 7 | 8 | 9 | 1 |

5 | 56 | 7 | 58 | 9 | 15 | 2 | 3 | 4 |

8 | 9 | 1 | 2 | 3 | 24 | 5 | 6 | 7 |

39 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 |

6 | 72 | 28 | 9 | 1 | 2 | 3 | 4 | 5 |

379 | 1 | 49 | 23 | 24 | 5 | 6 | 7 | 8 |

In this example, you can see smaller digits noted in some of the cells. They represent the possible numbers that could fit into each respective cell according to the current state of the puzzle.

Commented numbers effectively organise thoughts and strategies, helping players see the relationships between cells and make logical deductions. It is a tool especially utilised in solving more challenging puzzles, where keeping track of possibilities is crucial for success.

And here is the solution to this puzzle:

### Solution

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 |

7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 |

5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 |

8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 |

6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 |

9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

## Strategies to help solve Sudokus

Sudoku is not a game of luck but one of strategy. Whether starting with simple or highly complex difficulty levels, every Sudoku puzzle is a unique opportunity to exercise strategic thinking. Here are some techniques to help you become a Sudoku pro.

### 1. Uniqueness in Box, Row, or Column

A number can only appear once in each box, row, or column across the board. Note below in red the number 3 placements that are **not** possible due to the existing blue item. This strategy is foundational in solving Sudoku puzzles and the key rule to keep in mind when adding all of the pencil marks for each number.

1 | 3 | 4 | 3 | ||

4 | 3 | 8 | |||

8 | 1 | ||||

2 | 3 | ||||

7 | |||||

8 | 9 | 1 |

**Complexity rating: ★☆☆**

**Time rating: ★☆☆**

## 2. Only one number possible in a cell

One of the easiest strategies, certainly if you have added pencil marks for all of the numbers, as it will be immediately apparent which cells only have a single pencil mark in them. Somewhat unnecessary, but the green numbers below indicate where this is the case.

1 | 265 | 3 | 4 | 25 | 2569 | 7 | 8 | 69 |

4 | 2675 | 269 | 2579 | 8 | 2569 | 1 | 2 | 3 |

79 | 8 | 26 | 1 | 2 | 3 | 469 | 5 | 9 |

2 | 3 | 4 | 58 | 6 | 7 | 8 | 9 | 1 |

5 | 56 | 7 | 58 | 9 | 15 | 2 | 3 | 4 |

8 | 9 | 1 | 2 | 3 | 24 | 5 | 6 | 7 |

39 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 |

6 | 7 | 28 | 9 | 1 | 2 | 3 | 4 | 5 |

379 | 1 | 49 | 23 | 24 | 5 | 6 | 7 | 8 |

**Complexity rating: ★☆☆**

**Time rating: ★☆☆**

## 3. The only possible cell

The previous strategy is relatively easy, as you can easily see when there is only one pencil mark in a cell. Similar, but slightly harder to spot, is when there is only one possible cell within a box where a given number can go. This is typically easiest to see after adding all of the pencil marks. The green pencil marks below indicate numbers that can only fit in that cell. The green number may be alone or with other pencil marks in a given cell. However, the green number has no other suitable cell in that box, while the black pencil marks will have other options.

1 | 265 | 3 | 4 | 25 | 2569 | 7 | 8 | 69 |

4 | 2675 | 269 | 2579 | 8 | 2569 | 1 | 2 | 3 |

79 | 8 | 26 | 1 | 2 | 3 | 469 | 5 | 9 |

2 | 3 | 4 | 58 | 6 | 7 | 8 | 9 | 1 |

5 | 56 | 7 | 58 | 9 | 15 | 2 | 3 | 4 |

8 | 9 | 1 | 2 | 3 | 24 | 5 | 6 | 7 |

39 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 |

6 | 72 | 28 | 9 | 1 | 2 | 3 | 4 | 5 |

379 | 1 | 49 | 23 | 24 | 5 | 6 | 7 | 8 |

**Complexity rating: ★☆☆**

**Time rating: ★★☆**

## 4. Single row/column impact on consecutive boxes

If the pencil marks of a number are only possible in a single row or column within a box, then that number cannot appear in that same row or column in the two consecutive boxes in the same direction.

Below, the red number 1 on the top row of the first box and the orange number 1 pencil marks on the middle row in the middle box means that the number 1 must be somewhere on the **bottom** row in box 3 — indicated with green pencil marks. The same logic can also be applied vertically in columns.

1 | 3 | 4 | 7 | 8 | ||||

4 | 1 | 7 | 1 | |||||

8 | 6 | 2 | 3 | 1 | 1 | 1 |

**Complexity rating: ★★☆**

**Time rating: ★☆☆**

## 5. Double row/column impact on consecutive boxes

Similar to single row impact, if a number is only possible in the same two columns or rows in two consecutive boxes, then the third box in that direction must have the number in the remaining column or row.

Below, the red number 1 pencil marks in the first box and the orange number 1 pencil marks in the second box cover the top and middle rows of the first two boxes. This means that the number 1 must be on the **bottom** row of the third box — indicated with green pencil marks. The same logic can also be applied vertically in columns.

1 | 1 | 3 | 4 | 1 | 1 | 7 | 8 | |

1 | 1 | 6 | 1 | 7 | 1 | |||

4 | 8 | 9 | 6 | 2 | 3 | 1 | 8 | 1 |

**Complexity rating: ★★☆**

**Time rating: ★☆☆**

## 6. Exclusive twins in box

If there are two cells in a box that only contain pencil marks for the same two numbers, then those numbers cannot appear anywhere else in the box. Note that this only applies if pencil marks for **all** numbers have been added to the board.

Below, the two pairs have been indicated in green. They are the only two numbers and the same numbers in two cells. The red pencil marks of those numbers on other cells within the box can be safely removed in this case.

12 | 1259 | 3 |

6 | 12 | 12597 |

4 | 8 | 597 |

**Complexity rating: ★★☆**

**Time rating: ★★☆**

## 7. Exclusive twins in row/column

If there are two cells on a single row or column across the board that only contain pencil marks for the same two numbers, then those numbers cannot appear anywhere else in the same row or column across the board. The numbers may still appear on another row or column within the box. Note that this only applies if pencil marks for **all** numbers have been added to the board.

Below, the two pairs have been indicated in green. These are the only two numbers and the same numbers in two cells along the same row. The red pencil marks of those numbers on other cells within the box can be safely removed in this case. Note that the orange pencil marks must not be removed, as those are still viable options.

12 | 129 | 3 | 4 | 12 | 1268 | 7 | 5 | 1269 |

6 | 12749 | 5 | 127 | |||||

8 | 9 | 3 |

**Complexity rating: ★★☆**

**Time rating: ★★☆**

## 8. Twin pair exclusion in box

If two cells in a box have the same two pencil mark numbers, and those numbers don't appear in any other cell in the box, then those two cells will not accept any other numbers. This can be tricky to identify in some cases because the two cells could have pencil marks for additional numbers.

Below, the two pairs have been indicated in green. The numbers 1 and 2 only appear in these two cells. The red pencil marks on those two cells can be safely removed in this case, as the cells must contain one of the paired numbers each.

1259 | 59 | 3 |

6 | 127 | 597 |

4 | 8 | 597 |

**Complexity rating: ★★☆**

**Time rating: ★★★**

## 9. Twin pair exclusion in row/column

If two cells in a row or column have the same two pencil mark numbers, and those numbers don't appear in any other cell in the same row or column, then those two cells will not accept any other numbers. This can be tricky to identify in some cases because the two cells could have pencil marks for additional numbers.

Below, the two pairs have been indicated in green. The numbers 1 and 2 only appear in these two cells. The red pencil marks on those two cells can be safely removed in this case, as the cells must contain one of the paired numbers each. Note that the orange pencil marks must not be removed, as those are still viable options.

1259 | 59 | 3 | 4 | 128 | 68 | 7 | 5 | 69 |

6 | 1257 | 12749 | 5 | 127 | ||||

8 | 9 | 3 |

**Complexity rating: ★★☆**

**Time rating: ★★★**

## 10. Triplets

Somewhat harder, but strategies 6 to 9 can also work with three of the same numbers.

For example, for strategy 7, when there are three cells on a single row or column across the board that only contain pencil marks for the same three numbers. Those numbers cannot appear anywhere else in the same row or column across the board. The numbers may still appear on another row or column within the box. Note that this only applies if pencil marks for all numbers have been added to the board.

Below, the three pairs have been indicated in green. The red pencil marks of those numbers on other cells within the box can be safely removed in this case. Note that the orange pencil marks must not be removed, as those are still viable options.

129 | 1259 | 3 | 4 | 129 | 1268 | 7 | 129 | 1269 |

6 | 12749 | 5 | 127 | |||||

8 | 7 | 3 |

**Complexity rating: ★★★**

**Time rating: ★★★**