Sudoku Reasoning: Think Like a Mathematician
🧩 Sudoku Reasoning: Think Like a Mathematician
Sudoku is more than a number puzzle. It is a way to practise careful thinking.
In this lesson, we will solve a $4 \times 4$ Sudoku by checking rows, columns, and small $2 \times 2$ boxes.
We will also learn how to explain our answers. A good Sudoku solver does not just write a number; they can say why that number fits.
🎯 In this lesson you will
- Solve a $4 \times 4$ Sudoku step by step.
- Check rows, columns, and $2 \times 2$ boxes before writing a number.
- Use logical reasoning instead of guessing.
- Explain why a number belongs in a certain box.
- Create a simple Sudoku clue of your own.
🧠 Key Words
- Sudoku
- clue
- logic
- possibility
- eliminate
- justify
Show Definitions
- Sudoku: A number puzzle where each number must follow row, column, and box rules.
- clue: A number already given in the puzzle.
- logic: Careful thinking that helps us find the correct answer.
- possibility: A number that might fit in an empty box.
- eliminate: To remove an answer because it breaks a rule.
- justify: To explain why an answer is correct.
🔢 The Sudoku Rules
A $4 \times 4$ Sudoku uses the numbers $1$, $2$, $3$, and $4$.
Each row must contain each number once.
Each column must contain each number once.
Each small $2 \times 2$ box must contain each number once.
When we choose a number, we must check all three rules.
| $1$ | $3$ | ||
| $4$ | $2$ | ||
| $1$ | $4$ | ||
| $4$ | $1$ |
👀 From Guessing to Reasoning
Guessing means choosing a number without a clear reason.
Reasoning means using clues to decide which number must go in an empty box.
For example, if a row already has $1$, $3$, and $4$, then the missing number must be $2$.
But before we write $2$, we should also check the column and the small box.
✅ Worked Example
Look at the empty box in the first row and second column.
| $1$ | $3$ | $4$ | |
| $3$ | $4$ | $1$ | $2$ |
| $2$ | $1$ | $4$ | $3$ |
| $4$ | $3$ | $2$ | $1$ |
The first row has $1$, $3$, and $4$. So the row needs $2$.
The second column already has $4$, $1$, and $3$. It also needs $2$.
The top-left $2 \times 2$ box has $1$, $3$, and $4$. It needs $2$.
All three checks agree, so the empty box must be $2$.
🧠 Think like a Mathematician
Two students want to fill the same empty box.
Student A says, “I think it is $2$ because it feels right.”
Student B says, “It is $2$ because the row, column, and small box all need $2$.”
Which student is using mathematical reasoning? Explain why.
Show Answer
Student B is using mathematical reasoning because they explain the answer using the Sudoku rules. Student A is guessing.
❓ EXERCISES
1. Which number is missing from the first row?
| $1$ | $3$ | $4$ |
👀 Show answer
2. Use the row and column to decide the missing number.
| $1$ | $3$ | $4$ | |
| $3$ | $4$ | $1$ | $2$ |
| $2$ | $1$ | $4$ | $3$ |
| $4$ | $3$ | $2$ | $1$ |
👀 Show answer
3. Explain why the empty box cannot be $4$.
| $1$ | |
| $3$ | $4$ |
👀 Show answer
4. Complete the Sudoku puzzle.
| $1$ | $3$ | ||
| $4$ | $2$ | ||
| $1$ | $4$ | ||
| $4$ | $1$ |
👀 Show answer
| $1$ | $2$ | $3$ | $4$ |
| $3$ | $4$ | $1$ | $2$ |
| $2$ | $1$ | $4$ | $3$ |
| $4$ | $3$ | $2$ | $1$ |
5. Create one clue.
Start with this completed row: $1$, $2$, $3$, $4$. Hide one number and ask a friend to find it.
👀 Show answer
⚠️ Common Mistake
A common mistake is writing a number because it fits one rule only.
A Sudoku answer must fit the row, the column, and the small box.