Magic squares from ancient China

Quick answer

Magic squares from ancient China are special grids of numbers where every row, column, and diagonal adds up to the same total. The oldest known magic square, the Lo Shu, comes from ancient China and helps us understand patterns, addition, and number relationships.

Learning outcome

By the end of this page, you will understand how to make and solve simple magic squares, and you will see how ancient Chinese people used maths to discover patterns in numbers.

• You can complete a 3x3 magic square using numbers 1 to 9
• You can explain how magic squares show number patterns
• You can find the magic total in a given magic square
• You can create your own 3x3 magic square

Why this matters

Magic squares are more than just fun puzzles – they help us understand how numbers work together. Ancient Chinese people used them to study patterns, and today they help us learn addition, multiplication, and problem-solving. You can use magic squares to make games, solve riddles, or even design cool art. They also show how maths has been used for thousands of years!

How it works

Magic squares are grids of numbers where the total of every row, column, and diagonal is the same. The most famous one is the Lo Shu square from ancient China, which is a 3x3 grid.

1. Start with a 3x3 grid (3 rows and 3 columns)
2. Fill it with numbers from 1 to 9
3. Each row, column, and diagonal must add up to the same number
4. Find that magic number by adding up all numbers (1 to 9) and dividing by 3

Examples you can see

Let's look at the Lo Shu magic square, the first magic square ever recorded:

Try adding the numbers in each row:

• Row 1: 4 + 9 + 2 = 15
• Row 2: 3 + 5 + 7 = 15
• Row 3: 8 + 1 + 6 = 15

Now check the columns:

• Column 1: 4 + 3 + 8 = 15
• Column 2: 9 + 5 + 1 = 15
• Column 3: 2 + 7 + 6 = 15

And the diagonals:

• Diagonal 1: 4 + 5 + 6 = 15
• Diagonal 2: 2 + 5 + 8 = 15

Yes! Every row, column, and diagonal adds up to 15. That’s the magic number!

Try it as a mini adventure

Adventure: The Magic Square Quest

Imagine you are a young explorer in ancient China. You find a scroll with a mysterious 3x3 grid. It has numbers but they are out of order. Your task is to place the numbers 1 to 9 in the grid so that every row, column, and diagonal adds up to 15.

Here’s your starting grid:

Step 1: You know that the total must be 15. What number goes in the top right?

Step 2: What number goes in the middle of the grid?

Step 3: What number goes in the bottom left?

Step 4: Complete the rest of the grid and check that all rows, columns, and diagonals add to 15.

Good luck, explorer! You’re solving a puzzle that ancient Chinese people loved.

Common mistakes

• Mistake: Trying to use the same number twice in the grid.
  Why it happens: Confusion about how to place numbers.
  Fix: Remember that you can only use each number from 1 to 9 once.
• Mistake: Not checking that all rows, columns, and diagonals add to the same total.
  Why it happens: Rushing through the puzzle.
  Fix: Double-check each row, column, and diagonal carefully.
• Mistake: Thinking that the magic number is always 15.
  Why it happens: Not understanding how to calculate it.
  Fix: Add all numbers (1 to 9 = 45) and divide by 3 to find the magic total.

Parent and carer guide

Help your child understand magic squares by:

1. Using a simple 3x3 grid and writing numbers 1 to 9 on small cards
2. Letting them place the cards in different positions and checking if rows, columns, and diagonals add to the same number
3. Encouraging them to think about patterns and relationships between numbers
4. Playing the magic square game together – take turns placing numbers and checking the totals
5. Not worrying if they get it wrong – mistakes are part of learning!

What to say: “Try placing the number 5 in the middle – it’s often a good place to start.”

What not to do: Don’t rush them or say they are wrong if they make a mistake. Instead, guide them to find the right answer themselves.

Practice questions

1. What is the magic total for a 3x3 magic square using numbers 1 to 9?
   Answer: 15
2. Complete this magic square:

   8	1	6
   3	5	7
   4	9	2

   
   Answer: This is the Lo Shu square, all rows, columns, and diagonals add to 15.
3. What is the magic total for a 4x4 magic square using numbers 1 to 16?
   Answer: 34
4. Can you place numbers 1 to 9 in a 3x3 grid so that each row adds to 12?
   Answer: No, because the total of numbers 1 to 9 is 45, and 45 ÷ 3 = 15.
5. Which number is in the middle of a 3x3 magic square?
   Answer: 5

FAQ

What is a magic square?

A magic square is a grid of numbers where the sum of each row, column, and diagonal is the same.

Where did magic squares come from?

They were first recorded in ancient China, especially in a story called the Lo Shu, which dates back over 4,000 years.

Why are magic squares important?

They help us understand patterns, addition, and number relationships. They are also fun puzzles and have been used in art, games, and even science.

Can you make magic squares with other numbers?

Yes! You can make magic squares with any set of numbers, as long as the sums work out the same.

Are there magic squares bigger than 3x3?

Yes, there are 4x4, 5x5, and even larger magic squares, but they are more complex to make.