Example 1
A 3 by 3 magic square uses the numbers 1 to 9. What does each row add up to?
- 1 + 2 + … + 9 = 45.
- 3 rows share the 45 equally: 45 ÷ 3 = 15.
Answer: 15
🦘 Kangaroo Grade 3 · session 4 of 24 · 60 minutes · ⭐ Plus
Magic squares and number grids
Fill a square so every row, column and diagonal adds to the same number, put numbers on a triangle so every side matches, and build number pyramids. The key idea: add everything up first.
A complete lesson · 60 minutes
Teach it, see it, practise it, all on this page.
Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.
- Warm-up4 min
- Magic sum8 min
- Missing numbers9 min
- Triangles7 min
- Pyramids and grids7 min
- Activity 17 min
- Activity 27 min
- Activity 36 min
- Quiz4 min
- Set the project1 min
By the end you can
- Find the magic sum by adding all the numbers first
- Fill the missing numbers of a magic square
- Solve number triangles where corners count twice
- Solve number pyramids and 1-2-3 grids
Before you start
- Adding numbers to 50
- Session 2 (missing numbers) helps
Have ready
- Paper and pencil
- Number cards 1 to 9
Grade 3 · ages 8 to 9 · contest level 3–4
Warm-up · think first
Can you split the numbers 1, 2, 3, 4, 5 and 6 into two groups with the same sum?
Show the answer
No. 1 + 2 + 3 + 4 + 5 + 6 = 21, which is odd, so it cannot be cut into two equal whole-number halves.
Teacher tip: Let them try for a minute. When they get stuck, ask them to add all the numbers. This 'total first' idea runs through the whole lesson.
Part 1 of 4 · learn, see, practise
The magic sum
In a magic square every row, every column and both diagonals add up to the same number, the magic sum.
Add every number in the square and share the total between the rows. For 1 to 9 the total is 45, and 3 rows share it, so the magic sum is 15. In a 3 by 3 magic square the middle number is always one third of the magic sum.
Example 2
A magic square's top row is 2, 7, 6 and its middle row is 9, ☐, 1. What is ☐?
- The magic sum is 2 + 7 + 6 = 15.
- 9 + ☐ + 1 = 15, so ☐ = 5.
Answer: 5
Magic square builder
Type a number in each box. The sums of every row, column and diagonal update as you go.
- Make a magic square with the numbers 1 to 9 (every line adds to 15).
- Make one with 2 in the top-left corner.
- Use the numbers 2 to 10 so every line adds to 18.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Core
Stretch 🔥
Stretch 🔥
Answers (for the teacher)
- 45
- 1
- 18
- 5
- 21
- 34
⭐ Part of Pure Faith Plus
The rest of this lesson is in Plus
Already have Plus? Log in and the whole lesson opens.
You have seen the plan, the warm-up and part 1. With Plus you also get:
- Part 2: Filling in the missing numbers, with practice at three levels
- Part 3: Number triangles, with practice at three levels
- Part 4: Pyramids and 1-2-3 grids, with practice at three levels
- Class activity 1: Finish the magic square
- Class activity 2: Triangle puzzles
- Class activity 3: Pyramid builder
- A 8-question quiz
- After-class project: My magic square poster
- Homework and a printable practice sheet with answers
- Teach mode and the minute-by-minute teacher's plan
The first 2 lessons of every course, every game and all maths practice stay free.
That is ₹208 a month for every lesson of every course. Prefer a teacher? Ask about live classes.
FAQ
Magic squares and number grids: questions
How do you solve a 3 by 3 magic square?
Add all the numbers and divide by 3 to get the magic sum, put the middle number (a third of the sum) in the centre, then fill lines that have one box left.
What is the magic number of a magic square with 1 to 9?
15. The numbers 1 to 9 add to 45 and the three rows share it equally.
Are magic squares in Math Kangaroo?
Yes, along with number triangles, pyramids and grids where each row has the numbers once.
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