Topics: Addition, Subtraction, Logic
Materials: Pencil and paper,
Recommended Grades: 2, 3, 4
Put the numbers 1-9 into a square so that every column, row, and diagonal adds to 15.
Why We Love Magic Squares
This is a truly historical problem, with roots in cultures around the world. The arithmetic looks fairly simple at first, but the problem requires tenacity to solve. Scaffolded correctly, this is a great problem to encourage and reward perseverance.
The Launch (5-10 minutes)
Describe the problem: how can you put the numbers 1-9 into a square so that every column, row, and diagonal adds to 15? To start, the teacher should build a wrong solution with help from the students.
Teacher: Where could I start? Maybe I’ll try putting a 3 in the middle square. and a 7 in one of the side squares. What number should I put in on the last spot in the row?
Student 1: It has to be a 5, since 7 + 3 = 10, and you need 5 more to get to 15.
Teacher: Ok. What other numbers should I put in? What if an 8 went in the upper right corner? Then what?
Student 2: Then you’d need to put a 2 in the bottom right, since 8 + 5= 13, and you need 2 more to get to 15.
Student 3: And there needs to be a 4 in the bottom left, since 8 + 3 =11, and you need 4 more to get to 15.
Teacher: Looking good. So what else could we put in?
Student 4: The 9 has to go in the bottom, because 4 + 2 + 9 = 15.
Student 5: And then a 3 has to go in the middle top, and a 4 in the top left.
Teacher: [Completing the square] Uh-oh. Anyone see any problems?
Student 6: We used the 3 twice. And the 4 twice.
Student 7: And the diagonals don’t both add up to 15.
Teacher: So this is a pretty good start. We’ve got all the rows and columns adding up to 15, and one of the diagonals. But we didn’t use the numbers from 1-9 exactly once. We used 3 and 4 twice and left out 1 and 6. So here’s your job: Use the numbers from 1-9 exactly once, and try to build a magic square little bit by little bit:
Challenge 1: Make a square with all the rows summing to 15
Challenge 2: Make a square with all the rows and columns summing to 15
Challenge 3: Make a magic square, meaning all the rows and columns and diagonals sum to 15.
There are also larger squares. If anyone finishes these three challenges, I’ll have some extra challenges for you.
The Work (20-30 minutes)
Students can work individually or in pairs to work through the challenges and try to make a magic square. If students are becoming too frustrated, give them the worksheet with the partially filled-in 3 by 3 squares. If students finish and need a new challenge, give them the worksheet with the partially filled-in 4 by 4 square.
The Wrap Up
- Ask students to share their solutions. Some questions for the wrap up conversation:
- How many different solutions are there to the 3 by 3 magic square?
- How are the different solutions related?
- Is it possible to make the numbers add up to a different number than 15?