Multiplication squares
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
Problem
Multiplication Squares printable sheet
In the $2 \times 2$ multiplication square below, the boxes at the end of each row and the foot of each column give the result of multiplying the two numbers in that row or column.
| 7 | 5 | 35 |
| 3 | 4 | 12 |
| 21 | 20 |
The $3 \times3$ multiplication square below works in the same way. The boxes at the end of each row and the foot of each column give the result of multiplying the three numbers in that row or column.
| 15 | |||
| 108 | |||
| 224 | |||
| 144 | 8 | 315 |
The numbers $1 - 9$ may be used once and once only.
Can you work out the arrangement of the digits in the square so that the given products are correct?
Getting Started
Try placing the numbers $1$, $5$ and $7$ first.
Which rows/columns have an odd product?
Which are not divisible by $3$?
Student Solutions
This solution came from Adam from Moorfield Junior School.
This is how I arranged the numbers from one to nine in a multiplication square once and once only:
| 1st row: 3x1x5=15 | 2nd row: 6x2x9=108 | 3rd row: 8x4x7=224 |
| 1st column: 3x6x8=144 | 2nd column: 1x2x4=8 | 3rd column: 5x9x7=315 |
He is right but doesn't tell us how he worked it out. This solution is from James, Terry, John and Indiya from Sabden Primary School, Clitheroe, Lancashire.
This solution is from Ailsa from Romsey Abbey Primary School, Romsey, Hampshire. She says:
I managed to crack the Multiplication Square, below, here is how I did it:
| 3 | 1 | 5 | 15 |
| 6 | 2 | 9 | 108 |
| 8 | 4 | 7 | 224 |
| 144 | 8 | 315 |
First I worked out that the only way to make 15 was to use the numbers 1, 3 and 5. I then discovered that the only way to make 8 was to use the numbers 1, 2 and 4. Therefore the 1 had to go in the top row and second column. I then worked out 5 had to go in the top row and third column because 315 is a multiple of 5 and 144 is not, so that meant that the 3 had to go in the top row in the first column.
Then I divided 108 by 2, the answer was 54, and then made 54 using the numbers I had left, 6, 7, 8 and 9, the answer was 9 x 6, so I knew that 9 and 6 were the 2 other numbers that went in the second row. 6 is not a multiple of 315 so 6 went in the first column and 9 went in the third column.
So now the only 2 numbers left were 7 and 8 so I worked the answer out by discovering how to make 315. I multiplied 5 by 9 to give me the answer 45 and then tried multiplying it by both 7 and 8. 7 was the correct number so 7 went in the third column and 8 went in the first.
This is a really clear description of your method, well done!
Kaden from Coleridge Community College, Cambridge also sent a correct solution. Well done!
We have had some other solutions, from a different school, that had got the 6 and the 9 swapped over. Will it work then?
Teachers' Resources
Why do this problem?
This problem requires learners to find common factors and offers a context in which they can explain their reasoning.
Possible approach
You could begin by having a partially completed multiplication square (or squares) on the board, with just a few gaps, for example:
| 7 | 5 | 35 |
| 3 | 12 | |
| 20 |
| 8 | 24 | |
| 12 | ||
| 18 |
Ask the class to describe what they see. Can they fill in the spaces? Invite some children to explain how they know what goes in each empty cell and encourage use of appropriate vocabulary.
Key questions
Possible extension
Lee Gardiner, the head of a school in Cornwall, liked this activity and extended it by creating some extra challenges in a Word document or extra challenges in a PDF document. Thank you, Lee.
Possible support
Calculators might be useful.