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Multiplication Squares

Age 7 to 11
Challenge Level

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.

Scan of Sabden Primary School proof.

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?