Magic Matrix
2 out of 3
Find out why these matrices are magic. Can you work out how they were made? Can you make your own Magic Matrix?
Problem
Here is a "magic" matrix:
It doesn't look very magical does it?
This is how you find out the "magic" in the matrix:
Circle any number in the matrix, for example, $5$. Draw a line through all the squares that lie in the same row and column as your selected number:
Repeat for a third time, for example:
Then circle only the remaining number that has no line through it:
Add all the circled numbers together and note your answer.
Try again with a different starting number. What do you notice?
Try the same thing with these two slightly harder matrices:
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This problem was made to celebrate NRICH's tenth birthday - perhaps you can see the connection!
Let's try a different one with larger numbers.
Now make an addition table like this:
You can download a sheet of them here.
Put your numbers in the cells on the outside and add them to make the matrix:
Finally, copy the square without the numbered outside cells:
Now you know how the matrix works, you are ready for the real problem.
Can you work out what numbers were used to make any of the original three matrices?
Getting Started
You could start with the numbers in the top row of the matrix. What two numbers could be added together to make $1$?
Why not write them in, one to the left of the row and one above the number $1$ in the matrix. Now, you could fill in the rest of the numbers which will add to make that top row. Test out your numbers by looking at the second row.
Be prepared to start again if it doesn't work!
Student Solutions
Several of you noticed that the numbers you end up with in the matrix always add up to 10 - to fit in with NRICH's 10th anniversary! A whole group of pupils from Gorseland Primary School worked hard on the first matrix in pairs, discovering how it works, making their own and solving this problem. They found this solution for the first matrix:
George from Greenwich solved the second and third matrices as well. For the second one, he says:
The numbers on the sides are the eight magic numbers, they are 1.7, 0.3, 0.1, 2.6, 0.2, 1.7, 1 and 2.4. The eight numbers add up to 10 and they fit in place:
| 0.2 | 1.7 | 1 | 2.4 | |
| 1.7 | 1.9 | 3.4 | 2.7 | 4.1 |
| 0.3 | 0.5 | 2 | 1.3 | 2.7 |
| 0.1 | 0.3 | 1.8 | 1.1 | 2.5 |
| 2.6 | 2.8 | 4.3 | 3.6 | 5 |
For the third matrix, George says:
What you have to do in the first step is convert all the fractions to a common denominator to make it easier. The common number is 12. This is the solved grid below. As you can see, the eight numbers are 8/12, 6/12, 19/12, 27/12, 5/12, 9/12, 30/12 and 16/12. They add up to 10 as well.
| 6/12 | 19/12 | 27/12 | 5/12 | |
| 8/12 | 14/12 | 27/12 | 35/12 | 13/12 |
| 9/12 | 15/12 | 28/12 | 36/12 | 14/12 |
| 30/12 | 36/12 | 49/12 | 57/12 | 35/12 |
| 16/12 | 22/12 | 35/12 | 43/12 | 21/12 |
You can check that this is the same as the matrix in the problem! Excellent George, thank you for sending in your solutions.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem starts very easily but becomes harder with the second and third matrices. The last part, discovering how to work out how the matrix is made up, is really challenging because there are so few clues.
Possible approach
Key questions
Possible extension
When the numbers that were used to make up all three of the original matrices have been found, learners could make a larger matrix such as one on a $5$ by $5$ grid.
Possible support
Suggest sticking with the first matrix and then trying the one which adds to $100$.