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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:
Then circle another number that has not got a line through it, for example, the $1$ in the top right hand corner, and again cross out all squares in the same row and column:
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:
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.
What is the magic total this time?
I will show you how this kind of matrix works. You can then invent one to try on your friends!
First you need to choose your 'magic total'. As you know, I chose $100$ for the matrix above.
I have chosen: $1, 16, 9, 23, 18, 4, 2$ and $27$. [You can check that together they add to $100$.]
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?