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 themĀ 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?