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Magic Matrix

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2
Here is a "magic" matrix:

four by four grid

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:

circled five

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:

one circled

Repeat for a third time, for example:

third number circled

Then circle only the remaining number that has no line through it:

last number circled

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:

matrix of 16 numbers matrix with fractions

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:

addition square

You can download a sheet of themĀ here.
Put your numbers in the cells on the outside and add them to make the matrix:

completed addition square

Finally, copy the square without the numbered outside cells:

completed matrix

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?