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Magic Matrix

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Magic Matrix

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 them 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?

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

You could start by doing the first magic matrix on the board together. It is not very surprising when all get the same answer! If anyone gets $11$, for example, instead of $10$, you can check quickly that they are doing it correctly. Then let the class try another couple for themselves, perhaps suggesting that they make sure they choose different numbers from their neighbours. Those confident with decimals and fractions could try then the next two matrices. This sheet gives six copies of the first matrix and three copies for each of the second and third ones.

After that you could introduce the next matrix with the magic total of $100$, and then continue to show the class how it is built up. For this purpose you can use this sheet with the illustrations from the problem.

After this, the group could be challenged to work in pairs making their own magic matrices. It is advisable to limit these to magic totals between $12$ and $200$. This sheet has six copies of the frame for the matrix. However, squared paper can easily be used instead.

The final challenge, to discover what numbers were used to make any of the original three matrices is hard, and could, perhaps, be reserved for the most able learners.

Key questions

Why not try one of the other matrices?
How can the same number result whichever choices are made?
What number are you choosing for your own matrix?
How are you going to divide it into eight numbers?
You could start with the numbers in the top row of the matrix.
What two numbers could be added together to make $1$?

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 $100$.