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Magic Matrix

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

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 in the final matrix.

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 two copies 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. This sheet has the illustrations from the problem for this.

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 $100$.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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