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# W Mates

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Age 16 to 18

Challenge Level

Here is one of the magic labellings of the W shape from Magic W problem with a magic total of $14$ on each of the four lines making the W shape.

Prove that for every labelling with a magic total $T$ there is a corresponding labelling with a magic total $30-T$.

Find the values of $T$ for which magic labellings exist and show that there are a total of $12$ magic labellings altogether.

**Note** that two labellings are considered to be the same if they are reflections of each other or if the two numbers at the ends of the outside legs are interchanged.