Copyright © University of Cambridge. All rights reserved.

## 'W Mates' printed from http://nrich.maths.org/

Here is one of the magic labellings of the W shape from Magic W problem May 2003 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.