Given the nets of 4 cubes with the faces coloured in 4 colours,
build a tower so that on each vertical wall no colour is repeated,
that is all 4 colours appear.
Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.
Label this plum tree graph to make it totally magic!
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
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.