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.
A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has exactly one more vertex than it has edges.
Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.
To make a Magic W you have to put the numbers 1, 2, ...,9, one in each circle, so that each row of three numbers has the same total. Experiment to find as many different magic totals as you can using the interactivity from the Magic W problem.
Prove that if you have a Magic W and you replace each of the numbers by 10 minus that number then the new W is also magic. What will its magic total be?