The four overlapping regions could be edges in a graph and the remaining regions vertices. Each vertex and each edge has a unique number assigned to it. This is a problem on vertex magic graphs! A graph is vertex magic if, for each vertex, we get the same magic sum when we add the number at that vertex to the numbers on all the edges joined to that vertex.

What is the grand total of the sums at the 5 vertices? The numbers on the edges are counted twice so add this to the total of the numbers 1 to 8. What does this tell you about the total of the numbers on the edges (that is in the overlapping regions)? This gives you information about the possible values for the magic sum.

Be systematic and you?ll be able to check all the possibilities. What pairs of numbers can you use in the outer circles? What numbers can you then use in the overlapping regions (the edges of the graph).

There are several solutions. How will you know when you have found them all?