Division


Problem:

The three lines in the figure divide the circle into 7 regions. Can you place the numbers 1-7 in these regions, such that for each of the lines, the sums of the numbers on either side are all the same?


Solution:

Yes. Because

$$1 + 2 + 3 + 4 + 5 + 6 + 7 = 28,\)

the sum of the numbers on each side of a line must be \(28 \div 2 = 14.\) However, it must also be shown that such an arrangement is possible. One solution is pictured below. There may be others, I'm not sure.

Going further:

If we take these 7 regions and draw their graph, where vertices are placed within each region, and an edge is drawn between vertices that are across a line from one another, then we can ask the same question about graphs, which makes this problem much more general. That is, given some graph with n vertices, can we label each vertex with 1-n, such that every vertex is labelled uniquely, and the sum of every pair of adjacent vertices is equal? Another interesting question: If we have some number of lines in a circle, creating some number of regions, as in the original question, can we always determine whether a labelling is possible? What if we had intersecting circles. Could be place numbers 1-n where the sums across edges are equal? What about on faces of polyhedra?