5. When she drew two intersecting circles, as shown, Tatiana divided the space inside the circles into three regions. When drawing two intersecting squares, what is the largest number of regions inside one or both of the squares that Tatiana could create?
A 4
B 6
С 7
D 8
E 9
Solution
E

Suppose one square has been drawn. This creates one region. Now think about what happens when you draw the second square starting at a point on one side of the first square. One extra region is created each time a side of the second square intersects the first square. Therefore, if there are $\mathrm{k}$ points of intersection, there will be $\mathrm{k}+1$ regions when you have finished drawing the second square. However, each side of the second square can intersect at most two sides of the first square. So there can be at most 8 intersection points. Therefore there can be at most 9 regions. The diagram below shows what such an arrangement would look like with 9 regions.