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Regional Division

Stage: 3 Short Challenge Level: Challenge Level:1

the answerSome trial and error will produce a solution like that on the right, where there are $9$ different areas enclosed.

To see that this is indeed the maximum, there is always one central region ($9$ on the diagram), and then the others must be separated from this by one of the sides of one of the rectangles. Each side can only separate one region, and as there are a total of $8$ sides, this means at most $9$ regions in total.

This problem is taken from the UKMT Mathematical Challenges.