Houses and Wells

Can you connect each of the houses with each of the wells using continuous curves, so that no two curves intersect each other?

No, it is impossible. Here is one convincing, albeit a bit informal proof.

Imagine the problem is solvable. Then you can connect House 1 to Well 1, then Well 1 to House 2, then House 2 to Well 2, then Well 2 to House 3, then House 3 to Well 3, and finally Well 3 to House 1. Thus you will create one loop with 6 points on it, in alternating types (houses and wells). Now you must connect Point 1 with Point 4, Point 2 with Point 5, Point 3 with Point 6, such that the new three curves do not intersect each other. However, you can easily see that you can draw more than one such curve neither on the inside, nor the outside of the loop. Therefore the task is indeed impossible.

More rigorous, mathematical proof can be made using Euler’s formula for planar graphs.

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