The Three Utilities Problem

https://commons.wikimedia.org/wiki/File:3_utilities_problem_blank.svg
Image: Wikimedia Commons

Each of three houses must receive water, gas, and electricity. Is it possible to arrange the connections so that no lines cross?

No, it ain’t. Remove one house and draw connections to the other two:

https://commons.wikimedia.org/wiki/File:3_utilities_problem_proof.svg
Image: Wikimedia Commons

This divides the plane into three regions, here colored red, yellow, and blue. Placing the third house into any of these regions denies it access to the correspondingly colored utility. So the task is impossible.

Pleasingly, the task can be accomplished on a Möbius strip:

https://commons.wikimedia.org/wiki/File:3_utilities_problem_moebius.svg
Image: Wikimedia Commons

And a torus can accommodate four houses and four utilities:

https://commons.wikimedia.org/wiki/File:4_utilities_problem_torus.svg
Image: Wikimedia Commons

(By Wikimedia user CMG Lee.)

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