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:

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:

And a torus can accommodate up to four houses and four utilities:

(By Wikimedia user CMG Lee.)

08/31/2025 UPDATE: Reader Guy Bolton King points out that Mathsgear sells a mug embossed with the puzzle. The joke here is that this makes the puzzle solvable — like the torus, the mug is of topological genus 1, “a blob with 1 hole in it,” so it admits the same solution.

And reader Shane Speck writes, “My sneaky solution … has always hinged on the fact that in reality, houses don’t all have separate pipes, and popping on a shared water pipe instantly reduces the problem to the status of incredibly trivial”:

“The problem doesn’t, after all, say you can’t do that… :)”

(Thanks, Guy and Shane.)