A surprising solution to the (in)famous "Cross the Network" puzzle

Originally published at: https://boingboing.net/2020/07/06/a-surprising-solution-to-the.html

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Turning a 2 dimensional puzzle into a 3 dimensional puzzle is definitely a trick.


Is it a “trick?”

I defy you to present a single argument for why it isn’t.


“Is it a “trick?” That’s up for you to decide.”

I’ve decided; it is most definitely a trick.


Hey, if adding arbitrary dimensions until the math works is good enough for string theory, then it’s good enough for me.


Yes, if you alter the puzzle into a shape in which it can be solved, then you can solve it. QED.

Kind of a dumb puzzle, though, IMO.


The torus is two dimensional. You may often embed it in three, but you needn’t, and you may embed a plane in three just the same. Praise be to Gauss, and to actual mathematics. This one warrants the eye stabbing xkcd.


reminds me of this in terms of solutions to problems that don’t actually solve anything https://youtu.be/sKqt6e7EcCs


I mean, that’s like saying “if Cory leaves Austin at sunrise, how can they drive to Boston before the sun sets?” And saying the answer is to use a wormhole or a time machine.


“Everything is a 2-dimensional Torus if you are brave enough!” -Abraham Lincoln


Very reminiscent of this excellent mug. If you haven’t seen it, I challenge you: what simply stated problem is solvable on this mug?


Now if someone had just wrapped 18th century Königsberg around a torus…


Just to make it clear: This would traditionally be considered unsolvable, because there are 3 areas (4 counting the outside) with an odd number of edges, right ?

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I’m assuming the “right?” part is requesting confirmation rather than just being a rhetorical flourish.

Yes, you are correct.

In graph theory problems like this, the region outside the “brick wall” drawing is no less or more important than any region inside it, so mathematicians would usually say there are 4 regions with an odd number of edges, rather than saying there are 3 such regions excluding the outside.

On the other hand, the curved path you are trying to draw only has 2 ends, so as soon as you identify more than 2 regions with an odd number of edges you’ve proved the task is impossible, so you can stop counting.


Truly a Kirk-worthy solution, but ultimately it spoils it for everyone when the puzzle is reworded to specify the topological nature of the playing field. Another victory for the lawyers.

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You want to connect gas, water and electricity to all houses, and the pipes not allowed to cross? I think that is doable…

Edit: I got it! was much more challenging then I expected! Thanks!


Well, I mostly wanted to be sure I understood the rules correctly.

I typed “infamous cross the network puzzle” in my favorite search engine, and you wouldn’t believe the stuff that came up, but none of it was related to this puzzle. The wikipedia entry did show up later, but only after searching “famous cross the network puzzle”. Figures.

It’s even easier if you embed it in 73-dimensional space.

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