Think Outside the Euclidean Universe

You’ve probably all seen the brain-teaser that’s a perennial favorite with uncreative managers the world over. (Why an exercise in creative thinking is really only loved by incredibly boring people is a discussion beyond the scope of this blog post.) The brain-teaser goes like this:

Four dots in a square{.content width=”120” height=”120”}

Connect all four dots: draw three straight lines without lifting your pencil from the paper, and finish where you started. At first it looks impossible, but after your manager gets done chortling they’ll say your problem is that you need to “Think Outside the Box!”, show you the answer, and then go on to make tenous and tedious metaphors about creativity.

And that’s bullshit. An important part of creativity is working within constraints, such as the box pictured above, especially when they’re entirely arbitrary constraints on a meaningless project imposed by management fiat.

To truly solve this puzzle, we only need one straight line and the kind of mental agility that MBAs didn’t learn in a business marketing class. The correct answer looks like this:

Four dots connected by a circle

You may, at this very moment, be thinking I am also bullshitting because that is so not a straight line. Your problem is that you need to Think Outside the Euclidean Universe!

Space grid deformed by a planet

When we look at life, it seems that it takes place in a flat 3d space where two parallel lines will go on being parallel for their entire length. But Einstein’s Theory of Relativity showed our universe is way less boring and predictable than that. Gravity bends spacetime, allowing cool things like gravitational lenses where light (normally travelling in a straight line from our perspective) passes by a massive object like a galaxy or a black hole and bends towards it. We don’t notice it as we sit in go about our lives because we don’t spend a lot of time very near black holes.

So, when you recognize the four dots appear in a non-Euclidean space, the answer is obvious. There’s a massive object warping spacetime between the four points, and because they’re only an inch or so apart it must be a black hole. If you were to represent the geometry of this distortion, the solution would look be even clearer:

Solved from side{.content width=”180” height=”180”}

So there we are: our manager’s problem is trivially solved when precipitously near to a black hole, so you’d better hope that the poor bastard your manager picks to implement this solution isn’t you.

Thanks to Eric Sandalle for the 3d render.