Think Outside the Euclidean Universe «

Life: , , ,

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

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 planetWhen 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

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.


  1. That is a great solution to the problem but of course even with this wonderful explanation the manager would still be so pissed off he would not hire anyone who gave him this answer but then who would want to work for him.

    The “him” was used on purpose as it would be a man asking this type of question.

  2. Your solution is really clever. The three line method was simple enough given a 5 minute mull over but your explanation for the one line method really impressed me.

  3. Bah, there are a ton of holes in this problem.

    I mean, he doesn’t give a frame of reference. So instead of bothering with the warping of space time, you can just assume the whole thing was projected onto the surface of a sphere, with the dots aligned at the equator, and around you go. I mean, he didn’t specify Euclidean space, right? And it’s far more natural to assume that it’s on the surface of a sphere, cause we live on one, right?

    And while he said you can’t lift the pencil from the paper, he didn’t say a thing about lowering the paper from the pencil. Hold the pencil in mid-air, and move the paper about. Now draw the lines any which way you want.

    Alternately, roll the paper into a tube, and you should be able to do it in two lines using a kind of modified bow-tie shape. The dots should be on exact opposite sides of the tube. Start half the square’s diameter left of the dots and aligned with the center of the square horizontally. Draw downward at a 45 degree angle, crossing the lower-left and upper-right dots until you reach the symmetric position on the opposite side. Now turn 90 degrees clockwise and go back to the left, nabbing the lower-right and upper-left dots, and you’ll end where you began.

    If you roll the tube REALLY tightly, you can make it so the top and bottom row of dots touch. Then you can get two at a time from the same point. This makes 3 lines seem excessive. You brought scissors to the interview, right? Try not to threaten the manager with them.

    I also find this is a really handy time to have a blunt pencil with a 3-inch wide piece of lead. While such a thing probably doesn’t exist, I’m sure it could be built. Then you don’t need any lines at all, just a … um … blob. One of them. And you ended where you started up, too.

    Finally, you can also do it with a single, cyclic line if you move the whole mess to a torus. It’ll even work on an arbitrarily large torus (though you might have to cycle around many, many times), so it can take a while to finish drawing it, and the angle requires some precision. It’s also tricky to roll a piece of paper into a torus so you might want to pass on this one.

    Finally, there are many, many variations on the “classic” solution if you don’t bother to start at one of the dots. The set of such solutions is an entire class of triangles — symmetric triangles on both axes are possible.

    OK, I’m done.

  4. I really enjoyed your post Adam, but for the sphere one it doesn’t quite work that way. If you have it projected onto a sphere, then you are technically drawing a circle as you draw around the surface of the sphere. Pete is talking about actually warping space such that those four points are actually colinear.

    On a similar note, you could actually fold the paper such that all four points are co’point’er (I tried coining that term in Jr. High, but my geometry teacher didn’t particularly like that. Then any single line passing through 1 passes through all four points.

Leave a Reply

Your email address will not be published.