Solving the Flat Cube |

This looks like a Rubik’s Cube, but it’s not:

It seems that an open-cab truck filled to the brim with Rubik’s Cubes hit a bump in the road, causing some of the cubes to spill out of the truck bed and onto the roadway. When, mere moments later, those cubes got run over by a magic school bus, something unexpected happened: the plastic pieces of each flattened cube stuck together to form a puzzle of a different kind—one that only looks three-dimensional.

I call it the Flat Cube. Its pieces are rhombuses, also called lozenges, and wherever three lozenges come together to form a little hexagon, you can twist the three lozenges as a unit around the center of that hexagon by any multiple of 60 degrees, like this:

The challenge is to restore a scrambled Flat Cube to its unscrambled flat state using as few twists as possible. And that leads to the question: if the Devil gets to scramble the puzzle as maliciously as possible, how many twists will God need to unscramble it?

Here we are to assume that God, though omnipotent, is scrupulous about following the puzzle’s rules. We’ll also grant both God and the Devil unlimited computational resources; God can always find the most direct solution to any scramble the Devil devises, while the Devil wants to make that solution involve as many moves as possible.

In the case of Rubik’s Cube, this magic number—the number of moves God needs against the Devil’s most devilish scramble—has been dubbed God’s number . God’s number is known to be 20 if a 180-degree twist counts as one move and to be 26 if a 180-degree twist counts as two moves.

In the case of the Flat Cube, I don’t know God’s number. But whether you count twists in 60-degree increments or allow a twist by any angle to count as one move, I can show you that God’s number for the Flat Cube is at least 27.

There are two wonderfully visual ways to see why this is so; one works by adding a dimension, while the other works by taking a dimension away.

ADDING A DIMENSION

When you look closely at the Flat Cube, you may begin to experience dizziness. Lean into this feeling. It’s trying to help you.

Your eyes are telling you that the Flat Cube isn’t flat, and in a certain sense, your eyes are right. Even though the Flat Cube is flat in the real world, there’s nothing to stop you from seeing a Flat-Cube-that-isn’t-flat inside your head. That is, you can pretend those lozenges are squares living in three-dimensional space, viewed from an oblique angle, fitting together to form a sort of stepped surface. And that imaginary surface will show us where the number 27 comes from.

Take another look at the picture of the 60-degree clockwise twist. Let’s ignore the colors, which may distract you from seeing the image three-dimensionally, and instead use shading to make the illusion of three-dimensionality stronger.

Each way of tiling the big outer hexagon by lozenges—that is, each way of filling the hexagon with lozenges so that there are no gaps or overlaps—corresponds to a way of piling up cubes.

Don’t see it yet? Try imagining the three lozenges that form the little hexagon at the right, highlighted below, as the visible faces of a single 1-by-1-by-1 cube—a cube that isn’t there in the picture at the left. Let your brain interpret those lozenges as squares viewed obliquely. Since the faces point in different directions in three-dimensional space, light strikes them differently, accounting for the different shading.

Viewed in this light (quite literally), the hexagon twist becomes the operation of adding or removing a cube.

(Don’t confuse shading with the colors in the original puzzle: shading indicates a lozenge’s orientation, while each lozenge’s color travels with it as it moves. And anyway, we’re ignoring colors for now.)

There’s a slight problem with the way I’ve described the process of turning a tiling into a piling. To make the process work for every tiling of our big hexagon, you have to visualize many 1-by-1-by-1 cubes resting in a peculiar sort of tray. Picture an empty 3-by-3-by-3 box resting on one corner with its three top faces removed. The three remaining faces form our tray. But before we put any cubes in it, etch grooves into the three faces of the tray, dividing each face into nine 1-by-1 squares. Now the two lozenge tilings shown below correspond to an empty tray and a fully loaded tray. The empty tray contains no cubes, and the lines you see are the grooves in the tray; the fully loaded tray contains 3 × 3 × 3= 27 cubes, some hidden beneath others, and the lines you see are the edges of visible cubes.

Now you can see why it takes at least 27 moves to turn the uncolored lozenge tiling shown at the left into the one shown at the right. We start with an empty tray and end with a fully loaded one, and each move adds or removes just one cube. So, ignoring colors, we’ve found a state that’s at least 27 moves away from the solved state.

So far our argument has ignored colors. What...