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 if we put them back into the picture? Imposing the requirement that all the colors end up in the right places certainly can’t make the puzzle easier. So the Devil can force God to use at least 27 moves for the colored Flat Cube as well.
For Grant Sanderson’s animated version of the cube-tray argument, see the 3Blue1Brown video listed in the References.
SUBTRACTING A DIMENSION
For our second argument we’ll once again ignore the colors, so that we can’t keep track of individual lozenges when we perform a twist. From this color-blind point of view, a hexagon twist either leaves the uncolored tiling unchanged or else moves the three lozenges inside the hexagon to the other side of that little hexagon. So whenever a move changes the tiling, we may as well picture the move as a 180-degree twist.
Let’s look at these lozenge tilings in a new way by focusing on all the lozenges whose long axes are horizontal and pretending that the other lozenges aren’t there. Now a 180-degree twist amounts to nothing more than sliding one of those horizontal lozenges upward or downward by a single step, like a bead on an abacus wire:
Strictly speaking, when you do a 180-degree twist, that horizontal lozenge doesn’t simply slide; it rotates, and it arrives at its new location flipped upside down. But since the lozenge is featureless, we can’t tell that it’s flipped, so we can pretend that it merely slid.
Now consider the pair of lozenge tilings shown below.
If we replace the horizontal lozenges by beads, those tilings become these bead configurations.
Here I’ve assigned numbers to the positions that horizontal lozenges can occupy.
In the configuration at the left, the beads are at the bottom; in the one at the right, they’re at the top. Since a bead can’t jump from one wire to another, and since beads sharing a wire can’t pass through each other, each bead at a position marked 1 must move to the position marked 4 above it on the same wire, each bead at a position marked 2 must move to the position marked 5 above it, and the bead that starts at a position marked 3 must move to the position marked 6 above it. In short, each of the 9 beads must shift upward by 3 steps. But each twist shifts just one bead, and shifts it by just one step, so at least 9 × 3 = 27 twists are required.
In fact, we can push the bead-sliding argument or the cube-adding argument a little farther to prove that the worst-case number of twists for the uncolored puzzle is exactly 27. Or, equivalently, just solve the puzzle in 27 moves! As a practical matter, if you do this, you’ll probably want to use one-dimensional thinking from the second proof and just think about getting the horizontal lozenges to move upward; if you try to think three-dimensionally as in the first proof, there’s a risk you’ll undergo Necker reversal midway through and start taking away cubes instead of adding them!
BACK TO SCHOOL
There’s a serious problem with my suggestions for how to solve the Flat Cube: until recently, the Flat Cube didn’t actually exist—at least, not in the way that boring things like tables and chairs do.
When I learned that the Radcliffe Institute for Advanced Study had awarded me a Fellowship for the 2026–2027 academic year to pursue my ongoing work on tilings, I knew I wanted to furnish my new office with a Flat Cube. I’m the only mathematician in a class of fifty or so Radcliffe Fellows, and when one of my fellow Fellows stops by my office and asks “What are you working on?” I’ll want to have something concrete and fun to hand over.
Part of what drew me into studying tilings back in the late 1980s was an article by the mathematician William Thurston that emphasized the intuitive bridge between rotating hexagons and adding or subtracting cubes. Soon my imaginarium1 of mathematical objects, properties, and actions was populated by tilings morphing from one state to another. It frustrated me that in the real world, tilings don’t morph quite as easily as they do in the mind. For instance, take a fresh look at our hexagon twist; if you try to rotate the three lozenges in the hexagon, the six surrounding lozenges, marked 1 through 6, will get in the way.
Still, it seemed unfair that the laws of mere physics would rudely stand in the way of some beautiful mathematics. If those laws could not be overturned, then they would need to be circumvented. But how?
I took my problem to master puzzle designer Oskar van Deventer, asking: Can you make an object whose pieces really move this way? He came back with not one design, but three! His third design was my favorite, and it got even better when another puzzle designer, Dmitry Andreev (aka Pluton), tinkered with it. Here’s the result of that three-way collaboration, demonstrated by Oskar:
Oskar calls the Flat Cube the Propp Twist. While I’m happy to have my name associated with it, I also like the idea of a name for the puzzle that links it with the Rubik’s Cube.
Could there be a fourth version? Oskar doesn’t plan to make one, but I keep hoping for one that’s as smooth to operate as a modern speedcube. Perhaps one of you will come up with version four.
So what is God’s number for the Flat Cube? I don’t know. And there’s an even broader version of the question: what if we let the Devil choose both the initial tiling and the target tiling?2 Asking questions like these is really a way of asking about the cartography of the land of tilings, where each tiling is a town and each move is a road.3
Can the things we learn about the landscape of colored lozenge tilings help us understand landscapes of more complicated kinds of tilings, like the tiling shown below? That’s the sort of thing I’ll be thinking about at Radcliffe.4
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ENDNOTES
#1. I didn’t invent the word “imaginarium”, but I haven’t seen it in descriptions of the experience of doing mathematics; I’m hoping more mathematicians start to use it to describe the mathematical world we build inside our heads. I’ll have more to say about this in future essays.
#2: Mathematician Nicolau Saldanha has proved that every tiling that uses 9 red lozenges, 9 green lozenges, and 9 yellow lozenges to fill a hexagon of side length 3 can be obtained from every other such tiling by a sequence of hexagon rotations, and his argument should yield an upper bound on God’s number for colored lozenge tilings. I’m undecided as to whether every rotation should count as one move or whether rotations by 120 or 180 degrees should count as two or three moves respectively, so there are two versions of “God’s number” here.
#3. I gave a talk on this theme, entitled Spaces of tilings, in honor of Nicolau’s birthday a couple of years ago; I gave an updated version of this talk a year later, entitled Bringing tiling theory to new heights and vice versa, at a birthday conference in honor of Rick Kenyon.
#4. For some background on the proposed work, see my Radcliffe research proposal.
REFERENCES
Oskar van Deventer, Propp Tiles, Propp Lozenge, Propp Turner, and Propp Twist (videos).
God’s Number is 20 (website).
Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge, The Diameter of the Rubik’s Cube Group Is Twenty, SIAM Journal on Discrete Mathematics, vol. 27, no. 2 (2013).
Grant Sanderson (aka 3Blue1Brown), Solving problems by adding a dimension.
William Thurston, Conway’s Tiling Groups, American Mathematical Monthly 97 (1990), no. 8, pp. 757–773.