Pocket Cube Remix

Today we'll look at some reconfigurations of the 2x2x2 Rubik's cube that allow for different explorations and lower the floor.

Setup

For this exploration, I purchased this set of \(6\) 2x2x2 Rubik's cubes for $15. That set will definitely work, but I think every 2x2x2 speed cube I've handled has had a similar mechanism in the middle and way of attaching the face panels. The important thing is that the faces aren't glued in place. I began by removing the face panels like so:

Once the face panels are off, you can reattach them however you like. You can prepare interesting cubes in advance by removing and reattaching the faces and have students interact with them as though the faces are glued on. We'll look at a few reconfigurations with some interesting properties.

Reconfigurations

The colors on this cube set are a little nonstandard, since they have pink instead of red, so I'm going to describe things in terms of red (R), blue (B), green (G), yellow (Y), orange (O), and white (W).

We'll use the standard convention of naming the relative faces up (U), down (D), left (L), right (R), front (F), and back (B) and notate rotating face X \(90^\circ\) clockwise by X and rotating face X \(90^\circ\) counterclockwise by X'. For example, the maneuver RUR'U'L means rotate the right face clockwise, top face clockwise, right face counterclockwise, top face counterclockwise, and left face clockwise, in that order, all by \(90^\circ\).

Even though our rotation letters and color letters overlap, it should hopefully be clear what we're talking about from context.

Four colors, solid corners

This cube requires four different colors, six face panels of each:

Since this isn't a traditional solve-the-faces cube, it's debatable what the solved state is. Striped configurations like the one above are interesting. One thing to note is that there is an order to the colors, so there's not a canonical "striped" solution. In the case above, it's red, yellow, blue, green (RYBG) reading clockwise. A nice challenge is to make it so the order is instead RYGB, RGBY, RGYB, RBGY, or RBYG. Can you come up with a framework for getting from one to the other?

By rotating a face with two colors on it, you can interchange those stripes, and you can swap pairs of stripes until they're in the desired order. It's also the case that flipping the cube upside down reverses the order, so getting from an order to its reverse is even simpler.

Another way to think about it is more direct but less intuitive. By rotating the top layer \(180^\circ\) relative to the bottom, you get an arrangement where opposite corners are the same color:

It turns out that each of the six faces has a distinct color order from RYBG, RYGB, RGBY, RGYB, RBGY, RBYG, an interesting combinatorial property. If we choose the order we want (e.g., RGYB) and put the corresponding face face-up, we just have to rotate this new top layer \(180^\circ\) to get columns in the desired order.

Another interesting pattern to try to create is checkerboards. Below we have an RB checker on the front face and GY checker on the back face:

How do we get to this configuration from stripes? If we can get the strips in an order so that the colors we wish to see checkered together are neighboring, we can rotate one of the pairs \(90^\circ\) relative to the other:

Holding the cube so that the front face is two horizontal stripes, apply the rotation sequence RUR'U'. This will result in the checkerboard pattern, with the checkered layers being F and B. Holding a cube with checkered layers F and B, applying this same sequence will turn it back into stripes, with the front pair of stripes horizontal and the back pair vertical.

There is a relatively small number of distinguishable states compared to a standard cube puzzle, so it is easier to accident upon desired or nearly-desired states, making this a nice beginner puzzle.

Two colors, solid corners

I also played around with an easier version of the above configuration that used only two colors:

This one is too easy to be interesting, with only seven visually distinguishable states. Here are the other six:

I pretty quickly accidented into any state I wanted, but it might still be fun for very young learners to try and make the seven different configurations.

One color for a pair of opposite faces, a second color for the rest

This cube is interesting because every corner is identical, featuring two face panels of one color and one face panel of the other. The challenge is then in getting each piece rotated to have the correct orientation. Below is a cube with top and bottom faces in blue and the remaining four in orange:

While this cube is perhaps more mathematically interesting, the next cube is similar and more interesting from an instruction standpoint, so we'll push the discussion onto it.

Two colors for a pair of opposite faces, a third color for the rest

Below we have a cube in white, with top face colored green and the bottom face colored red:

As with the previous cube, I like this one because it solves like the standard 2x2x2 cube but with less complexity in each phase. There are two steps to solving this. First, pick one of the two opposite face colors and make a face with only that color. Here we've solved the red face:

Students can often figure out this step on their own with a standard cube, and this version doesn't even require making sure the edges are oriented correctly, making it easier. The complicated step is slotting the fourth square of the correct color, which is worth working out on one's own.

The next step is to solve the remaining of the opposite faces -- in our case, green. In order to do this, since every green piece can be thought of as occupying the correct position but with a possibly incorrect orientation, we only need to orient these green pieces. The standard move from solving a 2x2x2 cube works well here. We flip the cube upside down so the red face is down. If there is a single green face up, rotate so it's in the UFL corner. Otherwise, rotate so a white square is up in the UFL corner. The move sequence to do is RUR'URUUR'UU. Keeping red down, repeat this step, rotating the cube based on the green squares that are up until all four green squares are up.

Above we lucked out and had a single green up, but you can also have zero or two up if not four. It's a structural impossibility to have three greens up if all reds are down. If you note that our RUR'URUUR'UU maneuver rotates in place the three U corners not in the UFL position, you can be more deliberate about which non-green face you move into the UFL position before applying it, cutting down your solving time. Good to explore, but not necessary.

While less mathematically pleasing, I like the third color in the mix because it helps cut down on the choice paralysis of solving the first layer and gives nice visual cues for distinguishing the bottom and top layers of the cube, which is why I think I prefer this one for learning to solve cubes.

One color for one face, one for the rest

This is an even more watered-down version of the other two. Below we have an orange cube with only the top face in white:

It's not super interesting other than for learning to solve the first layer without the visual clutter of the second layer, but it might be nice for very young learners.

Three colors, one color for each pair of opposite faces

The cube below has pairs of opposite faces colored red, white, and blue:

It solves pretty much like the standard 2x2x2 cube, from what I can tell, but we have cut the number of possible configurations down. There are only two types of corner pieces, since each reads RWB or RBW clockwise, and we can actually permute them amongst each other with the correct orientations. To see this, note that on the standard cube we can interchange the RGW and OBW corners so that every face remains a combination of W/Y, R/O, or B/G:

Our cube repaints Y into W, O into R, and G into B, making the corners \(\{\textrm{RWB, OWG, RYG, OYB}\}\) all RWB and the corners \(\{\textrm{RBY, OGY, RGW, OBW}\}\) all RBW. Since the pocket cube has \(3,674,160\) states, by choosing a reference corner, this would overcount our cube's states by a factor of \( 3! \cdot 4!\), dropping the number of visually distinguishable arrangements to \(25,515\). While a much smaller configuration space, it is still pretty unlikely to accident into a solution.

Notes

• I explored a few other reconfigurations. While many of them dropped the space of configurations down, they occasionally counterintuitively made the problem of solving the cube harder. A couple such configurations were

  • Two colors: One for three faces around a corner, the other for the remaining three faces.

  • Two colors: One for three faces not around a corner, the other for the remaining three faces.

  While solving a standard cube, the six colors do a lot of work helping one get one's bearings. With two colors, keeping the cube oriented relative to what the solved state should look like becomes much tougher for a seasoned solver.

  I also created some cubes that were meant to be other sorts of stepping stones along the way to solving a standard cube but wasn't happy with them as stand-alone activities. For example, a cube with an all-yellow bottom layer, a white top face, and the remaining \(2 \times 1\) non-yellow strips along the sides in red, green, orange, and blue, as usual. This is a nice tool for learning to solve the top layer with correct edge orientations, but it feels like a nice stand-alone puzzle solution should have each face the same color or be more uniform in some way like the first couple "corner cube" examples.

• Six is a nice number of cubes to have to reconfigure, since cubes have six faces.

  If you want to make a set of our first example, you could have color variants and end up with six cubes with two each in the color schemes RBGY, RBOW, GYOW. Or you could make four of these cubes in the RBGY color scheme and either

  • an orange cube with white opposite faces and a white cube with orange opposite faces, or

  • two cubes each with four orange corners and four white corners

  If designing cubes with three colors, two for opposite faces and one for the rest, then some color schemes might be WRRRRY, WOOOOY, RBBBBO, RGGGGO, GWWWWB, and GYYYYB.

This post was sponsored by the Julia Robinson Mathematics Festival