In a 2×2×2 cube you twist half the puzzle with every twist. Same with the Pyramorphinx. With a 2×2×2 cube you can identify front, left, back, right, top, and bottom layers. Same with the Pyramorphinx, although it isn’t quite so obvious since there are only 4 sides to the puzzle. The 2×2×2 cube has 8 pieces. They are all corners. The Pyramorphinx has 8 pieces. 4 corners and 4 centers. The Pyramorphinx is a tetrahedron that can be solved exactly the same way you solve a 2×2×2. I usually solve the 2×2×2 using the Corners First Method. This isn’t so easy for me on the Pyramorphinx. I have trouble identifying the “white X” and the “yellow X.” So I solve it by getting a bottom edge using Up-Replace-Down moves. The bottom edge consists of 2 corners and 2 centers. Then I use the Up-Replace-Down Commutator to place the 4 pieces on top.
One interesting thing that happens sometimes is one corner needs to twist at the end. One way to deal with it is to twist it plus one of the adjacent centers using the typical corner twisting sequence explained in, say, the Edges First Method. But my favorite way to deal with it is to use either R U Ri U R U2 Ri U2 or the left-handed version, depending on which way the corner needs to twist. What happens is that the 2 adjacent centers are also twisted, but since the centers don’t show orientation, it doesn’t matter. So this is a quicker, more efficient way to twist a single corner. Hold the puzzle so the corner is at the top right back and if the sticker on the right needs to go to the left do the right-handed sequence. If the one on the left needs to go to the right do the left-handed sequence.
The same idea can be used if two corners both need to twist the same direction, but I haven’t worked out the details on how to hold it and when to do left-handed and when right-handed.
You might be thinking that the advantage of the Pyramorphinx over the 2×2×2 is that you don’t have to worry about twisting the centers correctly, so there are only 4 pieces to orient correctly as opposed to 8. That is right. Unless you want to line up all the grooves in the stickers. Look closely.
Although you can think of the Pyramorphinx as a 2×2×2, and solve it as such, it has a whole other personality of its own. There are several different shapes it can take on.
First of all it is a tetrahedron, or a triangular pyramid.
Turn one layer 90˚ and you have what I call a butterfly. I don’t know if this shape has a technical mathematical name or not.
From the butterfly turn a different layer 90˚ and you have what I could call a gopher. It doesn’t look like a gopher, but from this state you can Go Four different ways into 4 different shapes. Go Four. Gopher. I don’t know. I just made this up. I don’t know if it’ll really stick.
From the gopher you can go back to the butterfly, or you can go another 90˚ with the same layer and get what I call the rocket.
Also from the gopher you can twist a different layer 90˚ and end up with what I call a shuttle.
Lastly from the gopher you can twist that layer 90˚ the other way and get what I call a hat.