# Pyraminx Puzzle Solution Methods

Michael J. Ross

Compared to the famous Rubik's Cube, the Pyraminx puzzle is not well known. Anyone unfamiliar with it can think of it as the triangular version of Rubik's Cube. The most notable differences are that it is in the shape of a four-sided pyramid (a regular tetrahedron, in geometric terms) instead of a six-sided cube, and that its exterior is consequently divided into triangles instead of squares — usually referred to as "pieces".

Aside from that, the Pyraminx is quite similar to its more celebrated competitor. Each triangle is one of a limited number of colors, and the puzzle's interior mechanism allows those pieces to be moved from one side of the puzzle to another by rotating one or more pieces at a time. Commercially manufactured versions of the Pyraminx have various colors, but they tend to be primary colors. In the example puzzle I will use in this article, those colors are blue, red, green, and yellow.

As with the Rubik's Cube, the objective of the Pyraminx puzzle is to begin with all of the pieces scrambled by color, and then to move all of the pieces of each color to a single side, resulting in a solved puzzle.

One can rearrange the three pieces that compose a corner by rotating just those three.

However, solving a scrambled Pyraminx is mostly accomplished by rotating the 12 pieces closest to a corner, either in one possible direction or the other, around the imaginary axis passing through that corner and the center of the puzzle's interior.

The internal mechanism of the puzzle allows multiple corners to be rotated at the same time. However, when solving a puzzle, only one rotation is performed at a time.

As noted in its Wikipedia entry, there are several published methods for solving this puzzle, of varying levels of difficulty. This article presents two strategies for solving the Pyraminx — one that works by completing a full layer and the other by shuffling edge pairs.

## Terminology and Notation

Describing in written words how to perform the physical actions needed to solve a spatial puzzle, is difficult, but made much easier with some clear terms for the puzzle's components and some concise notation for the moves.

Imagine that you are looking at one side of a Pyraminx head-on, so only that side is visible to you, and one of the corners is pointed straight up, leaving the other two visible corners pointing to your lower left and lower right. Those three corners can be referred to as the top (T), the left (L), and the right (R), respectively. The remaining corner, in the back, is not visible from this perspective and is not referenced in this article.

Each side (a.k.a. face) has nine pieces: three tip pieces at the corners, three axial pieces adjacent to the tip pieces, and three edge pieces. Each edge piece is inseparable from another edge piece (of a different color) on an adjacent side. This can be thought of as a "pair" of edge pieces or an "edge pair".

Only two kinds of rotations are possible: 1) rotating the three tip pieces that form a corner (Figure 3), and 2) rotating, as a single unit, three tip pieces as well as their three adjacent axial pieces and the six edge pieces (three edge pairs) adjacent to those axial pieces (Figure 4). Such a group of 12 pieces I will call a "segment", and identify each one by its corner (T, L, or R). (The Wikipedia entry doesn't appear to provide any standard term.)

The process of solving a Pyraminx consists mostly of rotating the segments through one or two rotations, one after the other, in various combinations, in order to move pieces from one location to another.

Later in this article, I will have need to identify particular sides, as well as the edge pairs. When ordering things, most cultures look from left to right, top to bottom — which I will do here. Again viewing a Pyraminx head-on with a corner pointed up, the four sides can be designated: left (L), front (F), right (R), and bottom (B). The six edge pairs would thus be: LF, FR, LR, LB, FB, and RB. For instance, "RB" refers to edge pair consisting of the two edge pieces on the right side and the bottom. The order of those letters is significant in that it indicates the order of the pieces — in this example, the piece on the right side and then its neighbor on the bottom side. The two will always be of different colors.

To "flip" an edge pair is to end up with it, after all rotations are done, in the same position on the puzzle but with its two colors reversed, e.g., an edge pair LF becomes FL.

Two or more pieces are said to "match" if they have the same color after all rotations are completed.

To summarize this terminology and notation:

• pieces: tip, axial, edge
• edge pair: two adjacent edge pieces
• segment: a corner's tip pieces + axial pieces + edge pieces
• corners: top (T), left (L), right (R)
• sides: left (L), front (F), right (R), bottom (B)
• edge pairs: LF / FL, FR / RF, LR / RL, LB / BL, FB / BF, RB / BR
• flip an edge pair: reverse its colors
• pieces match: identical colors

A sequence of segment rotations can be concisely specified using the symbols of the segments. A rotation can be performed in the clockwise direction from the perspective of viewing the corner head-on (with the rest of the puzzle behind it). For instance, the sequence "R T L" would mean first rotating the right tip's segment clockwise, then rotating the top segment clockwise, and finally rotating the left segment clockwise. A counterclockwise rotation would be in the reverse direction and could be notated as rT, rL, or rR. For instance, the sequence "rR rL T" would be performed by rotating the right segment counterclockwise, the left segment counterclockwise, and lastly the top segment clockwise. The identical final result could be achieved entirely with only clockwise moves, "R R L L T", since two rotations in the same direction, one immediately after the other, is equivalent to a single rotation in the opposite direction.

## Layer Method

This first strategy for solving the Pyraminx is systematic and relatively straightforward, at least compared to many other published methods.

Step 1: Match the tip pieces: In other words, rotate each one of the four tips until all three pieces it comprises have the same color as the axial pieces next to them.

Because the tip pieces rotate independently of — and do not affect the positions of — all the other pieces of the puzzle, it is trivially easy to match them to their axial pieces. You may as well perform this step now, because there is no point in delaying it. Moreover, an advantage of doing it sooner rather than later, is that, when they are all matching, it becomes much faster to see at a glance the intended final colors of any given side.

Step 2: Match the axial pieces: For each of the four colors, find the three axial pieces of that color, and visually determine which side is the only one to which the three axial pieces could ever by moved. Rotate the segments as needed to bring those axial pieces to that side. It's actually easier to perform this step than it is to describe it.

Step 3: Complete the bottom layer: Choose any side and orient the puzzle so that side is facing down — i.e., it becomes the bottom side. One, two, or all three of its edge pieces may not be matching the tip and axial pieces of this "B" side. For each one of those mismatching edge pairs (let's call it "M"), you will need to replace it with the correct edge pair ("C") that belongs in its place, using this technique: Orient the puzzle so M is facing toward you. Rotate the top segment until C is either at the left edge LF or at the right one FR such that the piece of it on the F side matches the bottom layer pieces of F. Then, if the B colored piece is now on L, perform the sequence of rotations "R rT rR", or if the B colored piece is on R, do "rL T L". What each of these sequences is essentially doing is moving M up out of the bottom layer to the R or L edge, replacing it with C, and then moving C down into the bottom layer.

An example can illustrate this process. In the figure below, I have chosen the blue side to be the bottom one. On that side are two mismatching edge pairs. I choose the one with the colors red / green, and designate it "M". I orient the puzzle so M is facing forward (i.e., it is now located at FB).

It needs to be replaced with the red / blue pair (C), which in these two figures below is pointing forward in a half-rotation, so you can see both colors.

I finish the rotation so C is at LF. This brings the red piece of C to the red side (F). Then I perform the sequence R rT rR, which replaces M with C.

Now the entire bottom layer is complete — not just the nine B side pieces but also the fifteen L, F, and R pieces adjacent to the B side.

Step 4: Complete the top two layers: First of all, the rotations in Step 3 may have left the three axial pieces no longer matching their sides; if so, rotate the top segment until all the axial pieces are back in place.

At this point, assuming the puzzle is not already (and fortuitously!) solved, then the only pieces out of place are those of either two or three edge pairs, all in the top segment. There are five possible scenarios, and for each one there is a sequence that solves the puzzle. In scenarios 1, 4, and 5 — with two edge pairs that need to be flipped — begin by orienting the puzzle so those two edge pairs are facing forward (i.e., they are now located at LF and FB); otherwise, performing the given sequence results in two edge pairs needing to be flipped — just not the same ones as before.

Here are the five possible scenarios and their solution sequences:

• two edge pairs need to be flipped (but remain in place): rR L R rL T rL rT L
• three edge pairs need to be shuffled clockwise: rR rT R rT rR rT R
• three edge pairs need to be shuffled counterclockwise: rR T R T rR T R
• three edge pairs need to be shuffled clockwise; two need to be flipped: L T R rT rR rL
• three edge pairs need to be shuffled counterclockwise; two need to be flipped: rR rT rL T L R

## Edge Pair Shuffle Method

This second strategy for solving the Pyraminx is unique and potentially faster, but is less systematic and potentially slower. Based on my research, it has apparently not yet been published or perhaps even discovered until now.

For this method, implement Steps 1 and 2 above. At this point, all the axial pieces match their sides, but the edge pieces probably do not match.

Step 3: This step uses sequences of rotations that preserve the matching of the axial pieces achieved in Step 2. Consequently, you can perform any of them with that assurance.

Usually at least one edge pair consists of pieces that match their sides. Choose one of them. If none exists, perform one or more of the sequences below until at least one edge pair now matches. Orient the puzzle so that the chosen edge pair is at LB, because during this step, these sequences do not change the LB pair, and it can be used as an anchor point around which to work.

Examine the possible sequences and determine which one moves the maximum number of edge pairs into their desired locations. After performing the optimal sequence, reassess the puzzle's new configuration and again see which sequence does the most in moving closer to a completed puzzle.

If none of the possible sequences improve the configuration, then reorient the puzzle by pivoting it such that the matching edge pair at LB stays at LB. Consequently, tips R and T will switch places — i.e., tip R becomes tip T, and tip R becomes tip T. Then examine the possible sequences again to find the best one.

As noted above, for each pair, the order of their two letters is important, so pay attention to that, otherwise you may inadvertently shuffle an edge pair into the correct position on the puzzle but with the colors of its two pieces the opposite of what you intended.

Here are the eight possible sequences and the results of each:

• T R rT rR: LR → RF, FR → BF, FB → LR
• rT rR T R: LF → FR, FR → RB, RB → LF
• T rR rT R: LR → RF, FR → RB, RB → RL
• rT R T rR: LF → FR, FR → BF, FB → FL
• T R T R T R: LF → BF, LR → RF, FR → RB, RB → LF, FB → LR
• R T R T R T: LR → FB, LF → RB, FR → LF, RB → RL, FB → RF
• rT rR rT rR rT rR: LR → BR, LF → FR, FR → BF, RB → LF, FB → LR
• rR rT rR rT rR rT: LR → FB, LF → RB, FR → RL, RB → FR, FB → FL

This new method for solving the Pyraminx will be faster or slower than other methods depending on how many iterations of Step 3 you will need to do. If after completing Step 2 the puzzle's configuration conforms to one of the eight permutations listed here, then this approach will definitely take less time. Otherwise, you would need to do additional iterations of Step 3 — trying with each one to get closer to a solved configuration, but with no guarantee of success.

Arguably the overall best strategy for solving a scrambled Pyraminx would be to first see if it can be quickly solved using one iteration of this edge pair shuffle method, and if not, then use the layer method delineated earlier.