An excerpt from Puzzle Laboratory's A Guide to Twisting Puzzles
[This was our first published solution, which appeared in the inaugural issue of WGR in November of 1983. It has been considerably edited for this page.]
Pyraminx is a tetrahedral twisting puzzle devised by a German inventor, Uwe Mèffert, and originally manufactured by Tomy Corporation. Though conceived earlier, it was only manufactured in the wake of Rubik's Cube. Pyraminx is composed of 14 visible parts: four small corners (sometimes called tips), four large corners, and six edges (see photos above). It is a much easier puzzle: the Cube can be twisted into over 43 billion billion positions, while Pyraminx has only 75 million [75,582,720 = 2^5x3^8x6!/2], and once the small corners are (trivially) twisted into their correct positions, there are less than a million possible positions [933,120]. The object is to take a scrambled Pyraminx and return it to its original condition (called Start), in which each of the four faces is a single color. There is now a truncated version with no small corners, sold under the name Tetraminx.
The diagram above is a flattened aerial view (with the Down face hidden), showing the various parts of the Pyraminx. The large corners are shown in darker shades. Note that, from a vertical view, you can see all three facets of the Up corner and two of the three facets of the Left, Back, and Right corners. You can see both facets of the Left-Right, Front-Left, and Front-Right edges, and only one facet of the Front-Down, Left-Down, and Right-Down faces.
I believe that the solution we present here is
and is fairly easy to use. We will use the notation above. The four large corners (from now
on simply called corners) are called up, left, right, and back. Edge
locations are named by the capitalized initials of the two faces they
lie on. Edges are named by the two faces they should lie on when
Pyraminx is solved, in lower case initials (fd is the edge which
belongs at location FD). The faces are called Front, Left, Right, and
Down. Clockwise turns of the four corners are named by capital letters
U, L, R, and B. We will also use a turn of the base (the part of the
Pyraminx which is not part of the Up corner), holding the Up corner in
place. This is called D (Down). The five corresponding
turns are named by lower case letters (u, l, r, b, and d). We
only use the D and d turns during Phase 2, to keep the position of the
Up corner (and its two adjacent edges) fixed. We will only
need 8 turns in Phases 3
We will solve Pyraminx in five phases. Phase 1 turns the small corners so that their colors match the corners they are attached to. Phase 2 puts the two front edges fl and fr in place. Phase 3 turns the left and right corners, and Phase 4 finishes the front face by placing the edge fd. Phase 5 reorients the Pyraminx so the front face becomes the down face, turns the up corner, and finishes the solution by simultaneously placing the fl, fr, and lr edges.
Phase 1 is the easiest one. Simply turn each tip (small corner), if necessary, in the correct direction, so that its three colors match those of the corner it is connected to. After all four small corners are turned correctly, Phase 1 is finished, and we will not need to turn the small corners again. Phase 1 can take as many as four turns. (I usually don't bother twisting the small corners when scrambling the Pyraminx, since this phase is trivial anyway).
In Phase 2, we place the correct edges at FL and FR
so that their
colors match the up corner. We will assume that the Up corner is
correctly placed. The color of the Up corner which shows on the Front
face will be called the front color. For the rest of the solution, we
will assume that the front color is blue, but of course you may
the four colors in an actual solution. First find fl, the edge which
belongs at FL (check the
colors of the Up corner on the Front and Left faces -- blue and orange
in our example -- and find the edge
which has those colors). If fl is already at FL and it shows
blue on the Front face, continue with the edge fr. If not, we still
have work to do.
We need to get fl to one of the two target positions (above left) from which it can be placed correctly by turning the Left corner. We need to get fl either to FD with its blue side on the Down face (from where an l turn puts it in place), or to LD with its blue side on the left face (from where an L turn puts it in place). See the diagram above. If fl is at LR or FR, or already at FL but with the colors flipped, make a turn (B from LR, r from FR, or L from FL) which puts it at FD or LD. Half of the time, B from LR or r from FR will put it in a good position directly. Once fl is in the bottom layer, turn the base if needed so that fl goes into one of the two good positions described above. The base turn is always necessary always when fl starts at FR, and is also needed when fl is at FD or LD, but with blue on the wrong face. Once fl is in the correct position, make the Left turn that places fl in FL. The process of placing fl may take as many as three turns.The edge fr is now placed in the same way. The two key positions to aim for (above right) are FD with its blue side on the Down face (R now puts fr in place), and RD with its blue side on the Right face (r now puts fr in place). If fr is at LR, turn r to put it at FR (where either r or dr puts it in place). If it is at FR but flipped, rdR fixes it. If it is already in the Down layer, turn that layer if needed to put it in a good position and then turn R or r. Placing fr can also take up to three turns. Phase 2 has now been completed, taking a maximum of six turns. The Front face is now all blue except for the bottom row.
In Phase 4, we need to find and place edge fd. If fd is at LD or RD with blue on the Down face, turn b or B to get it to LR. If fd is at LD with blue on the Left face, place it with the sequence of turns rBR. If fd is at RD with blue on the Right face, place it with sequence Lbl (the mirror image of rBR). If fd started at LR, or you moved it there as instructed, you can place it with LBl if its blue side is on the Left face, or with rbR if its blue side is on the Right face.
If fd is at FD but flipped (i.e., with blue on the Down face), fix it by turning rBRLBl. The diagrams above summarizes these sequences. Phase 4 is completed in a maximum of six turns.
Phase 5Finally we come to Phase 5. The Front face is all blue, so we are finished with it. Turn the entire Pyraminx so that the blue face becomes the new Down face. Now turn the new Up corner so that its colors match those of the base. We are faced with four possibilities. We may be finished solving (an 11-to-1 longshot). If so, mix it up and solve it again -- practice makes perfect. If we are not so lucky, we need to find out whether we need to move the last three edges (FL, FR, LR) cyclically, and whether two of the edges need to be flipped. The easiest way to tell whether an edge needs to be flipped is to observe whether either of its facelets match the color of the adjacent Up corner facelet. An edge in the wrong position needs to be flipped if one of the colors match; an edge which does not need to be flipped does not match on either facelet. (If an edge is in the correct position, both colors match, of course, if it does not need to be flipped; the colors are reversed if it needs to be flipped).