As shown in the section on the Megaminx, a puzzle does not have to be a cube in order to be solved using the methods of The Ultimate Solution to Rubik's Cube. All that is required is that it has corners and edges. Recently I received an e-mail from Patrick Fincannon pointing out that the Ultimate Solution approach makes the solution to Ewe Meffert's Pyraminx and Tetraminx extremely simple. This is indeed true. We do not even need the Corner Piece Series. In the case of the Pyraminx, following a few preliminary moves, edge pieces are placed using the Edge Piece Series. Then the solution is completed by placing the three final edge pieces using a single Edge Piece Series and the principles of the End Game.
The Pyraminx sits on a triangular base and has three other triangular sides. It has three kinds of pieces. There is a three sided tip at each apex. There are four tips. Each of the three faces of a given tip is of a single color. This tip is like the face center of a Rubik's Cube in that it may be rotated but it cannot be moved to another part of the Pyraminx.
The tip is attached to a face piece. The face piece has three sides each of a different color. Like the tip it may be rotated but it also cannot be moved to another part of the Pyraminx. There are four face pieces.
Finally, there are six edge pieces. Each has two faces and each face has a different color. These edge pieces are the only pieces which may actually be moved from one part of the Pyraminx to another.
When solved each face of the Pyraminx will show a single solid color.
Let us assume you are presented with a scrambled Pyraminx. Its colors are orange, green, blue and yellow. The first step is to determine the target color of a single face. Look at one tip. It has three colors. The one I am looking at it is blue, green and yellow. Therefore the color of the opposite face is the color missing from this tip or orange.
Rotate each tip so that its colors match the colors of the face piece to which each is attached. Now rotate one of the apexes in the orange target face so that its orange color comes to that face. This means that you are to turn a pyramid which is two little triangles high. Repeat this maneuver for the other two apexes containing the orange color. Six of the nine pieces in the orange face are now properly placed.
Now look at the fourth apex, the one without an orange face. Rotate that apex so that its colors match the colors in the blue, green and yellow faces. A total of eight pieces are now in place. Six edge pieces remain although it is possible that one or more of them were accidentally moved into the proper position and orientation by these preliminary moves.
Look at the "orange" face of the Pyraminx. An Edge Piece Series (to be described) will cause the three edge pieces you see here to move about the orange face. As you look at this face one edge piece may be described as being on the right. Call it #1. One edge piece is on the bottom (#2) and one is on the left (#3).
Rotate the right hand apex counterclockwise so that edge piece #1 replaces #2. Now rotate the left hand apex clockwise so that #3 replaces #1. Reverse turn one and then reverse turn two.
Edge piece #1 has replaced #2, edge piece #2 has replaced #3 and #3 has replaced #1. Edge pieces #1 and #2 simply roll over into place. That is, #1 will wind up looking just as it did at the end of the first turn. Edge piece #2 will roll over in a similar manner. But #3 does not roll over. Rather, the color of that piece which was in the orange face at the beginning of the series is still in the orange face at the end of the series (although it has moved from the left side to the right side.
If you want the edge pieces to move the opposite way then rotate the left hand apex first and the right hand apex second. If you want a different edge piece to be the one which does not roll over then change the apex which points up, etc..
You may find that two pieces you want to use are in a given face but the third position is occupied by the correct piece with the correct orientation. No problem. Simply rotate a base or bases so as to move another incorrectly placed edge piece into the target face. Apply the Edge Piece Series and rotate the base (bases) back where they came from.
When only three edge pieces remain you are at the End Game. However, now you will use an Edge Piece Series rather than a Corner Piece Series as with the cubic puzzles. Two of the remaining edge pieces will be on the same face. If the third is not then you must bring it to that face with appropriate base turns so that a given Edge Piece Series will move all edge pieces into the correct position and with the correct orientation.
Don't overshoot the mark. You may find that the base turns to get the third edge piece in the proper face are the inverse of the final turns of the Edge Piece Series and you obtain a solved Pyraminx before you expect it.
Despite any appearances to the contrary, you will find that the Tetraminx and the Pyraminx are basically the same puzzle. Just take the four tips off of the Pyraminx. This will leave a space the size of one of the small triangles. Paint these four spaces with the four colors of the original Pyraminx and, behold, you have a Tetraminx. Omitting all of the references to the apex tips, the remainder of the directions given above may be applied to solve a scrambled Tetraminx.
There is one point where you may find an alternative approach to be more convenient and faster (it has fewer turns). Assume that you have turned the three apexes containing an orange face so that one face contains three orange triangles, each at 120o to each other. The other three triangles are some other color. You want to place new edge pieces on this surface such that the orange faces of the new edge pieces replace the non-orange colors on this surface and the colors of the faces hanging down match the colors on either side.
This can be done by a simple exchange. Assume an edge piece with a blue color on the orange surface and a green color hanging down between two yellow surfaces. (Check the orientation of the orange/yellow edge piece. It may be found in either of two different ways.) Turn the puzzle so that the top section divides along a diameter and the blue/green edge piece comes down to the lower section of the puzzle. There are two ways you can do this. One way will divide the top along a diameter at the right of the blue/green edge piece. The other will make this division along the left side of the blue/green edge piece. You want the one which will enable the orange/yellow edge piece to replace the blue/green edge piece (by a turn of the bottom section of the puzzle) and then have the orange color come to the top when the original turn is reversed.
When you have replaced the non-orange edge pieces in the orange face with the appropriate orange edge pieces you will be left with three edge pieces out of place. Move them into proper position and orientation as described in the End Game above.