Thursday, April 11, 2024

Robbie's Megaminx Tutorial

I'm going to hold off for the moment on writing a 6-cube and 7-cube tutorial because I'm no expert yet. I've only solved each of them four or five times now, but (at the risk of putting out spoilers) I've found 5-cube algorithms to be very helpful, with some creative adaptation—proving my assertion, earlier in this series of posts, that solving this kind of puzzle promotes deductive thinking. I also realize that I have yet to fulfill my promise to make the 5x5x5 "last two centers" (L2C) and "last two edges" (L2E) cases more intuitive, something that I feel will help me as well. Give me some time!

But meanwhile, let's talk about another one of my favorite 3D position puzzles, which I rank right up next to the 4-cube as one of my comfort cubes: The Megaminx.
Obviously, it isn't a cube, strictly speaking. It's a dodecahedron. That is to say, it has 12 sides, all of which happen to be identical, regular pentagons. Identical, that is, if you ignore the colors. (On my stickerless Megaminx, the colors of opposite pairs of sides are white-gray, yellow-cream, dark blue-light blue, red-orange, purple-pink and dark green-light green. Substitute words like "beige" or "cyan" if that flips your minnow.

This is another way interest in Rubik's Cube-type puzzles overlaps with an interest in mathematics, especially of the recreational variety. There's a small set of three-dimensional figures called Platonian Solids, including the cube and the regular dodecahedron. They're all convex polyhedra (meaning 3-D figures whose edges and vertices all project outward from a common center), and they're all perfectly regular (i.e. their faces are identical, regular polygons with identical sides and angles, all coming together at identical vertices and edges). And only five of them are possible. Besides the cube, the other Platonic solids are the tetrahedron (a pyramid with a triangular base, all made of equilateral triangles, three meeting at each vertex), the octahedron (a diamond shape with eight sides, all equilateral triangles, with four sides meeting at each vertex), and the icosahedron (a figure made of 20 equilateral triangles with five of them grouped around each vertex). And that's it.

Basically, we're talking the shapes you'd find in a set of polyhedral dice, used for roleplay games. Except one of them (the 10-sided die) isn't Platonic. And you'll also find Rubik's-type puzzles in all five Platonic solid categories, although some of them are harder to get. I have yet to score an icosahedral one, and it seems like any that I find offered for sale are either worth a mint or bait for an online scam. Also, there are non-Platonic-solid 3-D puzzles, including a pentahedron that I'll tell you all about one of these days. So, as I said, there seems to be a connection between an interest in these fascinating geometries and the twisty, turny puzzles that evolved from Rubik's Cube.

There are several other dodecahedral puzzles (some with more moving pieces, some with less), but the Megaminx is kind of the standard, the way the 3-cube holds the front line of six-sided puzzles. I think the thing was invented by a German guy named Uwe Meffert, but I've also ready that multiple people invented it independently. Each side is divided into five corners with three colors, five edges with two colors—viewed face-on, they seem to form a five-pointed star—and one pentagonal center piece that only has one color, and doesn't move in relation to the other centers. Each face is a separate layer that rotates in 72-degree, or 1/5-turn, increments. With 50 movable pieces, it has over 1.01x1068 possible permutations (a few less for the six-color variant, which has some duplicate pieces), and yet the current world record for the fastest single solve is 24.12 seconds.

NEW MOVES: You may have noticed that a dodecahedron is not a cube. Therefore, the available types of moves aren't quite the same. Viewing the Megaminx face-on, with one side lying flat on top, you could of course do an F move (rotating the front layer, facing toward you, 1/5 turn clockwise), kinda like this:
If you go 1/5 turn counterclockwise, that's F' (eff-prime). No surprises so far. However, with five possible positions, the face could go 2/5 turn both clockwise (F2) and counterclockwise (F2'), which adds a new wrinkle, and not the last one. And here, naturally, is the U move, which is also susceptible to U', U2 and U2' variants:
Still looking at the 'minx face-on, turn this side 1/5 clockwise (as if facing that side directly) and it's an R move.
No surprise there. And of course, 1/5 turn counterclockwise is R'. And you guessed it, both R2 and R2' are also possible. Without repeating all those permutations, which apply across the board, here's an L move:
Now, I'm sure you noticed, the top layer has five sides. This means there are two other sides you can turn just in the top half of the puzzle. Let's call this one BR as in "back right":
And this one, as you can probably guess, is BL:
Still in the face-on regime, there are two more moves you'll need to know: DR ("down right") and its counterclockwise version, DR', meaning this layer below the puzzle's waistline:
Don't worry about DL or any of the other lower-layer sides hidden around the back, because you won't need to turn them—at least, when you do, they'll be facing toward the front.

With me so far? OK, suppose you have to do a step that assumes you're looking at the puzzle edge-on; that is to say, with the edge between two faces at front, facing toward you. Early warning: This is gonna happen. And it flips the directions of the turns around. U is still up, of course, but now this is an R move:
And this is L:
Then we shall call this B, as in "back":
And since the front is now split between right and left, we'll call this FR for "front right":
And this, naturally, will be FL:
Theoretically, a D ("down") move is always possible, rotating the bottom layer, but in practice you never need it. Starting the scramble, white goes up and green to the front; during the solve, white generally goes down, but you're free to turn the puzzle every-which-way and, as you solve it step-by-step, you'll be looking at it from a different angle at every step.

SCRAMBLE: Getting this thing scrambled is another thing altogether. When you pull down the puzzle-type selector at this puzzle scrambler, you'll see a different notation again. For scrambling purposes, please don't ask me why, you only need six possible moves. D++ means you hold the top layer and twist the rest of the 'minx 2/5 of the way around, clockwise (as viewed from below). D-- is its counterclockwise counterpart. R++ and R-- mean you grab the left layer and do a 2/5 turn of the rest of the cube, clockwise and counterclockwise respectively. And U and U' are what they usually mean, a 1/5 turn only. Apparently a sequence of these turns, and only these turns, is sufficient to randomly scramble this puzzle, leading to a result like this:
STEP 1: White star. Actually, you could start with any color, but I make white my first layer every time. After this step, white will generally be facing down and gray will be facing up, with five sides on the lower tier (red, dark green, purple, yellow and dark blue) all pointy-end up, alternating with five points-down sides on the upper tier (orange, light green, pink, cream and light blue). To start—this bit is all intuitive, do-it-yourself-stuff—dial in the white edges (they look like triangles, viewed face-on) around the white center, making sure each edge's other color aligns with its matching center. I trust your problem-solving abilities to manage this, starting with one edge like white-yellow here::
...and continuing all the way around until you see a white star, like this:
STEP 2: Solve first layer. Which, again, generally means white. Find a corner piece that goes between two of the white star edges. Putting white at the lower front and viewing the upper tier edge on, put the edge piece directly above the spot where it belongs—like the purple, yellow and white corner here:
Now there are three cases for how to move this corner down to the slot between the purple-white and yellow-white edges. Notice in the case above that purple is on top of the piece in question, and the purple side is to the left. So you'll use the left corner algorithm: U R U' R'. In other words, you rotate the corner piece out of the way in the direction of its top color's center, dial the incorrect piece up and out of the corner, rotate your corner back in and dial it down into place: out, up, in, down. Result: Purple-yellow-white finds its spot, oriented correcty and all.
You can use that same mnemonic in the event the top color of your corner matches the center to the right, like in the case of purple-green-white in this example:
The move is U' L' U L but the reasoning is the same, as is the result:
For the third case, suppose the corner pieces comes into that staging position with the white side up, like this red-green-white corner. What do you do then?
Well, you do the algorithm R U R' U' x2. The "x2" means you do those four moves twice, which is easier than memorizing an 8-move algorithm. That'll flip the corner around so one of the other colors is facing up, and then you can do whichever algorithm applies. I feel like it's usually the right-corner one, but don't quote me on that.
After you've worked your way around the white star, you'll not only have a solid white side, but all its corners and edges will line up with the correct lower-tier centers.
STEP 3: Lower-tier edges. This is about putting the correct, two-colored edge pieces between the sides on the lower tier. Pay attention, because you're going to use this algorithm a lot, and it has both right-handed and left-handed versions. First, find an edge piece somewhere on the 'minx and hustle it around so that it's standing upright, above and to the right or left of the edge where it belongs. Holding the puzzle with white in the lower front and where the piece belongs at front, position your piece above the center that matches its lower color—like this purple and yellow edge at the left:
Then do this rather long algorithm, which (as I said) you'll be doing a lot, so expect to learn it well: U R U' R' U' L' U L. This is the left edge algorithm, and you can think of it as dialing the correct edge out of the way (away from its top-color center), moving incorrect edge up in the other direction, pulling the correct edge back in and dialing the other side back down, then dialing the correct piece across to the other side of the target edge, pulling that edge out in the other direction again, moving the edge back to where it was before (but now with the correct corner moving with it) and then dialing both pieces down into place. Away, up, in, down, across, up, in, down. Result: your edge is now snug against the corner piece it belongs above.
Again, you can use the same reasoning to explain and remember the right edge algorithm. So, take the blue-yellow edge at the upper right:
Again, stage the piece above the center that matches its bottom color and start the move by twisting it out of the way, away from its top-color center, doing U' L' U L U R U' R', or away-up-in-down-across-up-in-down. Result:
When you've gone all the way around the lower tier, you'll see something like this (viewed from the bottom):
STEP 4: Middle-layer valleys. A video tutorial I watched when I was setting out on my Megaminx journey called these pieces "bunny ears," but that's silly. With the white side facing down, this step is about color-matching the two edges and one corner piece in each of the five, V-shaped angles where the upper-tier sides point down between the lower-tier sides. You have to be careful while doing this to avoid scrambling parts of the puzzle you've already solved, which makes this step a little tricky. However, you can do this either left-handed or right-handed, by which I mean the direction you're going around the cube, and in my opinion it's good practice to switch directions from one puzzle-solve to another, to keep your technique tuned up going both ways. For the sake of simplicity, the example solve for the pictures below will go around the puzzle toward the right.

First, dial in an edge piece that goes between an upper-tier side and a lower-tier one, like the cream-red edge at right.
Then, turning the puzzle so the next edge to the right is at front, bring the corner piece that belongs in the valley to the top front, above where it belongs. Apply the left, right or center corner algorithm from Step 2, as the case requires. For example, this green-red-cream corner...
calls for the right corner algorithm (U' L' U L). Then the cream-green edge, staged here above the green center, will get the left-edge treatment from Step 3.
The result (in this example) is the three-piece, cream-red-green, edge-center-edge set all put together.
Now continue building these three-piece valley assemblies, moving from one side to the next, until only one side is left undone—I'm calling that a whole separate step, because it has a slight (cough) twist. Meanwhile, anytime you have to rotate a side whose valley-piece you've already assembled to free up a corner or an edge that you need elsewhere, don't forget to dial the valley back down to where it belongs. This will save you the maddening chore of having to re-solve something you previously solved.

STEP 5: The last valley. This is when you only have one middle-layer valley to fill in. Attach the first inverted bunny-ear (if you will) the same way as usual, making sure you right any other side that you have to twist out of alignment to do so. Like this light blue-dark green edge, for example. (I apologize for the overexposure in the photos that follow.)
Now, give that last side a 1/5 turn to bring the corner at the bottom of the valley one twist closer to the top. This is important, because it will enable you to put that purple-dark green-light blue corner (on top in the previous picture) where it belongs without scrambling the adjacent side. And ditto with the light blue-purple edge at the upper left in this picture, which will become the final bunny ear.
STEP 6: Middle-layer peaks. This is where you color-match the corner piece at the top of each lower-tier side, as well as the edge piece that sits on top of it like a floppy hat. It doesn't matter whether you do each corner and its neighboring edge before moving on to the next, or all the corners first and then all the edges. You're just going to use the same center and edge algorithms you've already learned, for example, to put this red-pink-cream corner in its proper spot...
...as well as this almost undetectably pink-cream edge (sorry again):
The result is everything solved except the top layer of the puzzle.
Yay! That means there are only four steps left! Four tricky, easy-to-mess-up steps that are, by themselves, half the challenge of solving the Megaminx. If you have the right attitude, you'll enjoy them ... and you won't begrudge the extra practice you get if/when you screw them up.

STEP 7: Gray star. Otherwise known as "orient last layer edges," meaning the object is to arrange all the two-colored edge pieces gray-side-up around the gray center, like the points of a star. If you reach the end of Step 6 and you don't already have a gray star (and I never have), you'll see one of three cases. Either you'll have one, lonely gray edge piece pointing away from its center; if so, point it toward the back left, viewing the puzzle face-on, like so:
Or you'll have three gray edges all bunched together, like the points of a crown; again, put the middle point toward the back left:
Or, finally, you may have three gray edges with two of them separated from the third, like an arrow. Let the tip of the arrow point to the right (or, again, the middle point to the back left):
And now do the algorithm—it's the same algorithm regardless—F R U R' U' F'. Seem familiar? Yes, it's a slight variation of the "yellow cross" move on the cube-shaped puzzles, and it does basically the same thing, only pentagonally. Like F U R U' R' F' on the cube, you may have to do it multiple times to get the desired gray star.
Also like that move, I find it crucially important to say the notation aloud so that I don't mess it up. Because I seriously want to do the cube/yellow cross algorithm, every time, and that's no good here. If it helps, say "fur urf" to yourself on the cube and "fru ruf" (froo roof) on the 'minx.

STEP 8: Gray-side up. I guess you could call this "orient last layer," in the sense that when this step is complete, all the pieces on the top laye will be oriented correctly, even if they are jumbled around in the wrong order. This is a deceptively simple yet dangerously tricky step. Screw up Step 7 and you'll only have to fix a couple pieces. Screw this up, and you'll be redoing practically the whole puzzle. So pay close attention now. (1) Holding the puzzle so you're viewing it face-on with an unsolved corner (i.e. not gray-side-up) at front right. (2) Do R' DR' R DR either x2 or x4, depending on which way the piece is oriented. I find it helpful to chant "down-down-up-up" during this move. DO NOT FORGET to complete that last DR ("up") move even after that corner moves into place gray-up. I mean it. Unless you need a lot of practice and fancy starting the puzzle over almost from the star. (3) Holding the puzzle in place and turning only the top layer, put another unsolved corner at front right and repeat step 8-2. (4) Trying not to break into a cold sweat as each move seems to be scrambling the puzzle, continue in this way until all five top-layer corners are gray-side-up. As a bonus, and as if by a miracle, all the lower layers will be restored to their previous, solved condition. Like this:
STEP 9: Permute top edges. Now that everything on the top layer is gray-up, you'll probably notice that the colors around the side are still scrambled. Don't worry about the corners for now; this step is about the edges. Again, there are a couple of cases but only one algorithm. In Case 1, you no more than one top edge aligns matches its side-color's center at a time, no matter how you turn the top layer. Put a side where the top edge matches its center at front (viewing the puzzle face-on). In Case 2, you can turn the top layer so that two adjacent edges line up with their side-color's center. In this case, you'll also find another adjacent pair of correctly aligned sides, but the fifth side will be the odd color out. Turn the top layer so that edge finds its center, and put that center at front. In Case 3, two non-adjacent sides have edges that line up with their side colors. Hold the puzzle with one of those colors at front and one at back right. Then, regardless of the case, do the following moves: R2 U2 R2' U R2 U2 R2'. It's very likely that you'll have to do this step multiple times, particularly if you have to cycle through Cases 1, 2 and 3. So, get used to seeing something like the following during this step:
Result: top edges permuted.
STEP 10: Permute top corners. This is the final solve, people! It's also a pretty intuitive step, by which I mean it isn't a simple formula; you have to think and use good judgment, and one of the cases conceals a pitfall that can also lead to (cough) additional practice. And it's another move where you need to hold the puzzle in one position throughout (viewed face-on), twisting only the top layer as you move from solved corner to unsolved corner. (1) Put an unsolved corner at front right, then dial it down and out of the way, and dial a dummy corner up into its place. For example, do R' DR' R. (2) Look at the gray corner you just pulled down and out. Determine the colors of the other two sides, which will tell you which edge pieces this corner belongs between. Then turn the top layer (only!) to put the corner between those two edges at front right. Dial that (wrong) corner piece down and out of the way, turning the DR layer in the opposite direction from before; this puts the "wrong" gray corner out of the way and pulls the "right" corner into position to dial up into place. Example: R' DR R. (3) Repeat 10-2 (changing the direction of the DR move each time) until either all five corners are correctly aligned, or you get stuck.

(4) I didn't tell you about this before, but there's a case—when this step starts with exactly four corners out of alignment—where at some point, you run out of correct edges to swap with the next incorrect edge, and you have to start over. This is the perilous moment when, only three or four moves from having the whole puzzle solved, I sometimes botch the whole thing and have to make repairs all the way down to the bottom layer. The key is not to panic. Make the next "dial the 'wrong' top corner and out of the way" move, only be sure to turn the DR layer in the direction that restores the puzzle somewhat, and don't forget to dial a dummy corner up into the space vacated by the wrong corner you'd just pulled out. Then continue repeating 10-2 until the top corners are aligned; and if you haven't botched it, you may only need to twist the top layer a 1/5 turn or two to complete the solve. Here are some illustrations of this crucial but complex step: First, here's the gray-blue-orange corner being pulled down off the top. (Believe it or not, the color at front is light green. Again, sorry about the overexposure.)
And here's that same corner being pulled out of the way, to the lower left.
Next, the blue, pink and (I think) light-green "dummy" corner is rotated up into that slot—an important step because otherwise, the adjacent top edge and corner on the right layer would also be knocked out of place.
Then, after twisting the top layer so the corner between the blue and orange edges is at front right, you twist that corner down ...
And you swap the gray-orange-blue corner into its spot:
And dial it inup to the top layer where it belongs:
Here, without further commentary, are photos of the third corner being twisted into front right and pulled out of the top layer and the second corner being pushed back in where it belongs:
The last corner goes into the spot where you stuck a dummy corner, earlier:
And behold, as you dial the last top corner into its proper place, everything else goes with it (give or take a final top-twist):
In summary:
  1. White star
  2. First layer
  3. Lower-tier edges
  4. Side valleys
  5. Side peaks
  6. Orient top edges
  7. Orient top corners
  8. Permute top edges
  9. Permute top corners
. And you can do all those lower- and middle-layer edeges and corners with right- or left-handed moves that are probably better learned by understanding what you're doing than by memorizing algorithms (i.e. reasoning not rote). The only algorithms it really pays to get down by heart are the case where you have to rotate a corner piece (R U R' U' x2) and F R U R' U' F'. Even the formulas for Steps 8 and 9 are probably better remembered by muscle memory (or "down, down, up, up," etc.) than by strings of letters, and of course the denouement is a magnificent move in which you don't have to memorize any algorithm, instead using your reasoning skills and an understanding of how the puzzle works to complete a solution.

Steps 7-10 above are different from how I first learned to solve the Megaminx. I probably wouldn't like it as much, and might even have quit playing with it by now, if I hadn't come across a different method. What was presented to me as a beginner's method was so complex that I needed a cheat-sheet to remind me of which way to turn the dodecahedron at each step, and different algorithms for each case. I am so glad that I kept shopping around until I found a method that (for example) required only one algorithm for Steps 7 and 9 and that gave me an intuitive handle on Step 10. So my final advice on this wonderful toy is: Don't give up if my instructions, or anybody's instructions, don't work for you. Keep searching and you may find the answers your need.