The solution to The Cube in the Cavern is not too bad once you know what to do. The puzzle is randomly generated each time, and there are forty-eight possibilities once you understand what you can put on the transponders.
You may notice that SUMMONing changes your mood ring to different colors. White, red, yellow and blue.
Also you may notice that the beacon colors in the center are red, yellow, blue, orange, green and purple.
You may also notice, by trial and error, that adjacent transponders cannot have colors other than black, so you have at most four transponders.
That should give you an idea. If you can combine colors right, you may be able to match all the transponder colors and combinations with their beacons.
Now there are two ways to do things.
You can probably just pick one color and guess the rest randomly. For instance, if the upper cube is red, then one corner is red, and the one opposite it is white. But there are four possibilities for where to place them. Then you can figure out the other two.
But you can be much more exact. Let's try something here.
Note the initial color. N.D.D gets you to another beacon. W.W.S gets you to another. Note the three colors will all intersect with the color of the northupwest beacon.
If they are, in order, red (up), purple (north), and orange (west), then these colors have an intersection of red, and thus the northupwest transponder will be red.
The northdowneast transponder will then be blue, since red + blue = purple, and the southdownwest is yellow, for orange.
That leaves the southdowneast transponder as white.
But if they are, for instance, yellow, green and blue, then there's no common color to these. Yellow = white + yellow, green = yellow + blue, blue = blue + white. (Note yellow/red/blue would have the common color white.)
That means the upnorthwest beacon should remain black. The north face must be yellow and blue, and the upper is yellow and white. That means the upnortheast transponder is yellow. The northdownwest is blue, and the upwestsouth is white.
That leaves the downsoutheast transponder as red.
There's a way to figure things out by exploring, too, once you know how the game works. Once you know the beacon colors on three non-opposing planes, you know the other three. Orange always opposes blue, purple opposes yellow, and green opposes red. You can notice this through experimentation or by noting that two opposite planes each have two transponders to color, and they must cover all four colors. There are three ways to pair off ring colors: red/yellow and white/blue, red/blue and white/yellow, and red/white and blue/yellow. Then you can note where, say, the red/yellow/blue planes meet, and that must be white. The red/yellow/blue transponders must be kitty corner from white, on the planes with beacons colored red, yellow and blue, respectively.
Now for the second part. You need to DROP ROPE at the start of a tunnel on the surface. You can go from any one.
You can enter at any hole where a beacon was. You need to go through all three perpendicular tunnels. But you need to go through a certain way. You may have noticed the rainbow. You can start with any color, as long as you go in the red-orange-yellow-green-blue-purple order or its reverse, but you can't go twice into any tunnel. Once you've been through all six, the game will see if you went in a rainbow-ly pattern. Go back to the start of your journey and TIE ROPE. I hope this is not too bad.
Note that you can also NEVER go straight in the tunnels with your rope.
An example for the above would be D.D.N.N.U.S.S.E.E.N.W.W.U.U.E.
You get a small set of clues if you cover all 54 squares. This is not too bad to do, even if you hate keeping track of things. From the up-face center you can go N.N.D.D.D.S.S.S.U.U.U.N to cover 12 squares on the up, down, north and south planes. Shifting west or east covers 12 more. To cover the east/west faces from your starting point on top of the cube, you just replace N/S with E/W.
The message varies, but that doesn't depend on the original configuration.
There is also one other way to clue what to do to get different messages. You need to weave your way through each square on the surface once without touching another twice, or using diagonals. The PATH command can track this, though it's not mentioned in-game. (Yes, that's a trivial bug to be fixed post-comp.) I think it's also a neat puzzle to figure out in general.
S W N N E E S S is a start. You can then cover each cube with an S-motion--well, all but one. The second-last should be, for instance, N N E E S W S E--you need to break from the usual S-maneuver. The reason for this is that if you just use a basic S everywhere, you go from one corner of the cube to another that is non-adjacent.
Here is a proof with examples. You may not find this relevant, but this bit of math popped out, so I hope you enjoy it if you're interested.
Graph theory can express covering each face with an S-move as 4 vertices all with 3 edges. For instance, you'd have the NWU, SEU, SWD, and NED vertices, and each edge would be a face of the cube you could traverse with an S. However, this can't be connected in a semi-Eulerian path, for which you must have all but two vertices with an even number of edges.
In plainer English, that means you can't draw over all the edges below without getting into a dead end. And because the graph is small, we can also note that no matter what the first two edges we take are, the room we visit after the first edge is a dead end, so our journey has to end there. But after another move, that leaves the *next* room as a mandatory dead end. We have two mandatory dead ends.
NWU---SEU
| \ / |
| x |
| / \ |
SWD---NED
Or, if you started elsewhere,
NWD---SED
| \ / |
| x |
| / \ |
SWU---NEU
(The X isn't really a meaningful intersection. It just would've been more awkward to draw a long loop from SWD to SEU.)