I remember that from when I was a kid.
Hmmm. I think that's too much for this; it'd take too long.
I think that's relative... how many tech problems have you solved in a crisis?
How long did it take? How many D&D turns was that (at 6 seconds a turn)?
TV makes those engineering scenes go quickly, but the actual pacing is minutes if not hours.
Mastermind (good observation Umbran) plays with 4 colors in 4 positions if I recall. Your tech list has quite a few more combinations...
I agree, it might take to win a game of Mastermind (10-20 rounds) might be longer than other players want to wait, but verisimilitude wise, it's kind of spot on for what a real engineer is doing. Which is trying changing one variable at a time, from the last test for a solution.
So we might have found a neat idea for simulating "problem solving engineering" but it's not necessarily "fun" for all players to sit through.
That's where I thought my tile idea would help, because it's tangible, and as a player, I can sort out valid combinations faster with my hands and eyes until I present a plausible solution on the table to the GM. Likely in real time minutes, which given 30+ seconds per player to take their actual turn, means it could be solved by the time you ask me what my PC does.
From a "feel" standpoint, I still think it is important that the player choose the combinations of the technobabble. It helps the player feel like they are "inventing" the solution. If you just say "make a science check" and I roll a 15, so you tell me "Rehydrating the fermostat with tachyon bursts solves the problem" all you've done is created a random NPC Name generator for tech.
Personally, if I play the Chief Engineer on a Star Trek game, I want to feel as if I am actually solving problems like on the TV show. That means some semblance of understanding the parts involved and how they connect (which is really just lists of nouns and their relationships and possible alternative connections). Maybe that's a goal for somebody else's game (not Morrus's thought experiment), but it'd be nice if some of that idea is reflected.
I do like Morrus's idea for custom sheets for topic/setting (steampunk, medical, sci-fi), that links to possibilities on what I'd like the idea to do.
Maybe a different viewpoint might be needed. Consider what is actually going on in a Trek engineering scene:
a good chunk of the time, it is about Rerouting Something From the XYZ to the ABC
Which is really trying to find a substitute from what we have, to fill in a gap on something we lack.
For instance, caught in a gravity well, we don't have enough power to break free. So we are lacking "more power". Somebody will think of alternate sources of "more power" like Life Support and something else, and viola, we have a plausible solution.
Another common tech problem is trying to find the right pattern, signature or energy type to satisfy the anomalies need/trap.
We might have tetryon, tachyon, neutrion sources. We need to figure out which one will work to satisfy the need.
Sounds like the same pattern I described previously. We need X, we have X,Y, and Z. Pick one of those to fulfill X. (gee, thats X).
So let's game that up.
Technologically, the parts of a machine are about inputs and outputs. Needs and Haves. We have stuff that outputs energy (by type, blood, water, steam, radio frequency range, mechanical force). And stuff that needs those energy types. Generally one thing supplies an energy type, another thing needs it. You could get fancy and assume that each thing has an input, and each thing has an output, but let's get to that later.
Let's say that We codified all the energy types by color (ex. Blue=electricity, green=gamma radiation, red=tachyons, black=neutrinos, white=plasma). I just made this stuff up, don't quote me on any of it.
Now we make a tech list of Things that either Supply or Need X units of a given color. We also make up some Inverter or Combiner tech that can flip the color or combine 2 colors to make a new one (I chose RGB for a reason, I ain't stupid). Which means we might have tech that needs Yellow. Or Cyan.
You ship should have the obvious parts that need obvious power sources (the engine, the warp core, the dylithium chamber, the deflector dish, etc). Throw in some inverts and Combiners as spare parts and the players should be able to mix and match things until they satisfy the Warp Core's need for 5 Purple when the Dylithium chamber is damaged (and thus not supplying 5 purple directly anymore).
Or when the space Leech is slurping off our hull. It turns out Yellow energy repels it. How do we make some of that using the parts we have?