Regarding Skill Swap Colour Change, unless I have made a mistake concerning game mechanics, I believe the analysis I did actually proves that it cannot work as a solution.
You can see it here.
Could it simply be that the quizmaster was mistaken? That he found a solution he thought would work, posted the question, and then found out it didn't work anyway? I mean, this thread has seen a couple of very thorough analyses exploring every possible option, and found exceptions that counter every possible strategy. A few solutions could work assuming a cooperative opponent, but unless the foe is tailored to be beaten by the chosen strategy, everything seems to fail. There simply is no correct answer to the question.
It is of course possible that there is no solution to the puzzle, but there is not yet proof that there is none.
However, it is also very possible that we are not working on a solution to the actual puzzle that the quizmaster provided, since it was translated from Japanese originally, and a small difference in how the question is phrased might have a large impact on the possibility of finding a solution.
In my original post that I linked above I read the requirement to "be able to hit every Pokemon super effectively" as being able to hit a Pokemon of any species, taking into account all available abilities and legal movesets, for super effective damage. However, I am not sure anymore if this reading of the requirements is possible to fulfill. In fact, I am relatively sure it is impossible, but for different reasons than one might expect: If every legal set is to be taken into account, what about an Electrode with only Explosion? None of the solutions discussed up until now could hit this Pokemon at all, since it will have fainted itself before they even have the ability to act. And to be able to hit a Pokemon super effectively certainly would necessitate to be able to hit it in the first place. This might sound like a bullshit objection, but it shows how careful one must be when wording a question like this to obtain the answer that one would expect.
There are other possible readings, however. Since it was explicitly mentioned that all possible abilities need to be taken into account, that part of the puzzle is not open to interpretation. However, nothing was said about the movesets. That means a few different alternative readings are possible:
1. Find a Pokemon using a set of up to three moves so that there existst a scenario in which it is able to hit an opposing Pokemon of any species, taking into account all available abilites, for super effective damage in a 1v1 situation.
The main difference between this reading and the first is that not every possible moveset needs to be taken into account, there only needs to be one moveset for each different Pokemon that can be hit super effectively. This actually changes the puzzle a lot. For example, Arceus, who usually stops Skill Swap Colour Change, gets Role Play. Under the condition that Arceus uses it this move, a standard Skill Swap Colour Change set will now work on it. However, I believe this reading of the puzzle was already solved by this set:
Kecleon
Ability: Protean
-Skill Swap
-Brick Break
-Flame Thrower
All Pokemon that cannot be affected by Skill Swap are hit super effectively by this set naturally. Everything else learns a normal type move with which to change itself to normal type, and make itself vulnerable to Brick Break, except Weedle, which is hit SE naturally as well.
This reading is sort of unfulfilling, though, since it does not really match up to the way we usually think about the question if one Pokemon can hit another super effectively. For example, if I asked "Can Basculin hit Greninja super effectively?", we would expect the answer no, since Basculin only gets Ice, Water, Dark and Ground attacks (outside of Hidden Power), which hit none of Greninjas types super effectively. However, if Greninja uses Extrasensory, Basculins Dark moves would of course be able to hit it super effectively, provided it is not fainted first.
But all those realities of an actual battle (if or if not certain moves are used by the opponent, if the attacker is fainted before being able to hit the opponent, etc.) are usually not of interest to us. This is why I prefer a different reading:
2. Find a Pokemon using a set of up to three moves so that there existst a scenario in which it is able to hit an opposing Pokemon of any species, taking into account all available abilites, for super effective damage in a 1v1 situation. The opposing Pokemon is assumed not to take any actions.
The difference between 1 and this is that under this reading, you can't rely on the opponent to use any of their moves, so the Protean solution would no longer work.
This reading might sound implausbile, since it is not how Pokemon battles usually work, but it models the way you would usually think about the phrase "a Pokemon can hit another Pokemon super effectively", since usually you do not think about the possibility that an opposing Pokemon might take actions to make itself unable to be hit when thinking about this question. The Electrode example above serves to illustrate this: If we ask, "Can Marowak hit Electrode super effectively?", the answer we would expect is yes, since Marowak learns several Ground moves, which Electrode is weak to. We only want to look at what the type chart would tell us, without taking into account any actual specific battle. Now, if we ask, "Can an Octillery with Soak and Charge Beam hit an Electrode super effectively?", we would again want to answer yes, since after using Soak, Electrode would be vulnerable to Charge Beam. However, in a real battle Electrode would be able to attack Octillery two times before it could attack with a super effective Charge Beam. This would give it enough time to realiably faint Octillery, or faint itself. But this is not something we are interested in when we ask the question. We only want to know if there is any way for Octillery to get a super effective hit, providing nothing it gets to execute all the attacks it wants. This is why I believe that this reading is legitimate, and also believe this reading matches the puzzle to which we are currently unable to find an answer.
To give an actual point to this post (other than making really clear what question it is that we are stumped by):
Has anyone thought about certain aspects of the battle that might help a Pokemon achieve a super effective hit, but that it might not be able to bring about by itself? Weathers, Terrain moves, Wonder Room and the like? A Pokemon might not be able to set them up themselves, but might be able to make use of them. For example, I believe Nature Power under effect of a Terrain move turns into that moves type. This might give a Pokemon the Electric, Grass or Fairy coverage it needs to pull off an SE hit it might otherwise miss.
