There are good answers already, @Matheus Jahnke's as an example
And on its basis, we can ask and have to answer another question, before we get back to the original one.
Note that 400 is 20 standard valuations, this is the chance of that happening is 2.75362E-89. Even if every atom in the universe(estimates from 1E+78 to 1E+82) had an IQ that followed human distribution, it still would be hard(1E+82 * 2.75362E-89 = 2.75362E-7) to find an atom with an IQ higher than 400
Horrific number, but we should understand that rarity does not mean significant superiority.
Let's say we talk about a human who will do all\any of your typical intellectual tasks 10 times faster than an average guy. Can it fit the role of that 2.75362E-89 chosen one? I would say, yeah, good enough. no? let's make him 100 faster - I bet there was no one that fast, in human history, for the last 4 billion years.
Can such a person replace the intellectual work of millions of people over the last 100 years - no. Sure he will have certain advantages, besides the speed, if we think about his judgments, but.
Another one, why 400 points maybe not so much when it is about rarity, how big is a difference between 200 and 400 ones - it may be not that big in absolute results in capacities to solve problems, probably there is some limit about calculation power for a human brain, just because of size of it and how it is build, the closer one is to such a limit rarer is the occurrence in nature/population. it is typical lim(smartness(to improbable)) = 10x as an example.
So intelligence in one meatbag can be qualitative and quantitative. quantitative can easily be measured, there may be some problems in assigning an exact number, but it easily measurable, and more than that can be attributed faster neural activity and be natural or artificial.
Can this guy outsmart you each and every time - not necessarily, will it be faster at everything than you - yes. Can it from scratch overnight reinvent the theory of general relativity - no.
- there are suggestions of "show, do not tell" - those are not great recommendations for reason that author has to understand the difference of possible and impossible, so as to have some measuring stick for what his character can or can't do. Because without having one, they often fail miserably on that "show" part.
qualitative superiority is harder to assess. But also if it is quantitative changes which in a combination become qualitative ones - like speed, memory, senses - then there is a chance to do some assessments. And I would say it quite a typical trope.
- chimps can count to 9-10, distinguish that number of objects and do that faster than a human does, so as they do have some treats we have - social interactions, linguistic aspects, knowledge transfer, etc - not a surprise, relatives after all. They can fly in space, but we have sea ships and airplanes and the agriculture industry - they do not.
10 times of speed, 100 times of typical memory - those can be measured, especially when we talk about human-type intelligence. Speed and details of visual perception, pattern recognition.
with pattern recognition, we can create some calculation difficulty measuring stick, as we can compare to not only humans, but to artificial algorithms as well, and thus assign some corresponding measurable value - iFLOPS or something.
Sooo, a character with everything is 10 times of usual norm. Can this guy outsmart you each and every time - probably, will it be faster at everything than you - yes. Can it from scratch overnight reinvent the theory of general relativity - no.
that general relativity stays no, and it will stay no for a very very high level of intelligence, despite it being invented by not so much out of ordinary(compared to the individual we discuss here) guy. layman example - let's go back to chimps - are we superior to them for the most time in therms of intelligence - yes, do we understand them fully - no, not a chance. we do understand them to some degree, but it is far from a full understanding and ease of manipulating them.
one can build a sound comprehensive theory encompassing the whole universe but the number of such theories is more than one, probably, at least we have more than one spacetime theories and we not sure which one is the correct one, if any. Sitting in a building one can do observations(thing around that person - sunshine, the wind blows, grass grows, etc) and it helps build and sort out theories that are not related to this specific universe. it like finding a proper book that fits the universe in The Library of Babel. And there are things which you can't figure out just because you like to do so, you have to measure or to test, do experiments to figure out their relations with each other. At the same time, you have to be able to produce that or similar Library of Babel and somewhat keep many books which later you sort out as you may find them to be not correct. it requires way more than 10x, 100x human intelligence.
this is the reason why one can't invent a super-duper wave gun sitting in a cell, to make an escape. Most likely escape strategy will be more mundane. Maybe smart, but not 400 IQ point smart, or you can't understand smart.
higher the intelligence is, then fewer experiments it may need, and further one experiment propels that process of universe model building. Maybe a stretch, but we can imagine an intelligence that glances casual look at your city and identify the universe in that Library of Babel - one observation act and firm and correct conclusion. But human measuring stick won't fit that intelligence, we as collective as humankind - maybe but barely.
There are examples of superior intelligence we know, as since not so long ago it became a perceivable reality, which you can touch and smell.
one is OpenAI Five - it plays Dota better than anyone on the planet
another AlphaGo - plays Go better than anyone on the planet.
Those are not general intelligence, obviously, but are they faster - yes they are, do they have a bigger memory, task-related - yes, they do. Do we understand what they do - yes, somewhat, as an afterthought, for those guys which do play those games. we have a hard time understanding how they come to the conclusion, what specifically did they noticed, we may miss bits and pieces, but our understanding isn't a total blank.
So, to outsmart, your guy does not need to be magical, breaking crypto as one suggested, it just needs to be one step ahead - that is a typical trope with strong roots in a reality, not surprisingly, as there are plenty of humans who are smarter than other humans, so it is not that we do not have a taste of that medicine.
impossible things stay impossible - if it is proven you need the energy of the whole universe to crack the encryption even with quantum computing, then it stays that way for 10x, 100x not matter what. Social engineering is a way much better way to fish things you need - passwords, money, pin codes, call to a president to request a nuke launch, messing with global politics - whatever you choose.
- there are examples of prankers that did call higherups in political circles fooling the whole chain through which the call has to go, and representing themselves as other party higherups.
So to wrap it up - distinguish impossible things and those which come from a better knowledge and ability to operate such to be a step ahead of situations, events, people, etc.
And you measure things as usual - by decomposing elements which combined create that intelligence. So in your case, you do it not with a general test, but each aspect you can separate in a separate way, as we do tests for animals or humans. So just more tests that are focused on their own aspects and as a whole are broader more encompassing measure, as we do with animals in our attempts to understand their intelligence, shape and strength of their intelligence, not a synthetic test you typically do to get your's 120 free internet points.