I disagree. A full third of all numbers are of the form 3x+1, but for numbers in that general range, only about 1% of them are perfect squares. Thus, the fact that all of the liked numbers are perfect square is far more significant than the fact that all of then are 1 mod 3.
Factors of 10,000 , or similarly 2^m5^n, are comparable in rarity, but are also a lot more arbitrary: One could just as well ask about numbers that are factors of 86753, or of the form 3^m7^n.
Agree w Chronos. Going back to my earlier post, the “right” or “best” answer depends critically on who the intended audience is.
Assuming it’s written for the slightly math-curious general public, IMO that means somebody who barely passed algebra 1. At best.
While 3x+1 is a lower-order polynomial than is x^2, I’ll argue it isn’t “simpler” for the metric the audience would apply. IMO x^2 is just “tricky” enough, without being too tricky.
Ultimately determining the “best” answer to any problem amounts to solving a multivariate meta-problem. The answer must be correct enough, exclusive enough, and obscure enough. Without being too much of any one of those things. Much less too much of all of them.
For well-written problems there’s a pretty obvious outlier in meta-problem space. Pick that one. For crappily-written problems the available answers and the rationale for them form a tight cluster in meta-problem space. Throw a dart; it’s the best you can do. Ideally aim your dart at the test-author’s eye; it’ll teach 'em a lesson. 
Oh, and
You can silver a piece of glass.
Why should the liked numbers be the rare ones? Why not have the disliked numbers be the rare ones?
Or, focusing on rarity of liked values, we might furthermore note the condition as “Squares next to multiples of three”. Now none of the provided answers work. But before we see them, why not consider this criterion? Should 4 be liked or not? It is a perfect square, but it’s not divisible by 100, which certainly seemed significant so far. Who knows? There’s no objective process of extracting a rule from such cases.
These questions rely heavily on “You think the way I think”; there’s no way around that. The very fact that some people find one possible extension of the rule most compelling, and others find another most compelling, is the brute empirical fact that there is ambiguity here.
Is the criterion rarity? Well, being one of either 100, 400, 1200, or 2500 is a very rare condition indeed!
Perhaps it is simplicity? 3x + 1 is simpler than x^2, in the same way that addition is simpler than multiplication, in the same way that being even vs being odd is considered very simple, or in the same way that simply reading off a last digit is very simple indeed.
Perhaps it is just what is most striking to most people in our contingent world? Base ten may be arbitrary, but a lot of people use it a lot, to say the least, and find “Is a round number, in the sense of evenly dividing a power of ten (thus, having a terminating decimal expansion for its reciprocal, etc)” more immediately striking than some non-obvious observation about the square root.
But what it really is is just “Do you think like me?”. There is no objective answer, intrinsic in the question; there is only the luck that you happen to strike upon the answer the grader decides to mark against.
Think of it statistically: If his criterion is just the number’s value mod 3, then it’s a truly remarkable coincidence that all of his stated liked numbers happen to be perfect squares. If, however, the criterion is being a perfect square, then all of the numbers being 1 mod 3 is still a coincidence, but a much smaller one. Unless one puts extreme weight on the mod 3 thing in one’s priors, then, it should be considered more likely that the condition is perfect squares.
The question of one’s priors, so to speak, IS the ambiguity here. Because again, even more rare than squares are elements of {100, 400, 1200, 2500}. Clearly, rarity of condition is not actually the criterion for plausibility of condition. So there is some other rubric one has in mind; some prior distribution on various conditions being the right one, if you like. And in the space of all possible priors, some will make different choices. There is no designated objective prior. This becomes again “How well can you guess what kind of thing the test designer is thinking?”.
But my point is that, while there is some uncertainty as to the priors, we are not completely in the dark about them. For instance, while one possible criterion for numbers that he likes could just be that he has a list of four specific numbers that he likes, there are many possible lists of four numbers that he could like. Even if we restrict it to numbers below some cap, there are a great many of them. And we have no reason to assume, a priori, that any such arbitrary set is any more likely than any other, and so the prior assigned to every such arbitrary set is extremely low.
Sure. I mean, I do find {100, 400, 1200, 2500} fairly unnatural a rule. But when it comes to judgements as of x^2 vs. 3x + 1 (vs. 100x^2 vs. …), I don’t feel the matter is at all clear-cut. I don’t see why, in the presence of debate on precisely this point, we should suppose it The One Right Choice to exclusively consider priors which assign x^2 a higher value than 3x + 1. That’s all. Anyway, I’ve already said all I keep saying, so I can shut up about it now.
Bah! I suspected you might be able to “silver” something but I couldn’t think of an example.
If you’ve used a soldering iron, you should tin a new tip. Copper, like silver, is also a verb. Ditto lead. (#11 in the 2nd def.)
Gold’s verb is “gild”. That just leaves cobalt and mercury among the non-“ium” metals that isn’t verbed. OTOH, mercury does have other forms like mercurial, for example. No one is ever cobaltial.
You can also tin vegetables, though the synonym “can” is more common.
EDIT: Tungsten (AKA wolfram) is also an non-ium metal.
Whoops, Tungsten. Thanks.
Anyway, I think the “density” people who favor a lower density answer are ignoring basically half of the original problem!
“John likes 400 but not 300; he likes 100 but not 99; he likes 2500 but not 2400. Which does he like?”
This suggests that there are both liked and disliked numbers and furthermore they might be ~evenly matched in some way.
This chucks the squares density argument right out the window. And even the 3x+1 idea might be overly sparse.
Any proposed solution to this that ignores any special properties of the disliked numbers may not work.
Indeed: