[Symmetry]

Usually the right next number in a sequence is the one with the lowest entropy, but of course the answer that the post's author says is the "correct" one is far from the simplest way to generate the same sequence of 7 numbers.

[jerf]

Entropy is relative to your encoding scheme plus or minus a constant. Usually you can ignore this fact when dealing with large amounts of information, because almost all human-interesting encoding schemes are related by relatively small constants, but when handed a small set of numbers it suddenly dominates as the constant for very reasonable schemes relative to other very reasonable schemes can exceed the size of the bits given to you in the form of the original numbers.

This argument basically collapses back down to, no, there still isn't enough information in a short sequence to identify the next number truly uniquely.

[Symmetry]

Yes, a function's Kolmogorov Complexity[1] does indeed depend on the encoding, but you would certainly have to go out of your way to find an encoding where the official answer can be expressed more easily than "f(n) = pi/2".

[1]http://en.wikipedia.org/wiki/Kolmogorov_complexity I didn't use the term because I didn't think most people would recognize it, but we're getting technical now.

[jerf]

First, I was speaking to the general case.

Second, for any given function there exists an encoding for which it is simply the shortest possible string, so no, you don't really have to go "out of your way" to find such an encoding. Trying to create criteria whereby that is somehow "illegal" or "cheating" hits some rocky shores very quickly.

[Symmetry]

By having to go out of your way, I meant that every encoding I could think of, whether English or Python or turing machine would rank these two solutions in the same order. In fact, I would be somewhat surprised if there were any encoding that had been used by more than one person in all of human history that would rank the algorithm offered by The Big Questions as simpler than pi/2.

Using unusual encodings like that isn't cheating in general, but when you play a guessing game with other people like the blog author did then social norms start to enter into the picture, and in this case I think it would be reasonable to say that the author did cheat. Either that or, for the reasons jl6 outlined, it was entirely meaningless.

But this isn't entirely confined to games people play. The only difference between the hypothesis that General Relativity is correct and that General Relativity has always been correct but in 5 seconds the rules of the universe will change" is the complexity of the two statements, and I don't think that the fact that you can specify an encoding where the second is simpler is going to convince them that its a equally valid scientific hypothesis.

[dedward]

or any sequence, no? not without context.

[dedward]

"right" needs a definition. true enough for these teasers in many settings, but mathematically, irrelevant, no?