# A Singular Application of Levels of Recursion

A friend of mine recently showed me the following question which I believe can be found online somewhere (besides here):

If an answer to this question is chosen at random, what is the percent chance that it will be the correct answer?

A. 25%

B. 15%

C. 50%

D. 25%

There is actually nothing wrong with this question.  If one looks at it at the trivial case level, it actually doesn’t have an answer, and therefore, an answer must be assigned arbitrarily in order to see the rest of the system work its way out, thus any answer given is ultimately arbitrary.  The question is, in this sense, like asking “What is the correct answer to this question?” which is really just nonsense.  However, ignoring that, lets suppose we assigned our trivial case the answer “B. 15%.”

This selection, while creating an arbitrary answer on this level, the trivial case, causes a relative correct answer of A or D on the next, lets call it the second, level of recursion.  There being two correct answers on the second level of recursion makes C the right answer on the third level, and thus on the fourth level we are back to A or D.

This is not a paradox, it is just a matter of an indeterminate level of recursion, which I find, as you probably could have deduced from the title of this website, quite fascinating.

Of course, the absolute answer to this question is that is does not have an intelligible answer anymore than does the aforementioned question, “What is the correct answer to this question?”  However, if we assume the trivial case for no reason (i. e. we chose it trivially˚), then I think the most convincing answer would be the “infiniteth” level of recursion, which, because the system has no limit, no end behavior, would be best put into the words “none of the above.”

In a later post, I might well invent a less trivial application or come across it by necessity; I just found this one interesting.

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˚ O dear.  I’m really not that funny am I.