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There are four options: everybody, somebody, anybody and nobody. Which one should be used in X place ?

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    Nobody would never be acceptable. Of the other three, somebody is unlikely, but just about feasible in certain contrived contexts - or maybe when somebody is a facetious usage meaning you (the person being addressed). The difference between using everybody and anybody is General Reference (if you use everybody, you're implying that some, but not all people can work it out; if you use anybody you're saying there aren't any people at all who can work it out). – FumbleFingers Nov 3 '13 at 16:22
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As FF has pointed out, only everybody and anybody are acceptable
(everyone and anyone also work). They work, and the others don't, for the same reasons:

  1. Too is a Negative -- too Adj to VP means 'so Adj that Not VP';
    therefore NPIs like anyone work within its scope,
    and so anybody is used here, instead of somebody.

    That's also why nobody is ungrammatical here, because that would mean
    *The math problem is so difficult that nobody can't work it out.
    which is ungrammatical (that's what the "*" means).

  2. Everybody, somebody, and anybody are all Quantifiers -- logically, and .
    By DeMorgan's Laws, propositions with both a negative and a quantifier are ambiguous.

    Note the opposite orders of not (logically ¬),
    compared to some/any () and every () below.

    I.e, Everybody doesn't like it
    can mean either
    It is true that there is somebody who does not like it.
    ¬(∀x) φ(x) ≡ (∃x) ¬φ(x)
    or
    It's not true that there is somebody who likes it.
    ¬(∃x) φ(x) ≡ ∀(x) ¬φ(x)

So the two choices that are grammatical do not mean the same thing.

  • The test is too difficult for everybody to figure out means
    'The test is so difficult that there are some people who can not figure it out'
  • The test is too difficult for anybody to figure out means
    'The test is so difficult that there are no people who can figure it out'
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Which of everybody, somebody, anybody and nobody should be used depends on the meaning to be conveyed. If the meaning is that the problem is so difficult that nobody can work it out, then (as other answers indicate) the best fit is:

The math problem is too difficult for anybody to work out.

If, however, the meaning is: Nobody finds the problem too difficult to work out, then nobody fits:

The math problem is too difficult for nobody.

Admittedly, adding back the (superfluous) to work out makes the sentence now sound a little awkward:

The math problem is too difficult for nobody to work out.

So I prefer:

The math problem is too difficult to work out for nobody

with appropriate prosody.

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The fifth option is the pronoun one, as in The problem is too difficult for one to work out.

This usage is synonymous with the usage described by FumbleFingers above of anybody, but I believe is a more likely usage in dialogue or conversation.

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