The problem is too difficult for ___ to work out

The problem is too difficult for ___ to work out

Could someone explain what the differences are between filling in the blank with the following options? Do the meanings perhaps overlap?

• everybody
• anybody
• somebody
• nobody

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'