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I see this sentence in this online article:

The following answers to this artful question each win a random book.

It seems ungrammatical to me because win should be a present tense verb in third person ending with s.

If I am wrong, then what is the rule applied here?

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The following answers to this artful question each win a random book. 
Each of the following answers to this artful question wins a random book. 

Only a handful of adjectives in English mark number.  When it is an adjective, "each" is singular.  It remains singular even when we treat it as a substantive adjective or an adjectival pronoun. 

In your example, we have reason enough to treat "each" as an adverb.  We can rephrase the sentiment of your original in this way: 

The following answers to this artful question individually win a random book. 

It's flexible placement is further evidence of its adverbial nature in context:

The following answers to this artful question win a random book each. 

As an adverb, "each" has no number and does not influence the number of the subject.  The keyword of the subject remains "answers".  The verb that agrees with this plural subject is "win", not "wins". 

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The word answer is a noun in that sentence. The main part of that sentence is : answers each win a random book.

Compared to: We each have our own skills.

  • Oh, fascinating.. So the answer can be found here. – Lerner Zhang Oct 28 '18 at 1:51

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