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".