TITLE: What is the $3$-SAT problem?
QUESTION [45 upvotes]: In the hopes of improving my knowledge on the question, could someone outline the inputs and outputs for the 3-SAT problem? It would also be helpful if you could express how this problem differs in structure to SAT, 2-SAT or 4-SAT problems.

REPLY [63 votes]: There are a bunch of teddy bears A, B, C, D and so on that are red on one side and blue on the other! (You choose how to color them)
AND there are a bunch of 3 armed aliens with really long arms. Each alien grabs 3 teddy bear hands!  (A teddy bear hand can be grabbed by more than one alien.)
3-SAT is the problem of whether you can color the teddy bears such that every alien is holding at least one blue hand!