This was a relatively easy puzzle and many people sent a correct answer. That included Siddhartha Goel, Suman Saraf, Pratik Poddar, Traveling Salesman (which I believe is a nick name), Narsimha Pai and Salil Panikkaveettil.
The puzzle used a popular and intuitive principle called the Pigeon Hole principle. At a simplistic level, it states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. This surprisingly simple principle is used very widely and finds many applications.
I am taking the liberty of reproducing Siddhartha Goel’s answer for this puzzle:
1) n – eaters, n – entrees
2) no duplicate entrees etc: each user has a 1:1 match with 1 entree
3) A permutation can be rotated ‘n’ times and a total of n matches (eater – entree) will happen in ‘n’ rotations
4) If in 1 permutation we have 0 matches => in remaining (n-1) rotations of that permutation there needs to be a total of n matches => at least 1 rotation of the permutation will have at least 2 matches… (0, 1, 1, 1, … 2)…
For people, who are interested in a more detailed treatment of Pigeon Hole principle, please refer to https://en.wikipedia.org/wiki/Pigeonhole_principle
Hope you enjoyed the puzzle!