Solution

It suffices to show that there can't be a constellation of 16 people so that every table position would result in exactly one hit. For all other cases the pigeonhole principle can be applied like in Dinner for 15.

Definitions

offset of a guest: distance to his/her intended seat (clockwise)
confusion of a constellation: sum of all offset numbers
When all guests sit on their intended seat, the confusion is 0.
For any other constellation the confusion always must be a multiple of 16. (Try to get this clear before you read the proof.)

Proof

Assume that C is a constellation where every table position results in exactly one hit.
Obviously no two guests can have the same offset, because then the table could be rotated in a way so that these two persons were seated correctly. So the offset numbers must be 0,1,2,...,14,15.
The confusion of C therefore is 120 and this is no multiple of 16. Thus C cannot exist.
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