Combinatorics Problem: Box Riddle A huge group of people live a bizarre box based existence. Every day, everyone changes the box that they're in, and every day they share their box with exactly one person, and never share a box with the same person twice.
One of the people of the boxes gets sick. The illness is spread by co-box-itation. What is the minimum number of people who are ill on day n?

Additional information (not originally included in problem):
Potentially relevant OEIS sequence: http://oeis.org/A007843

Just in case this helps someone:

(In each step we must cover a $N\times N$ board with $N$ non-self attacking rooks, diagonal forbidden).
This gives the sequence (I start numbering day 1 for N=2) : (2,4,4,6,8,8,8,10,12,12,14,16,16,16)
Updated:  a. Brief explanation: each column-row corresponds to a person; the numbered $n$ cells shows the pairings of sick people corresponding to day $n$ (day 1: pair {1,2}; day 2: pairs {1,4}, {2,3})
b. This, as most answers here, assumes that we are interested in a sequence of pairings that minimize the number of sick people for all $n$. But it can be argued that the question is not clear on this, and that might be interested in minimize the number of sick people for one fixed $n$.  In this case, the problem is simpler, see Daniel Martin's answer.
