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

Let $f(n)$ be the minimum number of ill people on day $n$ (sequence $1, 2, 4, 4, 6, 8, 8, 8,\ldots$ if we start with $n=0$), let $g(n)$ be the maximal number of days the illness can be kept $\le n$ (sequence $0, 1, 2, 2, 4, 4, 5, 5, 8, 8, \ldots$ if we start with $n=0$), so that $g(n)=\max \{k\mid f(k)\le n\}$ and $f(n)=\min\{k\mid g(k)\ge n\}$ holds.
We want to show for $n\ge 1$ that $g(n)=r(n)+1$ where
$$\tag1 r(n):=\left\lfloor\frac n2\right\rfloor+\left\lfloor\frac n4\right\rfloor+\left\lfloor\frac n8\right\rfloor+\ldots$$
is the exponent of $2$ in $n!$.
By inspection, we see that $g(n)=r(n)+1$ holds at least for $n\in\{1,2,3,\ldots,9\}$.
Lemma 0: $f(n)$ is even for $n>0$. If $n>1$ is odd then $g(n)=g(n-1)$.
Proof: Obviously, after patient $0$ being the only ill person inititally, the ill persons come in pairs as per the last box they were in. The statement about $g$ is dual to this.$_\square$
Lemma 1: $f(n+f(n))\le 2f(n)$ and $g(2n)\ge g(n)+n$.
Proof: Make sure that $f(n)$ people are ill on day $n$, enter $f(n)$ fresh people and let them cobox with the first $f(n)$ people in a circular manner fo the next $f(n)$ days. The statement about $g$ is dual to this.$_\square$
Corollary: $g(n)\ge r(n)+1$ for all $n\ge 1$. $f(n)\le \mathrm{A007843}(n)$.
Proof: Observe that $r(2n+1)=r(2n)=r(n)+n$ so that lemmas 0 and 1 make induction on $n$ work. The claim about $f$ is dual (using the description at OEIS).$_\square$
Lemma 2: $g(n)\le n$. 
Proof: Just note that patient 0 cannot coboxitate with more than $n-1$ other people. $_\square$
Corollay: $g(n)=r(n)+1$ if $n$ is a power of $2$.

(to be continued)
