Partition the natural numbers into countably many sets, $\{S_i\}_{i=0}^\infty$, where each $S_i=\{n_{i_1},n_{i_2},\dots,\}$ is countably infinite. Since we have countably many mathematicians, we may list them, and assign $S_i$ to the $i^{th}$ mathematician.

Thus the $i^{th}$ mathematician is assigned the sequence $u_{n_{i_j}}$, for $j=1,\dots$, and this sequence lies in some equivalence class. There will exist and integer $M_i$ such that for all $j>M_i$, the sequence $u_{n_{i_j}}$ is equal to its chosen representative. Applying the strategy as we did for the previous problem, the goal is to have mathematician $i$ guess an integer greater than $M_i$ by looking at every box except those in the set $S_i$, for all but possibly one value of $i$. If the sequence $M_i$ is bounded, then the problem is solved. The difficulty is handled an unbounded sequence $M_i$. 

Now, the sequence $M_i$ lies in some equivalence class of real numbers, and every mathematician playing knows what this equivalence class is. There exists an integer $N$ such that for every $i>N$, the sequence $M_i$ equals its chosen representative, say it is the sequence ${v_i}_{i=0}^\infty$. The strategy for guessing is as follows: Assign to mathematician $i$ with $i>N$ the integer $$H_i=1+\max\{v_i,M_{i-1},M_{i-2},\dots,M_1,M_0\},$$ and to each mathematician with $i\leq N$ integer $$H_i=1+\max\{M_{N},M_{N-1},\dots,M_{i+1},M_{i-1},\dots,M_1,M_0\}.$$ Then we must have $H_i>M_i$ for every mathematician except possibly one.