TITLE: Interesting tea-time problem
QUESTION [5 upvotes]: Problem A: Please fill each blank with a number such that all the statements are true:
0 appears in all these statements $____$ time(s)
1 appears in all these statements $____$ time(s)
2 appears in all these statements $____$ time(s)
3 appears in all these statements $____$ time(s)
4 appears in all these statements $____$ time(s)
5 appears in all these statements $____$ time(s)
6 appears in all these statements $____$ time(s)
7 appears in all these statements $____$ time(s)
8 appears in all these statements $____$ time(s)
9 appears in all these statements $____$ time(s)  
Note: they are treated as numbers, not digits. e.g. 11 counts as occurrence of 11, but not two 1.
EDIT
How do number of solutions behave, with respect to number of statements there are? I need a sketch of the proof.

REPLY [3 votes]: For the revised B there are exactly 3 solutions consisting of single digits: $173311121291$, $174121121291$, $191311111391$.  No single-digit solutions were found to the original B, and the solution to A is unique within this class.