Guess the number from its different base representations
Let's start with a simple algorithm. Go over all possible interpretations of the first string (using all possible bases). For each such interpretation, go over the remaining string, and check whether they equal the interpretation of the first string for one of the possible bases. If there are $m$ possible bases and $n$ digits in total, then the running time is $O(m^2 n)$, ignoring the cost of arithmetic.

We can improve this as follows. Suppose for simplicity that all strings are at least two digits long (if not, we know the value of the number, and can finish in $O(nm)$ time). Compute all possible interpretations for each string, put all of these $nm$ numbers together, and sort them. Check whether the sorted list contain $n$ identical elements. This algorithm takes $O(nm \log (nm))$ time.

Another improvement, with a slightly better running time, modifies the original approach using binary search. Go over all possible interpretations of the first string. For each such interpretation, go over the remaining string, and for each one, use binary search (on the base) to find whether they equal the interpretation of the first string for one of the possible bases. This algorithm runs in $O(nm \log m)$ time.