Many students have the attitude that a problem must be solved or a proof constructed by an algorithm. They become quite uncomfortable when faced with problem solutions that involve guessing or conceptual proofs that involve little or no calculation.

**Example**

Recently I gave a problem in my Theoretical Computer Science class that in order to solve it required finding the largest integer \(n\) for which \(n !<10^{9} .\) Most students solved it correctly, but several wrote apologies on their paper for doing it by trial and error. Of course, trial and error is a method.

Example 2 Students at a more advanced level may feel insecure in the case where they are faced with solving a problem for which they know there is no known feasible algorithm, a situation that occurs mostly in senior and graduate level classes. For example, there are no known feasible general algorithms for determining if two finite groups given by their multiplication tables are isomorphic, and there is no algorithm at all to determine if two presentations (generators and relations) give the same group. Even so, the question, "Are the dihedral group of order 8 and the quaternion group isomorphic?" is not hard. (Answer: No, they have different numbers of elements of order 2 and 4.) I have even known graduate students who reacted badly to questions like this, but none of them got through qualifiers!