At least once, I have seen a posted solution to a problem on Brilliant in which the following argument appears to have been used:
"The problem would not have been set unless it has a unique answer. In order to have a unique answer, the answer must be true for <a specific case of the problem>. Therefore, the solution to the problem [in its general form] is the same as that for <its specific case>."
This idea isn't a new one. Take a look at the Hole in the Sphere problem, which I originally came across in Martin Gardner's book Mathematical Puzzles and Diversions. Both the web page and the book actually solve it properly, but the book also notes this shortcut as a way to solve the problem.
I have used the same technique myself. For example, I solved this problem by noting that whatever the answer is will be true for an equilateral triange, and so was able to reduce the problem to this specific case and thereby solve it.
Once upon a time, there was an exchange on the solution board for a problem that went something like this:
- Person A: <a solution that appears to use the shortcut>
- Person B: Your solution isn't valid, as it makes up specific values for the variables for which no values have been specified.
- Me: I think this person has reasoned that the problem wouldn't have been set unless it has a unique answer, in which case the answer must be true for these values of the variables.
- Person B: So Person A gets the marks for the correct answer, but no marks for the method.
This begs the question: the method for doing what? Arriving at the answer, or proving that the problem is valid in the first place? Must one untie the Gordian knot, or will it do to grab one's sword with both hands and cut said knot?
Here's another example of solving a problem by taking advantage of the fact that there must be a unique solution. Every Sudoku has a unique solution. Some advanced strategy guides mention a 'unique rectangles' tactic. In this tactic, you eliminate a possible value for a square by realising that it will necessarily lead to a non-unique solution. By applying this technique, you lose the rigour whereby the solution process proves that that is the solution, but at the end of the day it's still a way of reaching the solution. The same goes for problems in algebra, geometry or potentially any branch of mathematics.
Of course a complete, rigorous solution would prove that the answer holds regardless of the values of the variables present in the problem. But at the same time, the skill at solving mathematical problems lies in arriving at the answer in the most efficient way. Seeing shortcuts is part of this skill. As such, I personally think that this shortcut is a legitimate tactic.