No offense taken, although im not really a fan of people saying "no offense" - criticism is at the core of self-improvement and learning, there's no point in sugar-coating it. Regardless, in the first example (math question), it's impossible: no value for x can satisfy both equations, no matter how hard you try. naturally, since both equations are given, x should be able to satisfy both equations, and since it cannot, the problem has no solution.
for the second example, however (chess puzzle), it's common sense - working backwards. there appears to be no solution, but if we make some assumptions, there is a logical solution.
going back to the first example, again, i don't see the connection. we aren't working backwards, there is no logical solutions, and the only assumption to be made is that the creator of the so-called riddle mixed something up, so there is no solution.
let me give you an analogy to explain.
it's like me saying this:
x + 1 = 2
(x(20173) + 218301283^921731!) = 39
and saying
"no body would spend the time to solve the second problem, so x =1"
This would be like the first question.
For the second, chess-related, example, it would be like one of those detective problems - where you are given clues, have to work backwards, and make assumptions. assumptions which, mind you, abide by the clues you were given.
i think that's the difference. the assumption you make in the first example is that the value of 0 for x satisfies the first equation, so it must satisfy the second as well. that makes no sense. you are given the complete information ("clues" if we are to draw parallels with the detective example) to verify your answer, but you aren't. that's not logic, thats laziness.
for the second one (detective one), you are making assumptions BASED ON the clues you are given, and only that, to come to a logical conclusion. in the math question, you are making assumptions, which don't fully abide by the "clues" (second equation), and instead make assumptions because the solver is too lazy to solve the second equation.
you aren't using the complete information at your disposal to solve the problem.
that's like me saying, "the earth is shaped like swiss cheese because there are holes in the ground", despite a plethora of evidence indicating otherwise. The other evidence is synonymous with the "second equation" in this analogy.
at least that's my take on what you're saying - feel free to elaborate on what you mean, because i don't see it. also, im 13, so you might have to explain it more simply, because clearly there's some deep idea i don't see.
You're overanalysing.
Edit: This thread has exactly 3.14x100 = 314 posts. This is history in the making of chess.com that I am able to acknowledge this fact. I can wholly attribute my pide to pi!
I dropped high school, I can just calculate basics, like %, after that, I'm lost for not knowing.
In France, if you drop mathematics at high school, you are still smarter than a vast proportion of Westerners.