@polfosol No, for the question; thanks for the suggestion. You know, I just wanted to be less involved personally. But then, it seems I would miss many intresting things like Project Euler.

Personally, I used Cut the Knot a lot and it really deserved whatever rewards it received.

The news of that "proof" was even given in an Iranian public newspaper at the time when a few Iranian (could) read anything but religious texts!

@ChristianRemling You dare to give a counterexample for "If P then Q" part of your statement :)

@NikWeaver Usually, I personally flag my question for the attention of the moderator to make it CW. In this particular case, I didn't, since I was under the impression that though there could be multiple answers, none of them would be opinion based, and each one could be only correct or wrong, like almost every other math question on MO.

Thank you, Adam, you made my day and the question crystal clear :)

@StanleyYaoXiao No, there are too many of these examples, in which you have an (if P, then Q) plausible but false conjecture. I am looking for something that both directions (if P, then Q) and (if Q, then P) seem plausible but false.

@StanleyYaoXiao That is exactly why I wrote "you might want to ignore my example" :) Please ignore it. I needed to mention it since it was in the original post. Please read just the last three lines of the post :)