The original asker of the question actually proposed that already.
> I know I could just take one piece out, and put it back in to feel it myself, but I am looking for a more logical way
Since that's the "puzzling" stack exchange, I think they were looking at this more as a logic problem than a real practical problem they needed to solve.
Right, that's why I'm not posting this as an answer to the stack exchange question. Though I'm pointing out that it's not necessary to remove any pieces, and also suggesting that there may be an interesting math problem still there in this case.
I guess the obvious question is "given x amount of slack per piece, after how many pieces can I fit in on piece the wrong way without tension", but that feels more like an engineering problem than a math puzzle.
In this case not-so-brute force of fitting actual pieces, since it won't take that many. And otherwise estimates based on highly simplified approximations.
> I know I could just take one piece out, and put it back in to feel it myself, but I am looking for a more logical way
Since that's the "puzzling" stack exchange, I think they were looking at this more as a logic problem than a real practical problem they needed to solve.