> First off, there will be a small lip to keep it from falling in flat. This means that the hole is actually smaller than the manhole cover by an inch or more.
Exactly, and with a big enough lip/flange, ANY shape will work.
Not a square. The diagonal dimension of a square is sqrt(2) * the edge length, so roughly 1.414*edge length. So if you rotate the square cover 45 degrees, you can kick it right down the hole no problem. Sure you could have a cover that was several inches bigger than the lip, but assuming the lip is only going to be on the order of .5-3" and the cover is going to be around 24-36" wide, you can always rotate the square one ot fall down there.
Your flange is not "big enough". For an absurd example, consider a hole 3" to a side, and a flange 48' in width.
That said, I honestly meant this as a mathematical/rhetorical statement, not a practical one. So yes, you're correct that for any reasonable flange width (given the assumption we're talking about steel/iron manhole covers, and accepted values of the material's strength, etc.), some shapes wouldn't qualify.
My point was to show that the question, taken at its face value, isn't even a good question. It assumes a flange, or no shape would work as a cover, as the cover would simply fall into its own hole. Thus, if we're to assume a flange (or a taper; it could be argued wine bottle stoppers and the holes they "cover" are both "circles"), we should be able to assume one of arbitrarily large size, and then ANY shape will work.
Exactly, and with a big enough lip/flange, ANY shape will work.