Too bad they won't ask about manhole covers anymore. I figured that one out on my own a long, long time ago. (free points!)
P.S. If you know why they make them round, then do you know why they don't make them triangular? ;)
P.S. If you know why they make them round, then do you know why they don't make them triangular? ;)
> Because all diameters are the same length, the Reuleaux triangle,
> with all other curves of constant width, is an answer to the question
> "Other than a circle, what shape can you make a manhole cover so that
> it cannot fall down through the hole?"
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.
Second, no matter how you orient an equilateral triangle with on point facing down the hole, there is an edge across the top that is too big to fall in. The way to visualize this, if you take one edge of an equilateral triangle and sweep it in a circle around one end, it will be outside of the triangle at all points except where the corners meet. So trying to drop it in any position other than with the edges lined up will make it impossible to fall through. And since there is a lip when the edges line up, it can't fall through.
In contrast with a square or rectangle, if you sweep one of the edges it will be inside of the square or rectangle at some points (on a rectangle, you have to use the small sides). This means all you have to do is shove one edge of the cover down the hole and it will fall in.
Exactly, and with a big enough lip/flange, ANY shape will work.
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.