Imagine a round manhole (a circular hole cover) and a square manhole.
Both of them have the same area, and both of them are designed such that the cover should not fall into the hole regardless of how you orient it.
Now, the round manhole cover won’t fall into the round hole because the diameter of the cover is larger than the diameter of the hole.
But for the square manhole, if you turn the square cover diagonally, it seems like it would fall into the hole because the diagonal of the square is longer than its side length.
The diagonal of the cover seems like it would be longer than the side length of the hole (which has to be equal to the side length of the cover for both to have the same area).
However, the engineer who designed the square manhole insists that the cover will not fall into the hole regardless of orientation, and he is correct.
How is this possible?
The trick to this riddle lies in a little detail: “Both of them have the same area.”
If both the round and square manhole covers have the same area, the square one would actually have a smaller diagonal than the diameter of the round one.
Here’s why: let’s say that the area of the round manhole cover is A. That means the radius of the round manhole (r) would be sqrt(A/π). So the diameter (which is 2r) would be 2 * sqrt(A/π).
The area of the square cover is also A, which means each side of the square (s) is sqrt(A). Therefore, the diagonal of the square would be sqrt(2) * sqrt(A), or sqrt(2A), which is less than the diameter of the round cover.
This means the square cover wouldn’t fall into the square hole when oriented diagonally because its diagonal would still be smaller than the diameter of the round hole – which is the side length of the square hole, given that both holes are designed to prevent the cover from falling in.
The brainteaser is designed to make you think that a square cover’s diagonal would naturally be larger than its side length, causing it to fall in.
But when the area of the square and round covers is the same, this isn’t the case!