RM Mentock - the material left in the sphere has a volume equal to the volume of an 8 cm diameter sphere.
Method: If there is a solution, the answer cannot depend on the diameter of the hole, because it is not given. Therefore the hole can have a diameter of zero. And if it does, we are left with an 8 cm diameter sphere. This is probably not the right way to solve it, but I like it. 
A non-intuitive logic puzzle that has been doing the rounds recently is this:
Three men are sitting in a room. Someone randomly places a red or blue hat on each of their heads, so that they can see the other two hats but not their own hat. On the count of three, each of them must either guess the colour of his own hat, or say nothing.
To win the prize, which will be shared among them, at least one of them must correctly identify the colour of his hat, and nobody must call out the wrong colour. How can they give themselves a better than 50% chance of winning the prize? (They can agree a plan beforehand, but are not allowed to communicate in any way during the ‘game’.)
Intuitively, you might wonder how seeing the other hats could possibly improve their chances of guessing their own colour, but there is a way.
Ummm… non-math adept WAG - you can only go precisely from true maximum edge ie “rim to rim” with an infinitely small diameter hole so material excised is infinitely small and thus volume unchanged. A hole of any measurable diameter would violate the true “rim to rim” requirement by taking in some portion of curvature from the “rim”?
It is important to add to this puzzle the fact that the three people have met beforehand and agreed on a common strategy. Given that they can increase their chances to 75%. But I will refrain from giving the solution. It can be found on the web, so no looking.
astro - the rim refers to the hole itself, not the original sphere.
Usram - Good puzzle. Think of the three men as A, B, and C. Before they enter the room they decide that if A cannot see any blue hats, he will guess blue. If he can see any blue hats, he will be silent. B will look only at C, and if he sees a red hat, he will guess blue. If C has a blue hat B remains silent. C will always guess blue if A and B were silent.
This strategy means they will win every time unless they all have red hats. I’m not sure if I violated the “no communication” clause though. Does seeing someone remain silent count as communication? That is an important factor.
But my strategy is better than 75%! Which means I must be wrong…
Here’s an old physics one.
A car is going 30 MPH on a collision course with another, absolutely identical ) car going at the same speed toward it on a perfectly straight road and crashes into the other car absolutely head on.
A car is going 30 MPH on a perfectly straight road on a collision course with an immovable concrete wall facing the side of a mountain and crashes into it absolutely head on.
In which scenario does the car suffer more damage?
While I understand where everyone is coming from when they say that the OP’s puzzle is counterintuitive, I think it actually makes sense when you think about it the right way. Imagine a cord 100 meters long lying on the ground. Take a second cord placed 16cm directly above the first one, with the ends directly above the ends of the original cord. How long would it have to be? Since the earth is practically flat over 100 meters, the answer is clearly “just a tiny little bit over 100 meters”. Extremely small addition. Now, for 200 meters, this extra length is twice as much, but still incredibly small. Now imagine how long the original cord would have to be for the extra length to be one whole meter! Pretty big, huh? As big as… the earth?
If nobody can call out the wrong color, then doesn’t that mean they all have to be correct to win? Or is it that they must guess red or blue and not guess something like yellow?
GTPhD1996
Nightime, they must make their guesses simultaneously i.e. with knowing what, if anything, the other two will say.
and GTPhD1996, they each have the option of remaining silent. They can say “Yellow” if they want to, but that would obviously count as a wrong guess.
Both the same.
I must admit I used no math just common sense/
Ok. I reread the problem and saw that they can remain silent. So if they do guess a color, they must be correct.
Nightime, I think they all need to call out the color at the same time, so your plan wouldn’t work.
GTPhD1996
Uh, I meant to say “i.e. without knowing what, if anything, the other two will say.”
Astro: If I remember my physics class correctly, the car striking the mountain will take more damage because that would be an inelastic collision.
Am I right? Do I win a no-prize? 
Ursam - You don’t say anything unless the other two are wearing identical hats, in which case you say the opposite color. Of course, this doesn’t work in the case that you all three got the same colored hat, but it gives you a 75% chance.
Unless I’m mistaken, astro’s question has been done to death in an old thread, which I’d rather not link to, for fear of its being resurrected. The thread ID is 30041, but please don’t post unless you have something really important to add.
For those of you into brain teasers ,logic problems etc. theres a web site called The Grey Labyrinth. The best puzzles are way back in the Archives, and you’ll find lots lots of good puzzles presented by that board’s members. I was heavily involved there for over a year until I found the SD message board .
What about the banana and camels problem?
The OP’s puzzle is “counterintuitive” because although we all agree that the length increase is infinitesimal compared to the length of the band, we tend to forget that the height increase will also be infinitesimal only compared to the diameter of the earth.