I’m confused by the question and answer. Essentially, if we remove a 6 inch long core from a sphere, how much of the sphere is left.
THe question, makes it seem like the sphere is a defined diameter of 6 inches, but that apparently plays into the answer. I think it can’t be explicitly solved because we don’t know the diameter of the core. We know the core is 6 inches long, but we don’t know how wide the core is, therefore we can’t solve it.
Can anyone help with an explanation, or describe how the puzzler should have been set up?
Since you seem to understand the mecanics he is talking about, coiuld you maybe phrase the question in a way that makes it more clear what he means? I think the majority of the problem is the question is not phrased very well, so I have no idea what he’s talking about with the answer.
I think it’s a question that tries to be clever, but fails miserably, because it’s almost impossible to describe clearly what the question means without giving the game away. The problem is that what it is describing as a “core six inches long” is a cylinder six inches long PLUS convex bulges at each end that from a larger sphere will make the entire core much larger than the six-inch-in-long cylindrical part.
In this extreme case, you can see how ridiculous it is to describe this as a “core six inches long” when the cylindrical part is 8,000 miles wide and six inches long, and the convex bulges at each “end” of the core are 4,000 miles deep. In other words, the “core” is the entire sphere of diameter 8,000 miles with a ring-shaped sliver six inches wide around the equator shaved off.
Here you go. The unshaded part is what he is calling the “core”. The claim is that the shaded part has the same volume whatever the size of the sphere. (For the larger one the shaded area is smaller in that diagram, but remember it’s 3-dimensional)
Yes, I am not sure what this has to do with cars, but in your illustration it is the shaded region which may be regarded as “six inches long”, that is, six inches high if you put it on the ground. In fact that leads to @K364 's answer almost immediately by e.g. integration (the radius of the sphere cancels out!)
It’s still convoluted, but perhaps a more elegant way to state the puzzler is something like this:
An exclusive jeweler is selling a wedding band that he claims is machined out of a perfect solid sphere of gold, with no distortion of the sphere in the manufacturing process in which the middle of the sphere is cored out. The inside surface of the resulting ring is a perfect cylinder. The diameter of the ring (i.e. the ring size) is unknown, but the height of the ring if laid flat on the table is 1cm. What volume of gold am I buying?
I suppose this is moderately interesting, because a ring of any diameter (ring size) made this way would contain the same amount of gold, even if the ring were made for a giant whose finger diameter is that of a tree.
Car Talk was a radio show hosted by a couple of brother mechanics who also happened to be MIT grads, and they took questions for listeners on all sorts of different subjects. They were very entertaining, and each episode featured a Puzzler. It would appear that their website continues the Puzzler tradition (the show’s original hosts have ended the show, as one of them passed away several years ago).
I think the misunderstanding is that I can see that the diameter of the sphere doesn’t matter. But the diameter of the corer should.
I have a 6 in sphere and I take a 6 inch long by 0.5 inch wide core out of it.
I have a 6 in sphere and I take a 6 inch long by 4 inch wide core out of it.
Scenario 1 has more material left in the sphere than scenario 2. THe diameter of the core is not specified in this puzzler making it difficult for me to see the answer as anything definite.
For any given sphere, only one diameter of corer will cut out a cylinder 6" long. So specifying the 6" length of the cylindrical part of the core is sufficient.
Neither of the core widths that you propose would result in the cylindrical part of the core being 6-in long, it would be shorter. This is what I pointed out is misleading in the way the problem is stated. The 6" length of the core that is discussed refers only to the cylindrical part of the core, it does not include the additional convex parts at each end of the cylinder.
This is why I was confused by the OP’s mention of a “recent” Car Talk puzzler. Is this from a recent rerun, or is Ray (the surviving Magliozzi brother), or someone else, still putting out new Puzzlers?
so are you saying if we could majically remove a true cylinder from the sphere but leave the convex ‘caps’ with the sphere, then it doesn’t matter what the diameter of the core is?
I think I can almost wrap my head around that then. It certainly makes more sense in that it leaves more material in the convex ‘caps’ with larger diameter cylinder cores.
If that is true than I agree that this is misleading, in that they did NOT clearly specify this, and in fact implied the convex caps come out with the cylindrical core.
No. You remove the cylinder plus the convex “caps”. But it is only the height of the cylindrical part that specifies exactly what you should remove.
And I’m not saying it doesn’t matter what the diameter of the core is. I’m saying that there is no additional degree of freedom. If you specify that the length of the cylindrical part of the core must be 6", for any given sphere there is only one possible core diameter that will achieve this.
Rerun. In October, Car Talk started releasing edited versions of their early NPR-era shows as podcasts, in chronological order. Previously, they released whole shows, mostly from later periods in its 26-year run.
The problem I have with this is that your 6 inch dimension is not how I understand the question. THe 6 inch dimension is perpendicular to your measurement.
I think my confusion stemmed from the different meanings behind “length.” When I think of the length of a core removed from a sphere, I think of it’s tip-to-tip end and in my mind any size core from a 6" diameter sphere would have to be 6" in length. I can see how it specifying the cylindrical length of the core adds the required constraints to solve the puzzle.
Vsauce did a video talking about this and they described it as the Napkin Ring Problem. Their video:
Huh. Well, on the one hand I feel smug that coming across the problem for the first time I drew almost exactly the same diagram that’s on that Wiki page. On the other hand, my diagram is so similar even in the details that nobody will believe I didn’t copy it from that Wiki page.
