Brain Teasers
Opaque Marble Balls
Science
Science brain teasers require understanding of the physical or biological world and the laws that govern it.Science
Lost in the woods, you come upon a mysterious stranger who hands you two large spheres made of two kinds of opaque marble. The spheres look the same, are the same size and have the same weight and apparent density; however, one is solid and the other hollow. The hollow sphere contains a tiny map that will enable you to navigate your way home. Unfortunately, you cannot break the spheres. The stranger agrees to open the sphere that contains the map only if you can determine which sphere is hollow. How do you determine which sphere is hollow?
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exactly where would you find a level flat surface in the woods. Since the balls have equal diameter and equal mass, their volume and
density are also equal. However, the mass distribution is not equal,
so they will have different moments of inertia - the hollow sphere has
its mass concentrated at the outer edge, so its moment of inertia will
be greater than the solid sphere. Applying a known torque and observing
which sphere has the largest angular acceleration will determine which
is which. An easy way to do this is to "race" the spheres down an
inclined plane with enough friction to prevent the spheres from sliding.
Then, by conservation of energy:
density are also equal. However, the mass distribution is not equal,
so they will have different moments of inertia - the hollow sphere has
its mass concentrated at the outer edge, so its moment of inertia will
be greater than the solid sphere. Applying a known torque and observing
which sphere has the largest angular acceleration will determine which
is which. An easy way to do this is to "race" the spheres down an
inclined plane with enough friction to prevent the spheres from sliding.
Then, by conservation of energy:
mgh = 1/2 mv^2 + 1/2 Iw^2
Since the spheres are rolling without sliding, there is a relationship
between velocity and angular velocity:
w = v / r
so
mgh = 1/2 mv^2 + 1/2 I (v^2 / r^2) = 1/2 (m + I/r^2) v^2
and
v^2 = 2mgh / (m + I / r^2)
From this we can see that the sphere with larger moment of inertia (I) will
have a smaller velocity when rolled from the same height, if mass and radius
are equal with the other sphere. Thus the solid sphere will roll faster.
Since the spheres are rolling without sliding, there is a relationship
between velocity and angular velocity:
w = v / r
so
mgh = 1/2 mv^2 + 1/2 I (v^2 / r^2) = 1/2 (m + I/r^2) v^2
and
v^2 = 2mgh / (m + I / r^2)
From this we can see that the sphere with larger moment of inertia (I) will
have a smaller velocity when rolled from the same height, if mass and radius
are equal with the other sphere. Thus the solid sphere will roll faster.
i have no idea why the comment above has split in two?
You could look for hours for a flat surface, then work on rolling both balls with equal force, which would be impossible to determine without instrumentation. In this situation, your best bet would be to knock on them with something. You'd probably be able to pick out the hollow one.
stares at the above comments
um.......... what does that mean (retorical question dont write even more stuff that will makr my head hurt)
um.......... what does that mean (retorical question dont write even more stuff that will makr my head hurt)
there is a easier way to find out. Just let, a few drops of lemon juice get into contact with the surface of the marble. You will see the acid in the lemon will leave a deep mark on the marble surface in 10-15 minutes time.
this comment is for the mad, it's just a riddle dude, don't get your panties in a wad, lol
Mad-Ades "comment" is copied directly from a maths site which explains the maths behind if you roll or push the balls. But the problem is he just copied it and can't do it because then he would understand that the answer given is the same as the one he has copied.
Its one thing to copy all your puzzle but to criticise someone else's by copying answers ......
Its one thing to copy all your puzzle but to criticise someone else's by copying answers ......
Posted by cathalmccabe Aug 07, 2002
Mad-Ades "comment" is copied directly from a maths site which explains the maths behind if you roll or push the balls. But the problem is he just copied it and can't do it because then he would understand that the answer given is the same as the one he has copied. Its one thing to copy all your puzzle but to criticise someone else's by copying answers ......
ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZz.......yawn oh, have you finished Cathal, hmmm, as usual very riviting and flattering that once again you have took the time to chase up my answers. I really don't understand this attraction you have for me, but hey if it means so much to you, then please, carry on, be my guest. After all we are all adults, we understand. What ever floats your boat! No need to be ashamed of your "supressed" proclivities. The world accepts people like you now, its ok, trust me. ;)
Mad-Ades "comment" is copied directly from a maths site which explains the maths behind if you roll or push the balls. But the problem is he just copied it and can't do it because then he would understand that the answer given is the same as the one he has copied. Its one thing to copy all your puzzle but to criticise someone else's by copying answers ......
ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZz.......yawn oh, have you finished Cathal, hmmm, as usual very riviting and flattering that once again you have took the time to chase up my answers. I really don't understand this attraction you have for me, but hey if it means so much to you, then please, carry on, be my guest. After all we are all adults, we understand. What ever floats your boat! No need to be ashamed of your "supressed" proclivities. The world accepts people like you now, its ok, trust me. ;)
LMAO. Come on now! How many ORIGINAL, non-copied teasers are there here. I'd wager one or two, maybe!!!
Oct 03, 2002
Never talk to a stranger in the woods!
how can you guys be such babies over a *riddle*? jesus...
But she is right, mad-ade. Why do you take the time to fill out two comments explaining exactly what they have already said when nobody has questioned it and nobody (I doubt even you) understands the equation the you posted??? Does it give you some sort of pleasure to imagine that you are smart?
you could tap them both on the same surface. one would sound hollow!!!!!!!!!!!!!
hey paranoid you could have just asked your alien buddies to stick you in the hollow one.. oh wait those are your other teasers. never mind 
Hit on them! The hollow one will be easy to tell =D
First off, the solid sphere would roll further.
Another option would be plausible if you had a thermometer of some sort. If you heat both balls for the same amount of time, then wouldn't the hollow one get hotter slower? Even if you heat them too long and they both become equal temperature, the faster cooling one would be the solid sphere.
Another option would be plausible if you had a thermometer of some sort. If you heat both balls for the same amount of time, then wouldn't the hollow one get hotter slower? Even if you heat them too long and they both become equal temperature, the faster cooling one would be the solid sphere.
Or, you could just shake each of them, the one that seems to have some movement at all is the hollow one! 
Anyone who thinks within the box would strike them. The one that sounds hollow would be the hollow one. So easy.
Or you could drop them both into water and see which one floats? But wouldn't it just be easier to take either sphere and knock the stranger out? Then you could do whatever you wanted with the spheres.
My first thought was indeed whacking the marble spheres with a stick or stone and listen for the one that is hollow.
Is it roll "further" or roll "farther" 
I thought about shaking or banging on them to see which was hollow, but I think I like Cubano's answer best! 
Even when I studied physics years ago, I don't think I could have solved this teaser.
so confused
For those who have suggested tapping the spheres and listening for an echo (and I admit I had the same thougth), it would depend on the thickness of the "shell". If for instance, the sphere is 12 inches in diameter and has, say, a 1-inch hollow space inside (it does say that the map is "tiny"), then the thickness of the outer shell would likely muffle any echo.
yeah and where would you find a flat surface in the woods? You could simply nock each sphere on a rock and listen to see which sounds hollow.
Hmp. I would have tried dipping them in water, but I guess we don't "know" if there's water nearby.
If you're in a magical woods where there are tiny maps inside of perfectly sealed marble spheres and perfectly frictionless ramps where you can conduct your experiment, then I'm guessing that you can find water somewhere to float the balls. And lemon juice to burn through them. And a saw that will cut through marble...
Dipping them in water wouldn't do any good. If they have the same density, they will both either sink or float, even if one of them is hollow.
I'd pull out my uzi and make the stranger guide me out of the woods.
You can get more of what you want with a kind word and a gun then you can with just a kind word.
--Al Capone
You can get more of what you want with a kind word and a gun then you can with just a kind word.
--Al Capone
I had the same thought as cubano!
The floating bit not the violent bit
The floating bit not the violent bit
To add to the answer, Just knock on the marbles. The one that sounds more hollow is the one. 
I was also thinking about knocking on them for a hollow sound, and also seeing if one of them would float; but I don't know how you would figure it out. Some of the solutions just make my head spin. It was an interesting teaser, though.
Feb 13, 2009
I second the float test. Irrespective of the fact that the density/mass of both balls is the same, the presence of any air in the hollow will provide buoyancy.
The teaser said "apparent density" probably because if one of them is solid and the other isn't and they weight the same, then one of them isn't marble.
Most importantly, hopefully your map isn't ruined after you submerge it.
Most importantly, hopefully your map isn't ruined after you submerge it.
One: If one marble is hollow, how can it be the same density? That being said brings me to number Two: Archemedes principle: Sink both marbles, obstensably, the solid one will displace more water than the hollow one.
Just a question, even if you did find the map, how would you know where you were on it? To use a map, you have to know where you are, and the teaser specifically tells us we're lost.
Look for the "You are here" icon on the map
May 13, 2009
since you said the hollow one had the map, i would guess that the one with the map in it was the hollow one.
I am like a lot of others. Just rap on them and listen, or find a stream and see which one floats. If you have to find a flat surface you can certainly find a stream in the woods. Not a very good teaser with so many ways to tell without all the figuring. Simple solutions work best.
Once again, love the comments! This teaser sure got a lot of wild responses! Thanks!
The second (alternative) answer is the obvious solution to me. It's the same basic way you test an egg to see if it's raw or cooked. Spin them. Just by applying a torque to each one you can feel the different way the angular momentum is being transferred. No need to bother with all the physics really. If you know the two possible constructions (hollow/solid or raw cooked for eggs) you can "feel" the spin of the object.
As long as the two examples being tested are distinct as described in the puzzle (or the eggs thing), there's no need for instruments or precision, just a reasonably flat bit of wood or rock.
Now, where DID I pack my perfectly level surface...
As long as the two examples being tested are distinct as described in the puzzle (or the eggs thing), there's no need for instruments or precision, just a reasonably flat bit of wood or rock.
Now, where DID I pack my perfectly level surface...
If these balls are both made of marble and weigh the same wouldn't the hollow one be larger in diameter?
Unless of course the map is made of lead. LOL .
I too enjoy the comments.
Unless of course the map is made of lead. LOL .
I too enjoy the comments.
The physical description of the balls as proposed by the teaser does not make sense. One of the balls would have to be larger than the other if the densities were the same and the weights were the same.
I understand the physical phenomena that the author is trying to demonstrate. It is a very measurable event.
But just stating that one of the spheres is hollow does not descriptive enough. Is the hollow sphere 12" in diameter with a 1" hollow core? Or does it have an 11" hollow core? Does the weight of the map make up for the lost mass?
If the hollow core were very small and the mass of the map sufficient enough, I'm not sure one could measure the rotational inertia to a significant degree.
If the hollow sphere has a relavively large hollow core, could one try to float them on a liquid?
How did the map get inside the marble sphere? How was the sphere hollowed out? Whould there be a seam or a detectible drill hole?
I would closely inspect the spheres for surface defects.
I understand the physical phenomena that the author is trying to demonstrate. It is a very measurable event.
But just stating that one of the spheres is hollow does not descriptive enough. Is the hollow sphere 12" in diameter with a 1" hollow core? Or does it have an 11" hollow core? Does the weight of the map make up for the lost mass?
If the hollow core were very small and the mass of the map sufficient enough, I'm not sure one could measure the rotational inertia to a significant degree.
If the hollow sphere has a relavively large hollow core, could one try to float them on a liquid?
How did the map get inside the marble sphere? How was the sphere hollowed out? Whould there be a seam or a detectible drill hole?
I would closely inspect the spheres for surface defects.
Just a question....
Would the resonance be different? I was thinking of hitting them with rocks or bottles and listening for the sound they make. The solid one would sound deeper than the hollow one.
Would the resonance be different? I was thinking of hitting them with rocks or bottles and listening for the sound they make. The solid one would sound deeper than the hollow one.
The las time I was lost in the woods and a strange man came up and offered me a way out......... I just ran til I found a genie with a lamp and wished my way out.
I did this one in 2012, but liked the comments afterwards. Especially auntiesis's Really funny and cute.
Three letters will get you out of the woods GPS.
Happy Friday!
Happy Friday!
I probably would have ended up breaking them both!!
I thought I was supposed to drop them both in water...
I can be a derp.
I thought I was supposed to drop them both in water...
I can be a derp.
Say, "it's that one!"
If he says you're right, you get the map. If he says you're wrong, you say, "ok, it's that one!" Now that you have determined which one, he will open it for you.
If he says you're right, you get the map. If he says you're wrong, you say, "ok, it's that one!" Now that you have determined which one, he will open it for you.
Easiest way to do this, pick one of them if you're correct congratulations, if not get him to prove it by opening the correct one at which point you then steal the map and run.
These comments should win the Braingle Academy Awards for Comedy! (If there was such a thing) The only things missing are comments from the bunnee bunch and the little green men!
As for the science problem, I thought of the putting them in water idea also.
As for the science problem, I thought of the putting them in water idea also.
Two other potential solutions:
Hold the spheres up to your ear and shake them to listen for the map rattling inside.
Drop the spheres in water. The hollow one will float.
Hold the spheres up to your ear and shake them to listen for the map rattling inside.
Drop the spheres in water. The hollow one will float.
I'm so not smart enough for this...
Wow. This generated a lot of confusion.
The hollow sphere is made of more dense material than the solid sphere. The volume of each divided by the total weight is the same.
Both should sink in water.
Both should sound pretty much the same when hit, unless the shell is very thin.
I'm confused by the rolling solution. The teaser answer says the hollow one rolls farther, while the comments say the solid rolls farther. I don't know which is right.
I went with the spin solution.
This was a decent science teaser made great by all the feedback and discussion!
The hollow sphere is made of more dense material than the solid sphere. The volume of each divided by the total weight is the same.
Both should sink in water.
Both should sound pretty much the same when hit, unless the shell is very thin.
I'm confused by the rolling solution. The teaser answer says the hollow one rolls farther, while the comments say the solid rolls farther. I don't know which is right.
I went with the spin solution.
This was a decent science teaser made great by all the feedback and discussion!
Follow Braingle!