[Guest]

Let us suppose that we have two absolutely same sacks.

The first sack is full with rice with round beans

The second sack is full with ellipse beans.

Let us suppose that volumes of round and ellipse beans are same

The question is: which sack contains more beans?

[pdqkemp]

This is a problem indeed. I think it needs to be re-thought and re-worded.

Were they "beans" or was it "rice". You said 2 different things. Or are you calling each individual kernel of rice a "bean"?

When you say "round" beans, do you mean they are perfect spheres?

Was the volume the same for 1 single bean or was the total volume the same? I think you are saying the volume of each bean is the same.

I think you are looking for us to say that the elliptical bean sack will contain more beans since there is less space for air between each bean. (They "fit" together more so than the spherical beans.) Still, I think this problem is a problem.

[Guest]

Would this not depend on how they are arranged?

[Guest]

Nice brain teaser Ilia. I think this is the answer...

there will be more beans in the sack with the spherical beans. I think this is because there will be less unused space between beans, allowing more beans to fit in the bag.

[Guest]

A very good question, Ilia.

I would say there are more elliptical beans because the packing efficiency for elliptical rice would be more than that for spherical beans.

Here's why:

Consider a can of each type of beans with beans filled such that there is no movement of the beans.

For a shperical bean, the possible movement is only lateral while for an elliptical bean, there is lateral as well as rotational. So, an elliptical bean would need more beans around it in order to restrict the movement. Which means that more elliptical beans can be packed in a can of same volume.

Using some maths, let us consider (in 2-D to start with) a container that can place sphere in a hexagonal arrangement (row of 5 followed by a row of 4 followed by 5 and follwoed by 4). Then in 3D, lets say the container is big enough to hold 4 layers of these arrangements (alternating 5-4 folloed by 4-5). Then, the width of this is 10r where r is the radius of the sphere, length and height is 2*(r*(2+root(3)) = 7.46r

Number of spheres placed = 72

Now, consider elliptical beans with one side 2r and other two sides r/root(2) = 0.7071 r for ease of calculations. Lets place it such that 10r width is covered by 2r side of ellipse. Then length and height are covered by sides measuring 0.7071 r. So, number of ellipse that can be placed along width will be 2 leaving space for 2r left. within this 2r, 2 layer can be placed along length such that height remains the same (3.73r).

Total number of ellipse placed will thus be greater = 74

This means that the packing efficiency with elliptical beans should be about 2% better. Meaning that there must be about 2% greater number of elliptical beans

Of course this answer is valid if both the sacs were very well shaken while putting the rice. In case they were not shaken at all, the number of at all, the elliptical beans will still be more but may be not as high as about 2%.

Edited September 25, 2009 by DeeGee

[EDM]

simple.....the one with thinner beans (the second sack).....

ellipse beans take less space than round beans. if you took a jar of rocks, would it be completely full..??? then add pebbles. you will see that it takes the spaces that the rocks couldn't fill.

similarly, ellipse beans can be considered as pebbles as they can take less space than round beans. they would 'fit' together better.

it's not about the volume; it's more about the shape...

i hope i explained it right...

[Guest]

This is a problem indeed. I think it needs to be re-thought and re-worded.

Were they "beans" or was it "rice". You said 2 different things. Or are you calling each individual kernel of rice a "bean"?

When you say "round" beans, do you mean they are perfect spheres?

Was the volume the same for 1 single bean or was the total volume the same? I think you are saying the volume of each bean is the same.

I think you are looking for us to say that the elliptical bean sack will contain more beans since there is less space for air between each bean. (They "fit" together more so than the spherical beans.) Still, I think this problem is a problem.

first of all sorry for my English...it is not very good i think.

Yes, I mean that volume of each "perfect sphere" beans and each elliptical beans are same.

By the way, I have no answer for this problem.

I can prefear you the "two dimension" variant of this problem:

Let us we have two absolutely same tables, and we are putting on tables round and elliptical figures cutted of paper.

So, which table will be more covered?

you know, this "two dimention" variant may be is more interesting, because in "sacks" variant we have chaos in distributing of beans and may be this forces the situation to equilibrium...may be. But in Table variant, there is no chaos because We are putting figures. So we must prove which method of covering of tables with figures "contains maximum figures". By the way this is another problem.

Even more, we can do this problem more difficuld if we take other figures, whatever.

Again sorry for my English

Edited September 25, 2009 by Ilia

[Guest]

I can prefear you the "two dimension" variant of this problem:

Let us we have two absolutely same tables, and we are putting on tables round and elliptical figures cutted of paper.

So, which table will be more covered?

you know, this "two dimention" variant may be is more interesting, because in "sacks" variant we have chaos in distributing of beans and may be this forces the situation to equilibrium...may be. But in Table variant, there is no chaos because We are putting figures. So we must prove which method of covering of tables with figures "contains maximum figures". By the way this is another problem.

Even more, we can do this problem more difficuld if we take other figures, whatever.

Again sorry for my English

For 2-D, it would depend on the dimension of the table. See the two figures below for circles. If the table dimension if an even multiple of radius, you wuold place circles as shown in the second figure whereas if it is slightly less than the an even multiple of radius, you would place it as shown in first firgure to place max number of circles... So, I guess the question is not complete yet

Edited September 25, 2009 by DeeGee

[Guest]

Here are my answers to both:

First, the answer to the sacks is they would be EQUAL since they would be filled based on weight and since each bean is the same volume and we can only assume the same density, each bean will weigh the same.

Second as for the table, no one said we can't let the shapes hang off the table (i.e. no loss on the table edges) so the question is just the wasted space in the packing.

For the Circle use DeeGee's 1st picture (with offset circles) and imagine an equilateral triangle fromed by the centers of 3 mutually touching circles. The area of the triangle is R*R*sqrt(3) where R is the Radius of the circle. The used space is 3*the area of a 60deg pie wedge from the circle which is 1/2 the area of the circle or 1/2*Pi*R*R. The fraction of used space is Pi/(2*sqrt(3)).

For ellipse, do the same offset and draw a line from the center up to the contact point of 2 other ellipses and then over to the center of one of those ellipses, back down to the contact point of the original ellipse with an adjacent one and finally back to the center of the original ellipse to make a rectangle. The area used in that rectangle is easy to see a quarter of each ellipse or 1/2 Pi*A*B, where A is the major axis and B is the minor axis. The area of the rectangle is harder, but if i'm not mistaken the point of contact must be symmetric so eually down along the A axis on one as the other and therefore must occur at 1/2 A. Now the equation for an ellipse is (x/A)^2 + (y/B)^2 = 1 so if x (the point of contact) is A/2 then y must be sqrt(3)/2 and again since it is same for both ellipses, the separation between the 2 centers (i.e. the height of the rectangle you drew) must be sqrt(3). So the fraction of used space is Pi/(2*sqrt(3)).

So again they are EQUAL

[Guest]

Using a slightly different method I evaluated the density to be pi/2root(3). As usually someone got there before me

.

However is this still the case in 3D?

Anyone know how to pack spheres as densely as possible?

[Guest]

I say the sack with the rice will have less beans in it because the other sack has no rice taking up space.

[akaslickster]

My first thought before reading the above.

Posted Today, 08:20 PM

Spoiler for Simply- Because there seems to be no rice at all in the second sack, there is that much more room left over that is occupied with beans in a perspectively full sack.

Edited September 26, 2009 by akaslickster

[EDM]

Using a slightly different method I evaluated the density to be pi/2root(3). As usually someone got there before me

.

However is this still the case in 3D?

Anyone know how to pack spheres as densely as possible?

yep....shake the sack so they become compact by filling spaces better.........

i have a question.....wouldn't the size of the ellipse beans be longer??? shouldn't it affect the number of beans anyway.....similar question for the paper.....

[Guest]

Since I think I understood the problem, I really do not think that the question setter needs to worry about his standards of English. My Hungarian, Japanese and Swahili are much more ropey.

However, the term "elliptical" really is far too vague. Does this mean ellipsoidal such that the cross-section is circular, or does it mean, as is normally the case, that the cross-section is an ellipse?

In any event, the packing for an ellipsoidal bean should be closer than that for a spherical bean, whose densest packing is either face centred cubic or close packed hexagonal (see any elementary book on crystal structure.)

As a result the latter sack should, in my view, contain more beans.

[Guest]

THE ELLIPTICAL ONE

[Guest]

Let us suppose that we have two absolutely same sacks.

The first sack is full with rice with round beans

The second sack is full with ellipse beans.

Let us suppose that volumes of round and ellipse beans are same

The question is: which sack contains more beans?

the "ellipse" beans do not have rice in the sack, so therefore, it can "contain" more than the round beans with rice

Edited October 1, 2009 by dontgivemetheanswer

[Guest]

I think that it is merely a difficulty with the English language, not a typo. If one substituted the word "grain" for "bean" in the original post, then the problem might have been clearer.