Packing a jar of marbles ```Name: roman zabicki Status: N/A Age: N/A Location: N/A Country: N/A Date: Around 1995 ``` Question: I saw one of those jars where you guess how many coins/jelly beans/marbles are in it to win a prize this summer. This got me thinking, how could it be calculated? I could easily enough measure the height and multiply it by the radius squared multiplied by pi. But doing this would only be accurate for finding the volume of water. With marbles/coins/jelly beans, there is empty space in between the marbles. Is there a constant or something like that to figure it out? If the marble (my question works better with marbles than coins) is as tall as the height of the occupied space there is a lot of wasted space. As the marbles get smaller, the wasted space decreases closing in on a "steady state" ( is that the right word) and eventually it would match the result one would get if one used water. Does anyone know of any research done on this? I know there are not any commercial applications so there probably is not much funding for it, but i was just wondering about this. Any explanations would be greatly appreciated. Replies: You can certainly put upper and lower limits on the number of marbles. For an upper limit you could assume that the marbles distributed their volume like a fluid, so just divide the volume of the jar by the volume of individ- ual marbles. For a lower limit, assume that the marbles sit a the center of cubes whose sides are the same length as the diameter of the marbles (i.e., the smallest cube that can completely contain the marble. Then divide the volume of the jar by the volume of the cube to get the number of marbles (lower limit). In reality the marbles will "pack" a little tighter than the perfect cube picture. Get a bunch of marbles of uniform size and try it. Make a layer of marbles as tightly packed as you can (just one layer). Now place another marble on top of the layer. It does not balance directly over one of the marbles in the first layer but nestles nicely into a pocket between three marbles in the first layer -- thus filling the space more completely than the cube picture above. For perfect, uniform spheres you will get a "closest packed" arrangement (either cubic closest packed or hexagonal closest packed) and the math for this has been worked out -- general chemistry texts or crystallography texts can give more details. For perfect spheres the result is that about 74% of the space is actually taken up by the marbles. Real packing will be slightly less. I do not know about other geometries (jelly beans). gregory r bradburnSo the maximum packing fraction is 74%. For random packing (what you get by just jumbling the marbles in) of spheres the packing fraction is 63-65%. Again, I do not know about jelly beans though, but in fact this probably has been investigated - I believe there is a part of materials science devoted to the ways in which agglomerations of materials pack together, and this kind of question is actually useful to companies who make large numbers of small objects - you really do not want to count all the nails in a bin, for example (although usually weighing the things is a better measure of the number). asmith Click here to return to the Mathematics Archives

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