## Monday, July 27, 2009

### How Many M&M's in a Jar?

I just had to copy this from GrrlScientist 'cause my favorite1 daughter loves M&M's and physics.

Here's the Science paper that calculates the packing fraction of M&M's.
Donev, A., Cisse, I., Sachs, D., Variano, E.A., Stillinger, F.H., Connelly, R., Torquato, S., Chaikin, P.M. (2004) Improving the density of jammed disordered packings using ellipsoids. Science 303:990-993. [PubMed] [doi: 10.1126/science.1093010]

1. and only

DK said...

That was tedious. Who is the intended audience of this guy?

The detail he is going about it belongs to a high school (not to mention that Cl- and Na+ should have been shown with different radii). When I was a kid, I had an almost identical "bean packing" question in math olympiad (8ths grade, IIRC). Never knew what the "correct" answer was but I remember doing it this way: take fractional volume of a sphere set inside a cube (that's pie/6), then assume that beans will pack right in the the middle between spheres and total volume. Then take off ~20% to account for an imperfect packing.

My calculator today says it is ~ 61%. The paper in Science says it is 66% for a random M&Ms packing.

In hindsight of what I know today, my old solution was totally clueless about symmetries (assumed tetragonal without even realizing a possibility of hexagonal). Still, goes to show: a back of the envelope calculation is good enough in most cases.

Real life lesson: complex models are overrated.

Anonymous said...

What a poor retrospective analysis of your past DK. It seems your ego was on full throttle and brain on standby when you typed this. So you got a good estimate to the Science paper but make no effort of even hinting that your ignorant guess would be wrong a majority of the time if you were not aware of the full picture. Also, since this is mathematics and may have further uses than guessing how many beans fit inside a space, precision is important.

DK said...

It seems your ego was on full throttle and brain on standby when you typed this. So you got a good estimate to the Science paper but make no effort of even hinting that your ignorant guess would be wrong a majority of the time if you were not aware of the full picture.

I don't know WTF you are talking about. The question was how to best guess M&Ms fractional packing. I recalled a similar beans packing question. Personally, I am rather amused how close the simplest educated guess turned out to be to the high-level math (or was it science? :-)).

As I *am* a big fan of simple models that can be understood analytically without the use of computers, I do find it gratifying that the crudest estimate seems to be almost as good as experiment. Is that a crime?

Also, since this is mathematics and may have further uses than guessing how many beans fit inside a space, precision is important.

It may be. But: 1) That is not my concern here and whenever this might be important is totally outside of my expertise, and 2) my experience as a biologist suggests that the precision of *this* kind is not very important. Rather, numbers that are orders of magnitude off the real life numbers (a common place) is a real problem.