Candy Conundrum: Newsletter Challenge (March 2013)

Likely all that you'll get is the pretzel ones, the heavier solid chocolate having settled to the bottom....

Since the pretzel are larger and less dense, they will "float" to the top and be dispensed last.

I would agree, in general, with everyone so far.

Granular Convection or the Brazil Nut Effect.

From what I can find, the pretzel M&M candies are heavier than their plain counterpart. This seems a little counterintuitive, can someone find better info about their respective densities?

If a gumball machine is filled with M&M candies, is it still a gumball machine?

Is a hot tub filled with ice and beer still a hot tube.

Oh, man, that's a party tub!

Neither. My kids will scoop them before I have a chance!

More of the pretzel M&Ms. As the weight is a factor others have covered. I do not see it as the biggest factor unless some one is shakeing the machine. There is the shape. The smaller solid M&M will fill smaller voids in the cup as they drop in. As the pretzel which are some what round and bigger. They will get wiped off the top as the cups turned to dispence.

Add water and salt to that tub and you'll cool those six packs in 5 minutes. Gets to about 25 degrees F.

Ok, what the heck is a pretzel m&m? here in Oz we have milk chocolate, peanut and recently released mint chocolate M&M's. We also have mini M&M's.

So are these Pretzel M&M's a pretzel with a hard candy coating or just an M&M in a pretzel shape?

Oh and a Hot tub with Ice and Beer , perfect after sauna respite

Likewise, another Aussie who has absolutely no idea of the size, weight, shape etc. of a Pretzel M&M, however, if they are the same size and shape yet lighter as the name would suggest, then granular convection would indicate that the heavier chocolate ones would gravitate to the bottom more readily.

If they are larger and heavier, then again granular gravitation rules would suggest that the larger particles will end up at the top due to the smaller ones more readily filling the smaller voids in the mass and therefore dropping to the bottom.

Depending on the shape of the container, the simple act of drawing the sweets from the machine may produce enough agitation for this process to occur.

A vertically sided container would produce the most turbulence whereas a tapered hopper style may allow a much more ordered drop and less separation of the product.

Then there is the question re whether the size difference (if there is one) will produce a situation where more of the smaller ones are dispensed at any one time simply due to the mechanical constraints of the gate.

50% Pretzel and 50% Plain

Chuck B

All right girls, into the bowl with ya.....research research and more research....

You would likely get the pretzel M&Ms because the continual sorting action during the use of the machine. The more dense chocolate M&Ms would gravitate to the bottom throughout the whole use cycle and therefore would be consumed at a higher percentage rate. What would remain at the end of the filling cycle would be mostly, if not totally, the lighter density Pretzel M&Ms.

And the Correct Answer is: "Pretzel M&Ms, being less dense than ordinary M&Ms, will float to the top of the gumball chamber and lift the machine clean off the floor, just like the way last month's antigravity device worked," or "Pretzel M&Ms are too large to fit through the chute of an ordinary gumball machine (not that we've ever seen one, mind you, hence the improbable scenario we describe here) and who the hell put M&Ms in a >>>gumball<<< machine, anyway?" or "Don't bother waiting around for the Correct Answer to this month's Challenge Question because we won't have bothered to put any effort into getting IT right either, so why should you?"

There, my work here is done.

Pretzel, because I'd much prefer the plain.

Wouldn't it be something to find that someone nearby has mechanical know-how; and just doesn't like pretzels? Is this a coin op machine? Is there a correct amount of coin?

The

answer

is:

YES.

An even number of each is the least likely result with an even handful, and impossible with an odd numbered handful.

I myself would end up with a hand full of broken peices of candy shells, some fuzzy chocolate, 1/2 a pretzel, and pieces of gumball from the last time the machine was filled. Although there is no actual reason why other than it was me putting coins into a machine and expecting something much different than I get.

This is particle flow problem involving small and large particles in funnel flow, which is defined as segregation of small particles in the center of the container and larger particles flowing to the outside of the container. Since the smaller particles are the plain M&M's they will tend to flow to the middle of the container and the larger sized particles the pretzel M&M's will flow to the outside of the container. The plain M&M's will be fed out of the container more quickly than the larger pretzel M&M's. Therefore, the container is more likely to feed out pretzel M&M's when almost empty.

If the container was a mass flow container where all the particles flowed down the sides and center at the same velocity, then we would expect the perfect mixture to stay perfectly mixed and the result would be 50/50, but this is not the case with this problem as presented.

Brazil nut effect dominates density almost always. Prolate vs oblate packing fraction effects and mixtures could play some interesting effects. The number of contacts goes up with the deformation. I wonder if there is some shear flow effect at the opening that forces collective orientation. There does not seem to be many opportunities for rattling for the Brazil nut effect to act for this machine. I suspect the ratio stays pretty constant. If the only rearrangements happen from shearing the frictional variations and irregularities of the surface might win out. http://www.math.cornell.edu/~connelly/JammedMM.pdf

The assumption that "half filled" means equal numbers of each is wrong. Equal weights or equal volumes, possibly. And in both cases means more pieces from the smaller, lighter, plain M&Ms. Now unless you live in a highly seismic area, the distribution will not change much from "perfectly mixed" until the machine is empty, so chance to get a plain M&M at the end is higher. S.M.

Interesting point. I'm not sure that is true. You have to check the packing fractions. Large spheres pack with the same density as small ones. Not sure how prolate vs oblate goes. I'll check the link above.

Wait, I think the density of pretzel M&Ms is less. They have more air. I also know that flattening spheres makes them pack more densely. If you mix to maximize density I think you will get a higher number and volume fraction of plain ones. Even if they were the same density I think this is true. Now I see that pretzel ones are close to spheres. If you poured half full of one then the other then mixed you would get a lower net volume and still have larger mass and number of plain ones.

The initial ratio is what counts, since candies are "perfectly mixed" initially and granular convection does NOT work without vibration of enough amplitude and time duration. (Thus the highly seismic area joke). Now the average weight per piece of each type, gives us the initial two types pieces ratio, which is NOT 1:1 as the OP tried to trick us to assume. In all cases favours the smaller and lighter (per piece) plain M&Ms, whether "half filled" means equal weights of two types, or equal volumes. Having dug out the right data, the challenge solution is now pretty much obvious. S.M.

I think the initial ratio does matter but it is not everything. It asks what are the odds after the machine has been mostly emptied. This is when there has been lots of agitation and granular flow at work. These do favor the Brazil nut effect. There will be a convective shear depression at the surface that catches larger pieces and won't let them carry downwards. The actual ratio is a really interesting problem but, because of this I think, you tend to get pretzels at the end even if start with a very small fraction of them.

because they are perfectli mixed, there are same numbers of both!

It says half filled with each then it says they are perfectly mixed. If this is temporal order of events there can be different amounts of each. If it means they are perfectly mixed to start with then it should say there are equal numbers or equal volume or equal mass of each. "Half filled" with one type is a little vague when referring to granular mixtures. How does one know how much of the interstitial spaces to associate with one or the other?

I had to google to find out what the Pretzel M&Ms are. So I've learned that the Pretzel M&Ms are something like irregular spheres (while the regular M&Ms are elliptical and smooth). So I suppose that because they experience more friction on their way down, it will be more difficult for them to reach the opening of the gumball machine. So, I think -when the machine is almost empty- it's more probable that there will be more Pretzel M&Ms inside the machine and more likely to get them.

Lots of variables to consider and without a lot of additional data i expect there is no clear answer - variables include bulk densities of both M&M's, angle of repose, size, shape, "M&M surface stickiness", internal friction angle of both M&Ms, the wall friction angle, how they were loaded into the machine in the first place, whether the machine flows in mass flow or funnel flow mode, whether the machine has been subject to vibration, relative size of the outlet opening to the M&Ms and there are probably others!!!

If the analysis is too hard then a large hole saw is likely to yield the results you want!!

I agree with your analysis, which corresponds to mine. I might add that there is little, if any, likelihood that the outlet will flow in mass flow. For mass flow to be achieved the proper minimum hopper angle must be developed prior to designing the hopper.

Ed Barker