Scaling, with M&Ms

While the undisputed leader of super-sized food items remains the United States, it's not impossible to find examples here in Europe. This week, I discovered limited edition packages of giant M&Ms on sale. What really caught my eye, however, was the label's claim to contain candies which are 3x larger than the original.

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Now, there's lots of ways to compare sizes. M&Ms are round, and round things often are defined by their diameter. But a circle with a diameter thrice as large as another actually has an area nine times larger. If it's two spheres being compared, the volume will be 27 times larger! Such is the nature of the geometric progression. People have used this to create sloppy visual aids (not always unintentionally) which skew the appearance of relative sizes. A representation of a 3D object printed on 2D paper where the scaled relationship is supposed to be based on a 1D linear dimension is practically criminal in the world of information graphics.

Now, when talking about chocolates, you'd think the volume would be the most important property, since this determines the actual amount of chocolate in each piece. To investigate just how much bigger these larger M&Ms are, I'd need to do some measuring....

At least they use the same letter stamp

Ordinarily, it would make sense to measure several different instances, to ensure that the variability of manufacture could be averaged out. However, in the interest of time, I simply took a shot at those dimensions which seemed to be good representatives. First up, the diameter.

Large M&M: 20.5 [mm]

Regular M&M: 13.4 [mm]

The ratio of diameters (which represents two of the three dimensions of these objects) is 20.5 / 13.4 = 1.53. Now to check the thickness.

Large M&M: 10.0 [mm]

Regular M&M: 7.0 [mm]

The ratio of thickness is 10.0 / 7.0 = 1.42.  This is not the same ratio that was calculated for the diameters. This may be because the two M&M sizes are not the same shape, or it may simply be due to the random variance in the M&M making process. Let us assume that the values are accurate, and that we can calculate the difference in volume from the product of the differences of the three dimensions.

1.53 x 1.53 x 1.42 = 3.324

And there it is. The larger M&Ms happen to be just over 3x larger than the normal ones.

Interestingly, a question arises about the thin candy shell. Just how thin is it, and how does the thickness compare between the sizes? After all, the ratio of a surface area will not be the same as the ratio of a volume between two objects. Some careful slicing with a chisel reveals the cross sections shown below.

The candy shells are about the same thickness.

The purpose of the candy shell is to prevent the candies from sticking together or from making a mess while you hold them. ("Melt in your mouth, not in your hand") Likely, the shell is more expensive to produce than the chocolate inside; the thicker it is, the longer it takes to harden and dry. Therefore, I'd guess that the shell of a normal M&M is about as thin as possible while still being able to remain in good shape by the time you pour them out. Having tuned the machines to produce at these thicknesses, the larger M&Ms wind up the same way. Thus, while the overall candy is 3.324 times more volume than the regular ones, the quantity of candy shell is only about 2.2 times more. This means that the proportion of chocolate to candy shell is higher in the larger M&Ms. This is something which was qualitatively tested by consuming a handful or two. Biting down on them, I noticed that there was distinctly less sharp crunchy bits floating amongst the chocolate. Still delicious though.