# What’s in a Stuf?

First, New Year’s Resolution = more blog posts.  In that spirit…

A few days ago, teacher of math teachers and all-around math dude Christopher Danielson posted an interesting argument with the conclusion that Double Stuf Oreos are not, in fact, stuffed double.  I’ll summarize it here.

Using the nutritional info on the packages, you can easily verify that one serving of originals (3 cookies) is 160 calories, and one serving of the Double Stuf variety (2 cookies) is 140 calories.  If we let w be the number of calories in a wafer, and f be the number of calories in one stuf’s worth of “creme” filling, you can set up the following system of equations:

$\begin{cases} 6w + 3f =160 \\ 4w+4f=140 \end{cases}$

After using your favorite system-solving method, you arrive at the following caloric values, rounded to two decimal places:

$\begin{bmatrix} w\\f \end{bmatrix} = \begin{bmatrix} 18.33\\16.67 \end{bmatrix}$

Simple enough, right?  Then Nabisco released the Triple Double, a Big-Mac-like creation with three wafers and two layers of creme, an Oreo so massive that the recommended serving size is a single Frankencookie with 100 calories.  The problem: the only ostensible difference between one Double Stuf (2 wafers, 2 stufs) and one Triple Double (3 wafers, 2 stufs) is one little ol’ wafer.  And 30 calories.

Wait, what?  How can one wafer account for a 30-calorie difference when we’ve already calculated a wafer to be less than 20 calories?  That’s more than a 50% discrepancy.  Ergo, the claim of a true Double Stuf is on shaky ground.

I have to admit that I was excited by this.  I mean, who doesn’t want to look Nabisco square in the eye, call them to account, and win one for the little guy?  I’m paying for (stuf x 2), and I want to get it.  Before beating down the door at corporate HQ, though, let’s take a closer look.

If we add this new information from the Triple Double package, we can update our system of equations:

$\begin{cases} 6w + 3f =160 \\ 4w+4f=140 \\ 3w + 2f = 100 \end{cases}$

Now, it’s true that there is no exact solution to this system (it’s overdetermined, i.e. more equations than unknowns).  However, we can  think of w and f as parameters and use least squares to get an approximate solution.  I’m not going to reproduce all the steps here, but after setting up some matrices, taking some transposes and inverses, and doing some multiplications, we can estimate w and f by:

$\begin{bmatrix} \hat{w}\\ \hat{f} \end{bmatrix} = \begin{bmatrix} 18.82\\16.80 \end{bmatrix}$

At first glance, that’s not too far off—less than half a calorie for each item—from our original solution.   In fact, plugging those values back into our equations gives us about 163 calories for a serving of original Oreos (advertised calories + 3), about 142 calories for a serving of Double Stuf (advertised calories + 2), and about 90 calories for a Triple Double (advertised calories – 10).

In order to makes sense of our estimates—and our errors—we need to have some idea about what manufacturers are doing, exactly, when they put calorie info on the nutrition label.  First of all, for any item with a per-serving calorie content north of 100, the government only requires that companies round their values to the nearest 10 calories (source).  That means our serving of original Oreos might actually have anywhere between 155 and 165 calories.  Likewise, our serving of Double Stuf can legally contain between 135 and 145 calories.  With that in mind, our 2- and 3-calorie variations from the advertised values aren’t even worth a second look; they’re well within the acceptable range.

But what about the Triple Double?  Based on the mandated interval, the T.D. must legally have between 95 and 105 calories, but our estimates for w and f place it at about 90.  Can we account for the apparent 5-calorie gap?

To begin with, 5 calories is a pretty small quantity.  In fact, according to the government, “amounts less than 5 calories may be expressed as zero” (cf. above).  Five equals zero!  (Incidentally, this explains how a product like nonstick cooking spray, which is essentially 100% fat, can legally be labeled as having zero calories: the serving size is so tiny that it contains < 5 calories.)  If 5 calories is small enough to be considered negligible for nutrition labels, it’s not unreasonable to think that, in calculating the calorie content of cookies, a company might err by that amount without malicious intent.

Second, the food scientists at Nabisco probably aren’t actually measuring calorie content of products directly.  They use pre-calculated tables of “specific factors” and “general factors” sanctioned by the USDA for different constituent ingredients.  They may, for instance, use a general calorie count (per gram) for glucose and just assume that the glucose in Oreos behaves, energetically, like glucose everywhere else on the planet.  This method saves a lot of time and money, but they sacrifice accuracy in the process.  These tables were created in 1973 and have only been “slightly revised” since then (cf. above, again).

Third, and maybe most interesting, is something I’ve been holding out on you.  In the Triple Double, one of the stufs is the familiar vanilla creme, and the other one is chocolate.  Big deal, you say?  It just might be.  It turns out that all carbs are not created equal.  In particular, the value of carbohydrate energy in chocolate is extremely hard to pin down, and can in fact vary widely, from 1.33 calories/g to 4 calories/g (source), potentially different from vanilla carbs by about a factor of 3.

Seriously?  Can flavored cremes really be that different?  I submit one final piece of evidence.  Golden Oreos have vanilla wafers instead of chocolate, but both have vanilla creme.  What’s more, they have the same calorie content.  In terms of the outer layers, it seems that we can ignore their composition; a wafer is a wafer is a wafer.  Besides the Triple Double, there is also a Neapolitan T.D., which has vanilla wafers (seemingly irrelevant), one strawberry stuf, and one chocolate stuf.  To clarify, once we ignore the apparently negligible difference in wafers, the only difference between the T.D. and the Neapolitan T.D. Oreos is a strawberry stuf in place of a vanilla one, but the Neapolitan has 10 more calories per cookie!  It seems that the strawberry, vanilla, and chocolate cremes contribute different numbers of calories.  Crazy, but true(?)

None of this is definitive, of course.  There may be other weird factors at play that can’t be ferreted out by nutrition labels alone.  At the very least, though, I think we can safely stow our pitchforks for the time being.  Double Stufs may, in fact, be short on stuf, but we can’t prove that beyond a reasonable doubt without breaking out a digital scale and a bomb calorimeter (I’m looking at you, science teachers!).

I know, I know.  I’m disappointed, too.