First, New Year's Resolution = more blog posts.\u00a0 In that spirit...<\/p>\n
A few days ago, teacher of math teachers and all-around math dude Christopher Danielson posted an interesting argument<\/a> with the conclusion that Double Stuf Oreos are not, in fact, stuffed double.\u00a0 I'll summarize it here.<\/p>\n 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.\u00a0 If we let w<\/em> be the number of calories in a wafer, and f<\/em> be the number of calories in one stuf's worth of \"creme\" filling, you can set up the following system of equations:<\/p>\n $latex begin{cases} 6w + 3f =160 \\ 4w+4f=140 end{cases}&s=2$<\/p>\n <\/p>\n After using your favorite system-solving method, you arrive at the following caloric values, rounded to two decimal places:<\/p>\n $latex begin{bmatrix} w\\f end{bmatrix} = begin{bmatrix} 18.33\\16.67 end{bmatrix}&s=2$<\/p>\n Simple enough, right?\u00a0 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.\u00a0 The problem: the only ostensible difference between one<\/strong> Double Stuf (2 wafers, 2 stufs) and one<\/strong> Triple Double (3 wafers, 2 stufs) is one little ol' wafer.\u00a0 And 30 calories.<\/p>\n Wait, what?\u00a0 How can one wafer account for a 30-calorie difference when we've already calculated a wafer to be less than 20 calories?\u00a0 That's more than a 50% discrepancy.\u00a0 Ergo, the claim of a true Double Stuf is on shaky ground.<\/p>\n I have to admit that I was excited by this.\u00a0 I mean, who doesn't want to look Nabisco square in the eye, call them to account, and win one for the little guy?\u00a0 I'm paying for (stuf x 2), and I want to get it.\u00a0 Before beating down the door at corporate HQ, though, let's take a closer look.<\/p>\n If we add this new information from the Triple Double package, we can update our system of equations:<\/p>\n $latex begin{cases} 6w + 3f =160 \\ 4w+4f=140 \\ 3w + 2f = 100 end{cases}&s=2$<\/p>\n Now, it's true that there is no exact solution to this system (it's overdetermined<\/strong>, i.e. more equations than unknowns).\u00a0 However, we can\u00a0 think of w<\/em> and f<\/em> as parameters and use least squares to get an approximate<\/em> solution.\u00a0 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 <\/em>and f<\/em> by:<\/p>\n $latex begin{bmatrix} hat{w}\\ hat{f} end{bmatrix} = begin{bmatrix} 18.82\\16.80 end{bmatrix}&s=2$<\/p>\n At first glance, that's not too far off---less than half a calorie for each item---from our original solution. \u00a0 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).<\/p>\n In order to makes sense of our estimates---and our errors---we need to have some idea about what manufacturers are doing<\/em>, exactly, when they put calorie info on the nutrition label.\u00a0 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<\/strong> (source<\/a>).\u00a0 That means our serving of original Oreos might actually have anywhere between 155 and 165 calories.\u00a0 Likewise, our serving of Double Stuf can legally contain between 135 and 145 calories.\u00a0 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.<\/p>\n But what about the Triple Double?\u00a0 Based on the mandated interval, the T.D. must legally have between 95 and 105 calories, but our estimates for w<\/em> and f<\/em> place it at about 90.\u00a0 Can we account for the apparent 5-calorie gap?<\/p>\n To begin with, 5 calories is a pretty small quantity.\u00a0 In fact, according to the government, \"amounts less than 5 calories may be expressed as zero\" (cf. above<\/a>).\u00a0 Five equals zero!\u00a0 (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.)\u00a0 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.<\/p>\n Second, the food scientists at Nabisco probably aren't actually measuring calorie content of products directly.\u00a0 They use pre-calculated tables of \"specific factors\" and \"general factors\" sanctioned by the USDA for different constituent ingredients.\u00a0 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.\u00a0 This method saves a lot of time and money, but they sacrifice accuracy in the process.\u00a0 These tables were created in 1973 and have only been \"slightly revised\" since then (cf. above, again<\/a>).<\/p>\n Third, and maybe most interesting, is something I've been holding out on you.\u00a0 In the Triple Double, one of the stufs is the familiar vanilla creme, and the other one is chocolate<\/strong>.\u00a0 Big deal, you say?\u00a0 It just might be.\u00a0 It turns out that all carbs are not created equal.\u00a0 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<\/a>), potentially different from vanilla carbs by about a factor of 3.<\/p>\n Seriously?\u00a0 Can flavored cremes really be that different?\u00a0 I submit one final piece of evidence.\u00a0 Golden Oreos have vanilla wafers instead of chocolate, but both have vanilla creme.\u00a0 What's more, they have the same calorie content.\u00a0 In terms of the outer layers, it seems that we can ignore their composition; a wafer is a wafer is a wafer.\u00a0 Besides the Triple Double, there is also a Neapolitan T.D., which has vanilla wafers (seemingly irrelevant), one strawberry stuf, and one chocolate stuf.\u00a0 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!\u00a0 It seems that the strawberry, vanilla, and chocolate cremes contribute different numbers of calories.\u00a0 Crazy, but true(?)<\/p>\n None of this is definitive, of course.\u00a0 There may be other weird factors at play that can't be ferreted out by nutrition labels alone.\u00a0 At the very least, though, I think we can safely stow our pitchforks for the time being.\u00a0 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!).<\/p>\n I know, I know.\u00a0 I'm disappointed, too.<\/p>\n","protected":false},"excerpt":{"rendered":" First, New Year's Resolution = more blog posts.\u00a0 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.\u00a0 I'll summarize it here. Using the nutritional info on the packages, you can […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[32,40,42,53],"class_list":["post-12","post","type-post","status-publish","format-standard","hentry","category-math-musing","tag-math","tag-nutrition","tag-oreos","tag-systems-of-equations"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/12","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=12"}],"version-history":[{"count":0,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/12\/revisions"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}