Just as I turned off the TV before heading out the door for a run last night, an unseen but jaunty-sounding fellow confided from within the idiot box that there were 1,537 ways to order a Burger King Whopper® (about 1.2 to 3 per calorie provided, I later ascertained). Being a numbers sort by inclinaton, I had no choice but to try to figure out, armed only with basic knowledge about the sandwich in question and lingering skills from my academic salad days, how the BK geek squad had come up with this figure.

After a few minutes of in-flight computation (during which I even turned off my MP3 player in order to hone my focus), I realized that the lowest factor of 1,537 was 29, a prime number, and that 29 needed to be multiplied by another prime, 53, in order to give the requisite total of advertised sandwich varieties. As I found this vexing, I gave up the mental chase until I got home, at which time I did a Google search that produced this information:

“With the addition of our new ingredient … reduced fat mayonnaise … the number of ways to order and enjoy a WHOPPER(R) sandwich now grows from 1024 to 1,537,” Clouser said. The new reduced fat mayonnaise contains only three grams of fat, bring the total grams of fat for the BK VEGGIE(TM) burger to just ten grams. The new BK VEGGIE(TM) burger is also made without soy.

With its nine original special ingredients (sesame seed bun heel and crown, beef patty, pickles, ketchup, onions, tomatoes, lettuce, and mayonnaise) and three optional ingredients (cheese, bacon and mustard), Burger King Corporation calculated in the mid-seventies that there were 1,024 ways to order a WHOPPER(R).

Now, 1,024 was a number I, like any binary-oriented being, could live with. I reckoned that any Whopper® worth its weight in atherosclerotic plaques had to contain, at a so-called minimum, a pattie and a complete bun, reducing the number of optional ingredients to nine. Given that these nine ingredients could either be present or absent (two choices), this yielded 2^{9} possible permutations, or 512. I then boldly assumed that one could order a bun either with or without sesame seeds in order to double this number to the proffered 1,024. So far, so finger-lickin’ good. Whoops, wrong grease-fest.

But I ran into a roadblock trying to figure out how the addition of a second type of mayonnaise added 513 more selections. I would think that one would have the option of either simply spooning the low-fat mayo onto the existing mess, implying a new total of 2,048, or substituting the low-fat mayo for the regular kind, keeping the number at 1,024. I also noticed that 1,537 is almost, but not quite, smack-gob between 1,024 and 2,048.

So either I’m missing something or the BK minions made a goof somewhere. Complicating matters is that the restaurant giant’s menu lists four types of “baseline” Whopper® choices — the original, the double, the triple and the Whopper Junior®.

Anybody feel like chewing on this one?

I think they are assuming no one would order both high fat and low fat mayo on the same sandwich, ergo it is an option for 2^9 combinations rather than 2^10.

If you’ve got 9 independant (two way) choices and then one three way choice (no mayo, high fat or low fat, but not both high & low fat), then that gives 2^9 * 3 = 1536 choices.

If you’ve got four baseline choices then it probably becomes:

4 * 2^7 * 3, which is the same (1536).

Where the other option to get it up to 1537 is I don’t know, a BigMac perhaps?

By my calculation, it should be 2^9 * 3 = 5136.

For all the other choices, you have one of two possibilities; for the mayonaise, you have 3 possibilities (no mayo, regular mayo, low-fat mayo). So, you replace one of the 2’s in your original calculation with a 3.

I have no idea why they added one more to that.

Not a math guy, really, it would seem to be that the low-fat mayo is something that can only be chosen when the regular mayo is not chosen, effectively turning the mayo option into a three-way (yes, no, low-fat): 2^9 * 3 instead of 2^9 or 2^10…

Obvious typo: 2^9 * 3 is 1536, not 5136. Duh!

Add 1 for the option of not eating a whopper and ta-da, 1537!

They are making the mayo a three valued option, I’m guessing: none, new mayo-like substance or mayo. Thus, the 9 non-mayo binary options, times the new three valued mayo option: (2^9)*3=1536. Ummm, plus one. Why plus one? Yo no s.

Thanks to the many (by nascent blog standards, at least) kind people who nudged this artery-teaser along the correct path. Hindsight with respect to problem parametrization is always keen, but I am nonetheless aghast at having interpreted the mayonaise issue as a 2*2 problem and not a 3*1 problem (obviously there is only one way to have no mayo, period). I’m now going to deny ever having minored in math without repeating any courses, and I’m sure I’ll soon find numerous egregious ways to publicly establish that my physics degree should be torn up as well.

I am still suspicious, as are others, that 1,537 — not 1,536 — is the actual answer and will be writing Burger King to inquire about this.