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An emoji puzzle that takes a little integral calculus to answer

You may have seen some of the emoji puzzles that make their way around Facebook and other social media sites. Usually they require that you do some simple algebra (like adding 3 monkey emojis equal the number 15, so what number is one emoji monkey representative of?), but I just saw this little gem that actually required a little bit of fun ol' integral calculus. Give it a look-see:

What do you think? Does it look like fun? Give it a go and then I'll post the answer below.

Okay, let's talk about the answer to this fun little puzzle. It starts off pretty easy, just using a little bit of algebra. The first part gives us three bottles of beer added together to equal 30. Easy enough, right? Each bottle must represent the number 10:

After that, we get to mix our beer bottle variable with a new variable, a cheeseburger! Again, pretty easy math. The cheeseburger must represent the number 5:

And then, again, we get to use the variable from the last bit to workout the next part of the problem, where we find that two glasses of foaming ginger beer (hey, it can be whatever you want it to be, really) will represent the number 2:

But after that, things get a little harder. Now we have the following integral:

We can start by plugging in the stuff we already have (in this case, our beer bottle, cheeseburger, and glasses of foaming ginger beer variables). That yields:

Which then can be rewritten as:

If you haven't had much experience with integral calculus, that expression above probably still looks pretty confusing. If this is the case, then you might want to check out Khan Academy's lessons on integral calculus, since that'll give you a good leg up on how this type of math works. But, assuming you already have some experience with integral calculus, you might notice that the above expression is very similar to the improper integral of the sinc function over the positive real numbers. This kind of function actually has a specific name and a well known solution. It's called a Dirichlet Integral, and, in this case, has a solution of pi over 2:

So, if we solve the same way using our previous expression from the problem at hand, we get:

So the answer to the original problem is numerically 5pi/2. But we started off with a mix of emojis and numbers, so why not go back to emojis. We already have emojis for 5 and 2 (cheeseburger and two frosty glasses of ginger beer, respectively), but we need one for the number pi (which is usually represented by the Greek lower case letter). Why not use pie?! We then get a final answer of:

And that makes the problem even more fun! Now I think I'll go enjoy a cheeseburger and some ginger beer and follow up with a little pie. Cheers!

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