Crack the Code!

Here's a fun one that I just saw on Twitter. Can you "crack the code"?

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!

Another order of operations adventure

Here's a gem I've seen floating about in Facebookland lately. A lot of people are answering with either 58 or 10. What do you think?

Find this puzzle and more awesome stuff at Curiosity

What do you think? Did you get 58? Did you maybe get 10? 

If your answer is 10, you likely made a mistake in using the order of operations. Let's talk about what that mistake was.

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The Order of Operations

As you may know from my previous posts (like Where the Math is Lacking), I love a good math problem. Problems like the one above are a great way to see whether people remember the Order of Operations for solving mathematical equations. 

In case you have forgotten the order of operations, then you might recall the mnemonic acronym "Please Excuse My Dear Aunt Sally", which stands for PEMDAS.

PEMDAS is an acronym for the order of operations and it stands for Parentheses-Exponents-Multiplication/Division-Addition/Subtraction:

This order of operations allows us to make sure that we're all following the same rules for solving or simplifying problems, or, at least, that's what it's supposed to do. Unfortunately, a lot of people have forgotten how to use the order of operations. Let's talk about that problem above again and how the order of operations applies to it.

We have the following problem:

6² ÷ 2(3) + 4 = ?

Let's step through each part of PEMDAS and see how the order of operations applies to this problem.

We have to start with the Parentheses part of the order. This is actually the step where most people get the wrong answer to this problem. The Parentheses step means that we have to solve terms or sets of terms within parentheses and brackets before solving the rest of the problem. 

For instance, if I had 2+(3+4), then I would solve the stuff inside of the parentheses first to get 2+(7). At that point, the parentheses are no longer necessary, since 2+(7)=2+7=9.

In our problem above, what's inside of the parentheses (the number 3) doesn't need to be operated on since it's already simplified. So, following the parentheses step, we still have the problem:

6² ÷ 2(3) + 4 = ?

Now moving on to the Exponents part of PEMDAS, we see that we have to solve the six squared term. That's pretty simple:

6² ÷ 2(3) + 4 = 36 ÷ 2(3) + 4 = ?

Then, the Multiplication and Division step tells us to solve all multiplications and divisions in the problem, but there's also a caveat there as well. Since many western languages read from left-to-right, we've also developed our modern mathematics to read left-to-right. This means that we have to sequentially solve multiplications and divisions reading from left to right in a problem. So for out current version of the problem,

36 ÷ 2(3) + 4 = ?

We have to solve the division of 36 by 2 first and then we can apply the multiplication by 3. Yielding:

36 ÷ 2(3) + 4 = 18(3) + 4 = ?

and then

18(3) + 4 = 54 + 4 = ?

And now we can take this thing home by solving the final step, Addition and Subtraction.

54 + 4 = 58

Bump bump baaaaaaah! The answer to this problem is 58. 

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So how did some people get 10? 

Like I said, most people who had trouble with this problem made their mistake in the first step. When solving for the parentheses part of PEMDAS, you only have to sequentially solve the parts that are within parentheses and/or brackets. However, if you apply the multiplication outside of the parentheses first, then you'd be solving the problem like this:

6² ÷ 2(3) + 4 = 

6² ÷ 6 + 4 = 

36 ÷ 6 + 4 = 

6 + 4 = 10

However, this is actually the wrong way to solve this problem. If you apply a multiplication outside of the parentheses first, you're basically then solving the operations out of order (MPEMDAS, in this case). The parentheses step in PEMDAS doesn't apply to operations outside of parentheses or brackets.

Here's another example I can throw at you, say we have this problem:

6 × (5 + 3) + 4 = ?

We could take away the multiplication symbol since the parentheses will already tell us to multiply, yielding:

6(5 + 3) + 4 = ?

Now it should be clear that we solve the parentheses first and then multiply, giving us:

6(5 + 3) + 4 = 6(8) + 4 = 48 + 4 = 52

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The same way that we removed the multiplication symbol in that problem could be applied to the original problem above. When you look at the problem and see it like this:

6² ÷ 2 × (3) + 4 = ?

Then it becomes far more apparent that the answer won't be 10. We can then see the right answer will be 58:

6² ÷ 2 × (3) + 4 = 58

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Element 73

Sheldon: I made tea. 
Leonard: I don't want tea. 
Sheldon: I didn't make tea for you. This is my tea. 
Leonard: Then why are you telling me? 
Sheldon: It's a conversation starter. 
Leonard: That's a lousy conversation starter. 
Sheldon: Oh, is it? We're conversing. Checkmate.

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Just watched the 5th season of The Big Bang theory.  In an early episode of this season, I noticed Sheldon Cooper (played by Jim Parsons) wearing a shirt with the number 73 imprinted on it:

I recalled there being an earlier episode (Season 4, Episode 10, "The Alien Parasite Hypothesis") where Sheldon avowed his love of the number 73, but I couldn't remember why.  Then I did a quick internet search (my how the world has changed) and found this quote from said episode: 

"The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3.  In binary, 73 is a palindrome, 1001001, which backwards is 1001001."

73: The number, the myth, the legend...

Ah hah!  Behold the power of the number 73.  If you check out the wikipedia entry for "73 (number)" you'll find all of this as well:

- Seventy-three is the 21st prime number. The previous is seventy-one, with which it comprises the 8th twin prime. It is also a permutable prime with thirty-seven. 73 is a star number.

- 73 is the largest minimal Primitive root in the first 100000 primes. In other words, if p is one of the first 100000 primes, then at least one of the primes 3, 5, 7, 11, 13, 17, ..., 73 is a primitive root modulo p.

- 73 is the smallest prime congruent to 1 modulo 24.

- 73 is an emirp.

- The mirror of 73, the 21st prime number, 37, is the 12th prime number. The number 21 includes factors 7 and 3. The number 21 in binary is 10101 and Seventy-three in binary, 1001001, both are a palindrome. In addition, of the 7 binary digits representing 73, there are 3 ones. Also, 37+12=49 (seven squared) and 73+21=94=47*2, 47+2 also being equal to seven squared. Additionally, both 73 and its mirror, 37, are sexy primes twice over, as 31, 43, 67 and 79 are all prime numbers (sexy primes are primes that differ from their next prime number by a value of 6).

Tantalum: the 73rd Element

Pretty cool, huh?!  73 is pretty awesome.  

But what about the 73rd element on the periodic table?  Is there anything cool about Tantalum? 

Tantalum (symbol: Ta) is the 73rd element on the periodic table.  It was discovered in1802 by Swedish chemist Anders Gustaf Ekeberg.  It has an atomic weight of 180.9479 and it's most abundant isotope has 108 neutrons in the nucleus.  

One could do an internet search and find their fill of chemical and physical information about Tantalum, but here are a few of the most interesting aspects (at least, the ones that I find most interesting!):

- Tantalum has a melting point of 3290 K!  Only tungsten and rhenium have higher melting points

- Ekeberg named Tantalum comes from King Tantalus, father of Niobe, in Greek mythology. Tantalum is almost always found with Niobium in nature and Niobium was named after Niobe.

- Almost all of our tech devices (computers, smart phones, HD TVs, etc) have capacitors containing small amounts of Tantalum

- Tantalum is commonly used in the production of surgical tools, metal sutures, and rods and plates for mending broken bones and other injuries (since Tantalum tends to resist chemical reaction with other agents)

So there you have it! Some interesting things about the number 73 and the element 73. Perhaps 73 really is the best number. Or, at least, maybe just one of the best.