There's a simple math meme going around Facebook that asks people to find the next element in a sequence—if we're being honest, there are lots of these going around. This one, like most of the others, is fairly straight forward and easily solvable by a brute force attack. Of course, as a mathematician, I took the problem a bit further by finding a function that would work for any value of $n$, not just the one the meme was asking for.

I'll give you both the brute force solution and mine as well later in the post. What I want to talk about first is why memes like this are both important and valuable tools.

If you guessed that most people would pass over my post without a second glance, you would be right. However, if I were to restate the problem and put it in a meme then people will jump all over it. Why?

If the goal is to get people thinking about math—and even working math problems for fun—then a meme like this is far more effective than stating the problem like it came out of a textbook. There are two things that make this meme effective. My original statement of the problem made it

The meme also offers a challenge, “97% Will fail this test.” While the statistic is likely inflated, the sentiment holds—most people

The good thing is that the meme not only got people thinking about math and working on a math problem, but also that it sparked discussions about math. But, I’m also interested in meme’s like this because they shed light on some actual problems in the United States—and probably around the world—when it comes to math education and attitudes about math.

The brute force solution is fairly straight forward. For each step we increase the first term by one, then increase the second term by one, and finally add the previous result. Thus we get…

$$1+4=5\\2+5=2+5+5=12\\3+6=3+6+12=21\\4+7=4+7+21=32\\5+8=5+8+32=45$$

Many of those who failed saw the basic pattern, but failed to account for the missing term.

Mathematics teaches us to think outside the box. It teaches us to look at the world a bit differently. So much so that mathematicians have been accused of practicing witchcraft in the past.

It seems that attitude may well be alive and kicking in the US still today. There is a general suspicion of modern mathematics that may not be as widespread as suspicions surrounding evolution, but one that exists none the less. Don’t believe me? Then consider the following quote…

There are so many things wrong with this quote and this attitude that it would take an entire post to discuss them all. For the moment it’s enough to point out that whoever wrote this knows precious little about mathematics, yet is somehow instructing generations of children in mathematics. The net result of this line of thinking is truly sad—people who will never be able to see the beauty in mathematics, because they were taught anything beyond the basics “is a creation of man and thus arbitrary and relative.”

That’s not something that I was ever taught, neither as an undergraduate nor as a graduate student. In fact, we are taught exactly the opposite. I can’t count the number of times I heard that “mathematics is the only body of knowledge that we can be 100% certain is completely true,” or something equivalent. Math is true whether or not you believe it. The integers are a ring but not a field—it doesn’t matter who you are, or what you think that fact remains a fact. There’s nothing arbitrary or relative about it.

Of course, the meme version of the problem is trying to make the problem accessible to everyone by assigning a new meaning to $+$ in this particular context. Which leads us into the general solution to the problem.

$$S(1)=5\\S(2)=12\\S(3)=21\\S(4)=32\\S(5)=45$$

We’re still at the brute force level, but the problem looks more mathematical.

The next step is to discover the function that making this thing tick. Notice that I said “discover” not create.

I know a couple things about this function. First of all, it’s recursive, i.e. to find $S(N)$ we have to first know $S(N-1)$, and to find $S(N-1)$ we must first know $S(N-2)$ and so on. We also know that for any $N\in\mathbb{N}$ we define $S(N)$ by adding three terms together.

From our brute force solution, we can quickly identify two of the three required terms.

$$S(N) = N+x+S(N-1)$$

A bit of basic observation shows us that we can define $x$ in terms of $N$ like this $x=N+3$ so that our function now looks like this…

$$S(N) = N+(N+3)+S(N-1)=2N+3+S(N-1)$$

But what about $S(1)$? It’s impossible to calculate $S(0)$ so we set it to what it must be, $S(0)=0$.

Finding the general solution is a fun exercise, and allows us write a simple computer program to calculate $S(N)$ for any $N\in\mathbb{N}$, if we were so inclined.

Better education is certainly a first step, but it’s a dead horse we’ve been beating since I was in school. We need some fresh ideas. What can we do to overcome these issues?

I'll give you both the brute force solution and mine as well later in the post. What I want to talk about first is why memes like this are both important and valuable tools.

## Meme Magic

Consider what the response would be if I posted a problem on social media that went something like this... "Given the sequence $5,12,21$, find the next two elements."If you guessed that most people would pass over my post without a second glance, you would be right. However, if I were to restate the problem and put it in a meme then people will jump all over it. Why?

Before we answer that, lets look at how the problem was stated in the meme versus how I stated the problem above.

*seem*difficult, while the restated version*seems*simple.The meme also offers a challenge, “97% Will fail this test.” While the statistic is likely inflated, the sentiment holds—most people

*will*fail to solve the problem. Judging by the comments on the original post, it seems closer to 75% failing, but that’s still high for such a simple problem.The good thing is that the meme not only got people thinking about math and working on a math problem, but also that it sparked discussions about math. But, I’m also interested in meme’s like this because they shed light on some actual problems in the United States—and probably around the world—when it comes to math education and attitudes about math.

## The Brute Force Solution

Among those who got it wrong, most did so because they missed a subtle point in the problem—there’s a missing term. There should be a $4+7=32$ line between the $3+6=21$ and the $5+8=?$ line. By missing this term, those who were on the right track failed to solve the problem because they calculated the following… $$5+8=5+8+21=34$$The brute force solution is fairly straight forward. For each step we increase the first term by one, then increase the second term by one, and finally add the previous result. Thus we get…

$$1+4=5\\2+5=2+5+5=12\\3+6=3+6+12=21\\4+7=4+7+21=32\\5+8=5+8+32=45$$

Many of those who failed saw the basic pattern, but failed to account for the missing term.

## What the Wrong Answers Tell Us

What really prompted me to write this blog post was not those individuals who failed to solve the problem because they missed the $4+7$ term. The motivation for this post came from those who disagreed with the problem in the first place. These are the people who answered $5+8=13$, and usually with quite a bit of attitude.Mathematics teaches us to think outside the box. It teaches us to look at the world a bit differently. So much so that mathematicians have been accused of practicing witchcraft in the past.

The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell. – Saint Augustine

It seems that attitude may well be alive and kicking in the US still today. There is a general suspicion of modern mathematics that may not be as widespread as suspicions surrounding evolution, but one that exists none the less. Don’t believe me? Then consider the following quote…

Unlike the "modern math" theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute....A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory. — Abeka.com

There are so many things wrong with this quote and this attitude that it would take an entire post to discuss them all. For the moment it’s enough to point out that whoever wrote this knows precious little about mathematics, yet is somehow instructing generations of children in mathematics. The net result of this line of thinking is truly sad—people who will never be able to see the beauty in mathematics, because they were taught anything beyond the basics “is a creation of man and thus arbitrary and relative.”

That’s not something that I was ever taught, neither as an undergraduate nor as a graduate student. In fact, we are taught exactly the opposite. I can’t count the number of times I heard that “mathematics is the only body of knowledge that we can be 100% certain is completely true,” or something equivalent. Math is true whether or not you believe it. The integers are a ring but not a field—it doesn’t matter who you are, or what you think that fact remains a fact. There’s nothing arbitrary or relative about it.

Of course, the meme version of the problem is trying to make the problem accessible to everyone by assigning a new meaning to $+$ in this particular context. Which leads us into the general solution to the problem.

## The General Solution to the Meme Problem

Anyone with a bit of mathematical sophistication can see that $+$ in the meme problem can be restated as a function $S:\mathbb{N}\to\mathbb{N}$. So now the problem looks like this…$$S(1)=5\\S(2)=12\\S(3)=21\\S(4)=32\\S(5)=45$$

We’re still at the brute force level, but the problem looks more mathematical.

The next step is to discover the function that making this thing tick. Notice that I said “discover” not create.

I know a couple things about this function. First of all, it’s recursive, i.e. to find $S(N)$ we have to first know $S(N-1)$, and to find $S(N-1)$ we must first know $S(N-2)$ and so on. We also know that for any $N\in\mathbb{N}$ we define $S(N)$ by adding three terms together.

From our brute force solution, we can quickly identify two of the three required terms.

$$S(N) = N+x+S(N-1)$$

A bit of basic observation shows us that we can define $x$ in terms of $N$ like this $x=N+3$ so that our function now looks like this…

$$S(N) = N+(N+3)+S(N-1)=2N+3+S(N-1)$$

But what about $S(1)$? It’s impossible to calculate $S(0)$ so we set it to what it must be, $S(0)=0$.

Finding the general solution is a fun exercise, and allows us write a simple computer program to calculate $S(N)$ for any $N\in\mathbb{N}$, if we were so inclined.

## Summing it Up

Problems like this are a great way to get people who would never spend their time working on math problems, to work on math problems. But they also uncover a general suspicion from the public towards mathematics as well as the usual issues with mathematics—hatred of math, math is too hard, math is useless, etc.Better education is certainly a first step, but it’s a dead horse we’ve been beating since I was in school. We need some fresh ideas. What can we do to overcome these issues?