The Problem With Viral Social Media Math "Problems."

 Alright, it's time for a lesson on mathematics and data analysis. 

If you've been on social media for a while, you've probably seen "math problems" like the one below: 

1+1+1+1 = 4

2+2+2+2 = 16

3+3+3+3 = ?

This is an attempt at what in mathematics is called a "sequence," and claims to have a singular answer. Many people guessing will just be told "no" or "you're wrong." This type of problem has been used a lot in recent years to "prove" how children are often more intelligent than those with a full education in mathematics, which would be very strange if true, if not outright depressing. If children are truly more intelligent than those with educations, then they should be running things, but every time we try that kind of experiment, it doesn't work well. Go ask parents about what happens when kids get to decide what's for dinner, how chores work, or what financial priorities are. 

But I digress, the point here is that this is not a legitimate mathematics problem with a single solution. There are multiple solutions, and in fact multiple versions of this exact problems. The way a sequence problem is supposed to work is that it provides all necessary data to extrapolate a single solution, not provide just enough data to stumble across the one "correct" solution among many. Let's look at this one here. 

So, it's clear the first line is directly accurate, but the second requires more. If we assume the base difference, that there's a multiplication by two, then we have to look at the first line and see how that compares. If we simply assume that each line is multiplied by the number present within it, then the solution would be 36, as (3+3+3+3)3 = 36. However, another solution that allows for the first and second lines is not to simply multiply by the same number, but rather to break down each line from x+x+x+x to (x+x)(x+x). The first two lines remain the same, but the solution becomes 81, as (3+3)(3+3) = (9)(9) = 81. 

Many such problems also claim to require one to extrapolate nonexistent previous lines, or extrapolate based on that lack of existence, so we could consider the rule to be (x+x+x+x) * (1/2 * previous sum), which would give us the answer of 48. Or perhaps (x+x+x+x)+(2 * previous sum), which would give us an answer of 44. And the problem is, due to the lack of mathematical logic common to these problems, those are all equally likely solutions. When these questions are asked, those asking them are given a solution that is not the most simple, which in this case would have likely been 36. If I had to guess, I would say that those using this particular "problem" would insist on 81. It's completely arbitrary. 

Why does this matter? Because otherwise well-meaning, but under-educated, teachers have been known to use this kind of problem in the classroom. This teaches the students that the rules of mathematics are not logical, but arbitrary. There *are* ways to use problems similar to this to teach students about the many different ways mathematics can work, but none of them involve a single arbitrary answer with no leeway. As proof, I will explain how to fix this problem so it's actually useful. 

Instead of it being as shown above, remove the operators, in this case, the plus signs, add in your arbitrarily chosen solution, and include necessary instructions, as shown below:

[ ] 1 [ ] 1 [  ] 1 [ ] 1 [ ] = 4

[ ] 2 [ ] 2 [  ] 2 [ ] 2 [ ] = 16

[ ] 3 [ ] 3 [  ] 3 [ ] 3 [ ] = 81

Each row of operators must be the same. 

That will give students the ability to demonstrate their knowledge of the order of operations and all mathematical operators to provide a reasonable solution, a single answer using mathematical logic. On the downside, you don't get to play "gotcha" with college graduates because your 8 year old students got the answer right and they didn't, but at least you're not teaching your students that mathematics only matters when it matches an arbitrary set of rules that depends on the person giving the problem.