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Sloppy Math on Facebook, or "Who The Heck Would Write A Function Like That Anyway?!"

Does this look familiar?

SOLVE IF YOU ARE A GENIUS! 99% OF PEOPLE WILL GET IT WRONG!

8 = 56
7 = 42
6 = 30
5 = 20
3 = ?

No doubt every time you've seen this on
Facebook, it's followed by thousands upon thousands of responses,
each indignant that other people are getting the wrong answer.
Generally there are two or three different numbers that keep coming
up, with nobody able to see how anyone else could have gotten a
different answer from their own.

I hate these things.

These things are designed to be vague. There is no answer, or rather,
there are an infinite number of answers. The crux of the issue here is
that they don't define the rule.

So these things are basically a weird
way of presenting a function. You remember functions from my
previous blog post, right? Well, essentially what this thing is
saying is “you take 8, do some mystery function to it, and you get
56. Et cetera, et cetera, what do you get when you use 3?” The
problem is that there are multiple rules that could apply here, so
you have no idea what function they actually gave you and therefore
cannot answer the question.

Here's our example from above, showing
two different rules that could be used to create the list:

FIRST OPTION
y = x * (x-1)

56 = 8 * (8-1) = 8 * 7
42 = 7 * (7-1) = 7 * 6
30 = 6 * (6-1) = 6 * 5
20 = 5 * (5-1) = 5 * 4
 
So
y = 3 * (3-1) = 3 * 2 = 6

This is the answer that a lot of people
get because it's a bit more obvious. It's also the first answer that
jumps out at me. But there's a second option:

SECOND OPTION
n1 = 7
nt = n1 - (t-1)
y = x * nt

This second option is more of a
programming-type sequence, but no less legitimate. In this case, nt
means “the value of n for any given term t.” The rule given for
finding nt works out to mean that after each term, the n
value counts down by 1. n1 would be set to 7 by the
puzzle, as is common with rules of this type. This means the rule is
no longer relative, but absolute. No matter what first x value you
choose, the first n is 7. The second n is 6, the third is 5, and so
on. By this logic the beginning is still correct:



56 = 8 * 7                      First Term: t = 1, nt = n1 – (1-1) = n1 - 0
42 = 7 * 6                      Second Term: t = 2, nt = n1 – (2-1) = n1 - 1
30 = 6 * 5                      Third Term: t = 3, nt = n1 – (3-1) = n1 - 2
20 = 5 * 4                      Fourth Term: t = 4, nt = n1 – (4-1) = n1 - 3
 
BUT THEN
 
y = 3 * 3 = 9                  Fifth Term: t = 5, nt = n1 – (5-1) = n1 – 4, so 7 – 4, or 3

With this version you get 9 instead of
6. Why is that, when the other terms are all the same?
 
Notice they left out x = 4 in the
puzzle – it jumps straight from 5 to 3. If we were using the first
rule it wouldn't matter, since the rule is only relative to the
current x value and will work no matter where in the sequence a given
number finds itself. You could shuffle the lines around to your
heart's delight and the answers would still be the same. But this
version is what's known as a “recursive” sequence, where the rule
depends on where the term is in the sequence, and moving the terms
around will change the results drastically. What looks like a
logical sixth step is actually the fifth term in the sequence.

If we put 4 in for x where it looks
like it should be, it changes how you find the result of each term
thereafter. Then we'd have:

20 = 5 * 4
12 = 4 * 3
6 = 3 * 2

So we'd get the same answer as the
first option. But that's assuming they've left out a term, which you
can't be sure of since they didn't tell you that explicitly. To
assume that is simply to be a sloppy mathematician, and sloppy math
leads to incorrect math. Here is what they would have had to show
you for that to be the case:

n1 : 8 = 56
n2 : 7 = 42
n3 : 6 = 30
n4 : 5 = 20
n6 : 3 = ?

By showing you that it jumps straight
from the fourth term to the sixth, they're letting you know the
counter should have ticked down one additional time, and at that
point you will be able to solve the problem. But not without that
additional piece of information.

So here I've shown two legitimate ways
to solve this badly-written problem, getting two completely different
answers. Since the issue is that the problem is not specific enough,
the only prudent answer to these questions is always “There is not
enough information given.”

I hate these things because they are
intentionally vague. The people who make these problems (or at the
very least the people who post them on Facebook) likely don't realize
just how complicated the problem is. Most people will only see one
solution, so it will appear easy and they won't be able to understand
how someone else got 9 when it's “obviously” 6. This leads to
arguing futilely over something which, to any mathematician, is just
a problem that's so badly-written it's useless. No mathematician
worth her salt would ever write a sequence that way anyway, so why
argue about it?

Comments

Perhaps it's worth pointing out that an infinite number of analytic f(x) vs. x functions can be written to match any finite set of point values; there are tables of them in math reference handbooks for given sequences of f(x) (assuming x's are uniformly spaced, such as positive integers).
What a thought: "positive integer energy"!
^Intergergy?
 
Thank you for this awesome breakdown. Now I have something to point to that is far more eloquent than I am when I answer a question about mathematical rules and the importance of defining them prior to beginning a problem. Much appreciated!
I agree that there is no "Correct" answer to questions of this type, because there are a infinite number of mathematically valid answers. However, the key to these types of puzzles is an aesthetic sense of the "simplest" or "most elegant" solution: the first answer given here is clearly much simpler than the second, and therefore the better answer, though still not the only one possible. See my blog entry "Complete the Series" (http://www.wyzant.com/resources/blogs/246872/complete_the_series) for a discussion of this using a different example.
Hint - first answer this: pop, mop, hop, so on a green light you ___.
If you got that one right ("go"), then you should be able to see why the only possible correct answer is 3.

Another clue would be that, as noted above, if you follow the herd in thinking that you are supposed to be looking for a pattern, then you can reasonably come up with almost an infinite numbers of equally reasonable answers.

There was no instruction to find or follow a pattern.  Just a series of statements made to create misdirection, to throw you off from answering a simple question - like the example I gave with hop, pop & mop ....