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I'm Glad I've Got a Proofreader

In addition to occasionally writing math problems for this website, I've also written math problems for several states' math competitions. When I sit down to write problems, it's usually because I have to crank out a set of 100 math problems for a competition somewhere. When I'm finished writing them, I send them off to my proofreader. When he's proofreading, what do you suppose he's most likely to find?

  1. spelling mistakes
  2. grammar mistakes
  3. arithmetic errors
  4. algebraic errors
  5. ambiguity errors

He does occasionally find spelling/grammar mistakes, but those are fairly infrequent. And yes, I do occasionally have arithmetic/algebraic errors! (When I'm writing a problem, sometimes I have to try two of three different things to make the problem work out correctly, and then I forget which things I've changed, and use numbers from the first version of the problem, and then my proofreader says, "Where in the world does this number come from?")

But the most common issue he finds is an ambiguity issue. Ambiguity simply means that students could reasonably interpret the problem in more than one way. If the student could interpret the problem in multiple ways, then it's a bad problem!

Here's a simple example:

In a right triangle, the sides are 5, x, and x + 1. Find the length of the hypotenuse.

Why is this ambiguous? Because I've implied that there's one answer ("find the length" instead of "find all possible lengths") but there are two possible answers, because the hypotenuse could be 5 (if x and x + 1 are less than 5) or it could be x + 1 (if x + 1 is greater than 5).

Here's another example:

Find the next term in this sequence: 5, 10, 15, ...

The obvious answer is 20, but that's not the only possibility; someone who is familiar with Fibonacci sequences could argue that this is the beginning of a Fib, and therefore the next term is 10 + 15 = 25.

These are just simple examples. The more complicated a problem is, the more likely it is to introduce something ambiguous without meaning to. So I'm glad I've got a problem proofreader.

The Internet Has No Proofreader

Now I put on my other hat - the "Professor Puzzler" hat. On any given day, this blog is a pretty quiet corner of the internet. This blog usually gets just a few hundred visitors per day. (Except on days we publish a new post, then the visitor rate climbs a bit).

But every once in awhile, something goes crazy. Back in February, for instance, there was one day where the traffic to the Professor Puzzler blog was 100 times it's normal rate. We had over 25 thousand page views on just one page of the blog.

Whenever this happens, it's invariably because someone posted an ambiguous (bad) math problem on the internet, and someone linked to our explanation of why it can't be solved.

And I realized something, after the most recent one: the majority of math problems posted on social media are ambiguous. The internet has no proofreader.

If you're interested to see some of the ambiguous/bad problems we've talked about in this blog, here are links to some of them. Just click on the image to jump to the corresponding blog post.

The most recent one was "The Flower Problem" or (as one site called it) "Pollen Pandemonium." This problem is straight-up ambiguous (and no, it was not part of a kindergarten exam in China; give the Chinese some credit, will you?). The most appropriate interpretation is to say that each distinct flower image represents a different variable. If that's the case, you have a system of three equations in four unknowns, which is unsolvable. If you operate from the assumption that the problem must have a solution, you reject the most appropriate interpretation, and then there is no way to solve it without making an assumption about what the writer intended. Each assumption will lead you to an answer, but depending on the assumption you make, you'll get a different answer. So the internet is full of people arguing for their interpretation, when the reality is, every solution is based on an unwarranted assumption.


The square problem is also ambiguous in the same way that my sequence problem above is ambiguous; there are multiple ways of interpreting the intent of the problem writer, and there isn't one right answer; there are many possible answers. But the problem is great for click-bait sites, because people will spend all day arguing their answer in the comments.



The "genius" problem is bad for a couple reasons. One is the horrible math notation. 8 does not equal 56. The problem writer should have used function notation, OR an arrow instead of an equals sign in order to indicate "8 maps to 56." Because of the way it's written, an answer of "3" is probably the most reasonable answer. 3 = 3, even if the problem writer got all the other equations wrong. But setting that aside, many of these types of problems are designed so there are multiple ways of interpreting the sequence, for precisely the same reason as I mentioned on the square problem: it gets people posting on their website or Facebook page!


 And then, our most popular "viral" post - the apples/bananas/coconuts problem. For the record, we didn't create this problem. Someone asked us about it, and we wrote up an explanation. This problem is the least ambiguous of them all, and can be reasonably solved without making assumptions. However, if you read our blog post on this problem, you'll note that we have an alternate interpretation from a very articulate fellow named Pany, who argues that there's another way of interpreting the problem. That's the sort of thing my proofreader looks for. More often than not, if my proofreader thinks something is ambiguous, we'll dump the problem, even if I don't agree with him, just to be on the safe side!



A new addition to this page: the "Rabbit Going to the River" problem, which you can read about by clicking the image below. This problem is also quite ambiguous.


Also, you may find the following post interesting, which is about ambiguous notation.

And there you have it: a history of bad math problems on the Ask Professor Puzzler blog. Click-bait sites love problems like this, because they generate arguments, and arguments turn into page views, comments, likes, and notoriety.

Math teachers, on the other hand, hate them...except as teaching tools about ambiguity!

Update June 20, 2016: One wise commenter responded to this blog post by saying "So people would rather argue, pretending unsolvable problems are solvable, than accept the simple fact that the problem was badly phrased to begin with. Seems this could have applications to Internet interactions far beyond the math realm!"

Can't argue with that!


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