Choices or No Choices

I am coaching my AMSA students for math competitions. Recently, I gave them the following problem from the 1964 MAML:

The difference of the squares of two odd numbers is always divisible by:
A) 3, B) 5, C) 6, D) 7, E) 8?

The fastest way to solve this problem is to check an example. If we choose 1 and 3 as two odd numbers, we see that the difference of their squares is 8, so the answer must be E. Unfortunately this solution doesn’t provide any useful insight; it is just a trivial calculation.

If we remove the choices, the problem immediately becomes more interesting. We can again plug in numbers 1 and 3 to see that the answer must be a factor of 8. But to really solve the problem, we need to do some reasoning. Suppose 2k + 1 is an odd integer. Its square can be written in the form 4n(n+1) + 1, from which you can see that every odd square has remainder 1 when divided by 8. A solution like this is a more profitable investment of your time. You understand what is going on. You master a method for solving many problems of this type. As a bonus, if students remember the conclusion, they can solve the competition problem above instantaneously.

This is why when I am teaching I often remove multiple choices from problems. To solve them, rough estimates and plugging numbers are not enough. To solve the problems the students really need to understand them. Frankly, some of the problems remain boring even if we remove the multiple choices, like this one from the 2009 AMC 10.

One can holds 12 ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?

It’s a shame that many math competitions do not reward deep analysis and big-picture understanding. They emphasize speed and accuracy. In such cases, plugging in numbers and rough estimates are useful skills, as I pointed out in my essay Solving Problems with Choices.

In addition, smart guessing can boost the score, but I already wrote about that, too, in How to Boost Your Guessing Accuracy During Tests and To Guess or Not to Guess?, as well as Metasolving AMC 8.

As the AMC 10 fast approaches, I am bracing myself for the necessity to include multiple choices once again, thereby training my students in mindless arithmetic.



  1. Jonathan:

    Careful, if we remove the choices from the 1964 problem, we need to reword to ask for the largest number that always divides the difference of two consecutive odd integers (thus creating, imho, a bit of a reading comprehension problem. Not a tremendous impediment, but not to be ignored either).

    I like, btw, (2k+1)^2 – (2k-1)^2

  2. Harry Stern:

    For me, the question is “why should I compete?” (besides the fact that I can’t because there isn’t anywhere for me to take it)

    As you noted, some of the problems are very boring, so why would I spend my time preparing for something that bores me? I would be interested to hear any reasons for competing in these math competitions besides, perhaps, to get to compete in the higher level ones.

    Also, @Jonathan, minor typo. (2k+1)^2 – (2k+1)^2 is always zero. It should be (2k+1)^2 – (2m+1)^2, or some other variable.

  3. misha:

    I recently tutored someone for GRE exam. The student complained that in the commercial GRE preparation class the teacher very often explained how to get the right choice by actually avoiding solving the problem. When somebody asked him about mathematics behind the problem he would usually reply, “don’t over-think it,” and refused to explain. It looks like this problem of “teaching to pass the math test” instead of teaching mathematics is endemic in the commercial test-prep classes. The questions come to mind: “What are these tests actually testing, does it have anything to do with the subject?” It looks like all too often the answers to these questions are “we are not sure” and “not too much.”

  4. JBL:

    @ Harry Stern: people who enjoy competing in math competitions should do so. People who don’t, shouldn’t. Is this a good enough answer for you?

  5. Jonathan:

    Hi JBL!
    And Harry, 2k+1 and 2k-1 are consecutive odd integers. Their squares are 4k^2 + 4k – 1 and 4k^2 – 4k – 1, and the difference of those squares is 8k.

  6. JBL:

    Hi Jonathan! Hope all is well!

  7. Harry Stern:

    @JBL Certainly if someone enjoys competing then they should compete. One way to interpret my question is as asking why they enjoy it. Surely there must be a reason why it is enjoyable, not “just because”.

    @Jonathan I see now. If you look on my comment, I misread yours and thought they were both 2k+1. My mistake.

  8. JBL:

    @Harry Stern, people enjoy doing all sorts of different things, and many people enjoy adding a competitive aspect to things that they also enjoy for other reasons. Why should this need a special defense, any more than any other human preference? It seems odd to me to wander onto a blog and demand that people justify the existence of math competitions to you.

  9. Harry Stern:

    @JBL I am curious as to why you think that I am demanding anything from anyone, or that they need a justification to exist. I am simply interested in knowing whether, from my lack of experience in these competitions, there are interesting parts that I have not seen. I did not say it needs a special defense, nor did I say that they shouldn’t exist (in fact, I preceded my original comment with “[f]or me”); however as you noted many people enjoy competing for the sake of competing, which I did not really consider. This is the kind of answer I was looking for. I apologize if I came off as demanding.

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