2021 National MathCounts Team Round #9 (video solution)

Gretchen labels each of the six faces of a cube with a distinct positive integer so that for each vertex of the cube, the product of the three numbers on the faces touching the vertex is a perfect square. What is the least possible value of the sum of the numbers on this cube?

Defeat the AMC: lather rinse repeat

I track the scores of my students who are prepping for various math contests. Here is an example of student who is hoping to qualify for AIME next year. My method is to assign an old AMC 10 as homework, and the student takes it under timed exam conditions. The student grades her exam at home, and then spends as much time as she likes to solve any remaining problems. We meet to discuss the problems that could not be solved, and I explain the solutions and offer quick proofs and explanations of underlying theorems.

There isn’t a lot of strategy or overthinking here. Old AMCs are freely available online, with 2 per year, there is no end of practice tests. Take an exam, study what you could not solve, take another exam, lather, rinse, repeat.

In just a few months this student has raised her score to very close to the AIME floor most years. Notice she’s more than doubled the number of correctly solved problems.

There’s no magic or secret way to prepare. Time spent researching books and classes is better spent taking as many exams as possible and most importantly, studying the problems you couldn’t solve. Good luck!

Math Contests and the Spacing Effect

The Spacing Effect is the idea that students are better able to recall information if they study and restudy the same ideas over multiple sessions in time. This is the opposite of cramming for a test, where that knowledge quickly degrades and becomes unavailable to our brains. Retrieving the same information trains our brains to assign greater importance to those facts.

Math generally lends itself well to the Spacing Effect since topics build on each other. Algebra is used to solve geometry problems, and geometry can be used to solve probability problems. If you need to find the area of square to solve a probability problem, then boom, you are recalling facts you learned last year. And now your understanding is deeper and you are less likely to forget.

Math contests are a great way to leverage the Spacing Effect. Typically students learn a math algorithm or property or theorem, complete a bunch of homework problems, take a test and move on to the next topic. Math contests include problems from all different subject areas: algebra, probability, geometry, number theory, etc. In order to answer a question, you may need to retrieve information you studied even 1-2 years ago! Take enough practice tests and you are using that information repeatedly in new situations.

I like to say that students who participate in math contests aren’t allowed to forget the math they’ve learned. They are recycling their knowledge and using it in novel problems.

These are not students who need to prepare for the math section of the SAT because they’ve been preparing for it their whole academic lives. Without putting a label on it, they’ve been using the Spacing Effect to their advantage.

Learning from AMC Statistics

I never waste an opportunity to teach using data, and a perfect example is this histogram of scores from the latest AMC 10. Notice the spikes at regular intervals among the lower scores to the left. What is going on here?

These students do not understand how the AMC is scored, in particular that you receive 1.5 points for each answer left blank. While you receive 6 points for each correct answer, you receive no points for each wrong answer.

Take a look at the spike occurring at 30 points. You can earn 30 points by answering 5 questions correctly and the rest are wrong, with no questions left unanswered. A student can choose the correct answer to 5 problems on a multiple choice exam with 25 questions and 5 answer choices, by selecting answers at random.

As an exercise, try to generate scores 24, 36, and 42 as a naive student who does not leave any question blank. There is no excuse to not know this scoring rubric, as it is explained on the front of each exam booklet:

I coach all my AMC students to write a “clean exam.” That is one where all answered questions are correct and no questions are answered incorrectly. All other questions are left unanswered to achieve the highest possible score. (Notice the higher scoring students to the right are savvy to this strategy.)