Mathematical Tricks in the Academic Hunger Games

February 14, 2024

Geometry problems, especially well-designed geometry problems are generally awesome. Sometimes, some of these problems are even more puzzle-like than these so-called math puzzles: you either get it, or you don’t.

I hate geometry problems.

The Academic Hunger Games

I spent most of my life living in China, and this meant that I got to experience the totally-not-Orwellian torment that is Zhongkao. ⊕Alternatively, the Junior High School Scholastic Aptitude Examination

Here’s the deal. You take a test, including sections on Mandarin, math, a foreign language (which is almost always English), various sciences, and various social studies topics. Based on the score and the only the score, 50% of the people get to go to high school, and the other 50% either drop out or go to a vocational high school.⊕

This is the real score distribution from when I graduated middle school, just to give you a mental scale.

There’s no need to stress, just do your best. Worst case scenario you go to an assembly line and work 14 hours every day until you die.

What do you do, then? Obviously, people start spending more time at school, and they get better at taking tests; and obviously the test gets harder each year because people became better at taking tests.⊕
Yes this is a school. Yes they are literally called the anti-suicide bars. Then people start spending even more time at school. And the cycle has never stopped. It is not ever going to stop, either.

Anyways, back to geometry problems.

Hard-for-grade-level geometry problems are always on the math section of Zhongkao. You are only allowed to test on a limited set of skills - this is a test for 9th graders, after all. However, with well-crafted puzzles, the time, effort, and cleverness that is needed to solve these problems are at a significantly higher level compared to the “base” skills they are supposedly testing on.

For example, this is a real problem from the math section:

Chengdu *Zhongkao* 2021, Math section, problem 27.

Seriously, try to solve it. It can be really fun if you are the type of person that enjoys puzzles, and it only requires high school geometry skills. It may be harder or easier than you think.

I used to really enjoy these. However, all the fun goes away when there is a 20-minute timer in front of you, and not being able to solve it means that you go to the factories. To be honest, I haven’t attempted these problems for a good while now, because I’m not in that system anymore. However, just recently, one of my experience reminded me of these types of problems, and perhaps more importantly, my relationship with these types of problems.

Problem six

Problem 6 from math league #5 this year felt quite similar to these problems:

Math League Press, 2024, Competition #5, Problem 6

I tried very hard to solve it during the competition, but I couldn’t. However, I demanded an answer from myself. Therefore, I spent 2 hours after school by myself playing with various ideas and methods. At last, I figured out that I shouldn’t waste my time like this, so I used an unholy method as the last resort. With unholy methods came unholy numbers:

\[\frac{-6(3\sqrt{3}cos(20^{\circ})-6cos(10^{\circ})+4sin(40^{\circ})-13sin(20^{\circ})-4(2sin(10^{\circ})+1)\sqrt{3})}{cos(40^{\circ})+1}\]

Then, at the next day, the answer was revealed to me. It was part beautiful and part twisted, but I couldn’t figure out why I thought of it that way.

You can rotate the skinny triangle upside down, and everything’s easy from here.

And, of course:

\[the\:horrid\:trig\:expression=41.5962...\] \[24\sqrt{3}=41.5962...\]

Would I get full credit if I had written the horrid trig expression? I don’t think it really matters now anyways. I had a test coming up, so I couldn’t really say anything even if I knew what to say. Also, of course I blundered two easier problems because I spent too much time on problem 6 during the competition.

Reflection

To be entirely honest, the answer was exactly what I was hoping it to be. I should be so excited and satisfied for this answer.

However, I just cannot keep telling my self the lie that I did all of this just for fun. I tried to find the solution because of a sense of urgency, and a sort of lingering, invisible expectation that I had no idea where it was coming from. Until I started to think about how I was conditioned to think about geometry problems, and how through this conditioning I started to hate them with the entirety of my heart.

This might be a tricky problem for current me, but I really believe that I will be able to work it out.