Suckers for Math

One thing I’ve noticed is that a lot of Chinese citizens (especially the college-age-ish) have great pride in their math skills. They have heard time and again that their math skills are superior to most other nations’ students’, and they believe it. This can be sort of fun to mess with.

I recently stumbled across the World’s Hardest Easy Geometry Problem. After working on it for about half an hour and realizing that it really was pretty tough, I decided to see if I could entice my wife to try it.

Now, my wife is intelligent, but she’s no math person. She earned her degree in law (and not the mathy kind). To be honest, I can’t remember her ever showing much interest in math or geometry. But I decided to tempt her with the problem.

She ended up spending the rest of the evening working on it. Ah, pride is a fun thing.

(Anyone got any other deceptively difficult math bait?)


John Pasden

John is a Shanghai-based linguist and entrepreneur, founder of AllSet Learning.


  1. On a more obvious note, did you solve it? His hint of drawing more lines just got my picture all messed up…

  2. I didn’t have as much fun as you did. The riddle only served to reinforce stereotypes. My girlfriend solved it within minutes, and I’m still sat trying to understand the solution.

  3. x is 40. That problem is evil.

  4. Don’t mean to gripe, but WHY, Jeffrey D., did you post an answer (or something) here? Are you trying to show off when everyone else had the sense to keep it to themselves?

    I’m sure I’m not the only person who just learned about this fun geometry problem and want to dash over to try it out after reading the comments.

  5. Yeah, it’s better not to post solutions to problems like those. People aren’t gonna like it.

    That said, in this particular problem it doesn’t help you much. You can see it’s around 40, but the way is the key.

  6. I didn’t solve it. I still think I could have if I put in enough effort, but I decided I didn’t want to spend the time.

    My wife also came up with x = 40, but I don’t think her proof was complete. I was too lazy to actually go through her process to try to poke holes in it though. I just insisted that the answer was actually 41. Ha!


    Your girlfriend may have found the answer, but did she have a real proof? That was the actual challenge. And, according to the webmaster of the site:

    Of the hundreds of people that have emailed me, I’d estimate only one or two percent (mostly math professionals and college students) have solved it without significant hints…. Most people who think they have found the solution are wrong.

    You should e-mail your girlfriend’s solution to the guy and see what he says. He might give you ammunition to torture her for a bit longer… 🙂

  7. I have to take issue with the presentation of the problem. We’re trained, I think, to look for a catch or a trick solution when we read something like “the hardest easy problem”, so we spend our time looking for a-ha moments rather than bisecting angles and dropping perpendiculars like this sort of geometry problem demands (the creator of the webpage basically admits his misdirection in the one hint that’s not totally useless).

    I also think that “in a sense, easy” is valid only if you also are willing to mock a random guy on the street for not being able to prove any given result from Euclid: “You can’t do this? Fool! It’s just a compass and a straightedge.” You know, like Barrow mocked Newton during his undergraduate studies….

  8. I wanted to solve the problem too, but like Gwen I read Jeffrey D’s comment and didn’t get a chance to :(.

  9. The point of the problem is to test your ability to find the answer, not the answer itself. So knowing what the answer is beforehand doesn’t really spoil it, at least not for me.

  10. The actual number is totally unimportant. The point is to work it out and make the proof. It’s not like I told you he’s a ghost all along, and ruined the movie’s plot twist. It’s more like I tell you that if you beat the game, the Princess actually is in the proper castle.

    But yes I was showing off to a bunch of Internet strangers. That question is an evil brain-teaser that got harder and harder, I felt a sense of accomplishment.

  11. 40 is incorrect, by the way. Looking at this link, the angle x looks closer to 20 degrees than it does to 40.

  12. sorry John but it’s not tough at all. These questions had been shown up on my high school test for thousands of times. I solved it in 5 seconds, literally.
    humm, this is not a good one, man!

  13. Ben, if your girlfriend really solved this in a few minutes she must be exceptionally talented in this area. No offense, but I personally doubt that it’s possible to just whip out a pen and paper and be done in a matter of minutes. The full, correct proof takes a good deal of ingenuity and time.

  14. Marco: Care to post a step by step proof? I failed to arrive at the solution myself, however I’ve seen a correct proof, and it’s quite detailed and would probably take at the very leasttwenty minutes or more, to come up with by yourself.

  15. with the process of good step-by-step proofs

  16. The answer is not 40. Eyeballing will get it about right, but the proof is really hard.

  17. There was no eyeballing, ha ha, definitely using a protractor would be cheating. Anyway I checked out the numbers for the other angles inside the triangle, and it all works out like sudoku.

    And yeah I just don’t think it’s possible to solve it in 5 seconds.

  18. Oh man, egg on my face – I googled the solution and got a different answer! Ha Ha – I’m still not so sure.

  19. I have ever read a news article on math in China. The journalist was regretting saying that Chinese students perform very well at the International Mathematics Olympics but most of them don’t want to study math anymore at university, and as the result China still has no Fields Medal winner.

  20. Haha. If you Google “World’s Hardest Easy Geometry Problem,” the first page is the one you link to.

    The second one is the answer with a Chinese “Jianghu” title:

    After reading your post, I found this hilarious. The American version says, I’m not going to give you the answer, you have to figure it out for yourself. The Chinese page seems to say, here’s the answer, now you can show it off to your American friends.

  21. Sorry, correction.

    Please neglect “saying” in my above comment.
    × “The journalist was regretting saying that….”
    ○ “The journalist was regretting that….”

  22. I think Marco illustrates the whole point well: a lot of young Chinese people are quite smug and overconfident in their math skills.

  23. The journalist was regretting saying that Chinese students perform very well at the International Mathematics Olympics but most of them don’t want to study math anymore at university, and as the result China still has no Fields Medal winner.

    This is due to the poor quality of Chinese Universities and top students going abroad. High school math olypmiad competitors in other countries study just as hard as the Chinese, or anyone else for that matter.

  24. China has produced a number of distinguished mathematicians including Shing-Tung Yau (a Fields medalist) and Shiing-Shen Chern,
    both of whom are/were differential geometers.

    In countries that perform very well at the International Mathematical
    Olympiad (IMO) every year, such as Russia, China, Romania and Korea, there is a strict selection of contestants and a long-term special training program specialized in IMO-type problem solving, which doesn’t have much to do with actual research in mathematics. Therefore, I agree that IMO contestants do not necessarily qualify as excellent professional mathematicians, although several past IMO contestants I personally know have become good professional mathematicians (I do not know yet if they will get a Fields Medal).

  25. “World’s Hardest Easy Geometry Problem” is generally called the Langley problem.

    352 Langley-type problems are listed below (Japanese fonts
    will be required to display all the contents, but if you just follow
    English texts, you will get the idea). Enjoy!

  26. The trick is to attack this problem from different angles 😀

  27. Hi pharmine,

    As you already know, both Shing-Tung Yau and Shiing-Shen Chern are Chinese-American. The two of them graduated from top-level universities in China and after that immigrated to the USA. I think what the writer of the article wants to say is that there has been no Fields Medal winner who has Chinese citizenship so far. Anyway, I agree with your opinion on IMO contestants.

  28. I hate you John.

    I concentrated too much on angles and not enough on congruent sides. Took a while with a proof in front of me to get it all.

    I wouldn’t exactly call this straight math, but problem solving. Some people are really good at math, but can’t find their way into their house with a key.

  29. nah…. You gotta let Chinese people smug their math skills since you know we’re not good at randomly talking sweet.:) what else can we do then?

  30. When you are in school, you can inevitably observe the fact (note: not truth). Yes, I am talking about average people, which standards for the mean of the distribution. and NO, I am comparing Chinese students with their counterparts in Department of Math of any school period.

  31. typo: I am not comparing Chinese students with …….
    sorry about the long thread I created

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