Although I empathize with the struggle of the journey, having gone through a similar path of math team -> Stanford (where I briefly ran into the author), I think this is a particularly uncharitable characterization of math competitions.
Yes, there are many students (even more these days) that are in the grind for the accolades and college admissions, but math competitions were genuinely my favorite part of high school. They helped hone my problem solving, grit (!), work ethic, social skills, and leadership skills in a way that I continue to see pay off 15 years later. I am forever grateful for my high school teacher who supported me during this time.
It helped that I was actually passionate about it, which I think is the underlying point here. Weird constructs like math competitions that help kids channel their passions? Incredible. Forced hoop-jumping for the purpose of college admissions? Horrible.
I also enjoyed math (and related) competitions and continued them in college and beyond. Though I don't think math or algorithm puzzles are good "technical" interview questions, I was still sad to see Google Code Jam come to an end.
Went to the same school, was on the same math team, but was a year ahead of the author. Quit math team in my junior year because I wasn't at a competitive level, but it was a great time while I was on it. As a SWE, I rarely get to exercise the tenacity and problem solving skills I learned on the math team, but when I do they are "transformative" to the rest of the team.
> Nights were for other worthless extracurriculars to pad out our applications. ... The worst part was knowing that it was all going to be extruded into a few lines in an application form, that a committee would review for about ninety seconds
This is the tail wagging the dog; work really hard for 4 years in HS so that the next 4 years will be spent at a comparatively more prestigious college. I don't think we should expect high school students to be able to optimize this correctly themselves, and from my experience guidance counselors weren't particularly helpful. College admissions feel like a local maxima that have a lot of unintended side effects but would be difficult to change.
Personally, I didn't do the extracurriculars that would look good for college, but rather the ones I enjoyed; I didn't study very hard either. Didn't get into Stanford or an Ivy, but I look upon my experience in high school/college fondly rather than bitterly. Life seems to have turned out ok too.
I can relate; I didn't study quite as hard as my peers, didn't get into the prestigious private colleges that they got into (still went to an excellent public school across the Bay from Stanford), studied a fair bit of math, and ended up with a decent job where I've found that my math and problem solving skills have paid dividends. I'm certainly not making tech or F-U money, but I have a comfortable life a few years out of college and I'm lucky that I didn't have to sacrifice my youth or my interests to get here.
It's also interesting to follow the trajectories of people who were in the group that over-optimized for outcomes but have fallen off that path. I think the author was able to mitigate his burnout (I presume he had a good tech career), but I know a fair amount of folks with good pedigrees who haven't, and are still in limbo (unemployed, underemployed, or taking an extended break early in their career/schooling). I know they have the potential to do great things, but it seems the stakes of burnout are much higher today and harder to recover from financially.
My experience, a few decades later, is that everyone who optimized for STEM and long hours in high school made the right choice. The only downside I’ve seen is some wistfulness about lost romances and the preciousness of childhood. The upside is much higher earning power and greater opportunity later in life. I’d actually be interested in counter examples. In my circle I’m not aware of any, which is different than saying they don’t exist.
The author and I have discussed this elsewhere, but we had dramatically different experiences with competitive math, and I think part of the key is that he experienced it as a part of the college admissions grind, while I experienced it as an escape from the classroom to go do cool puzzles.
My major takeaway from his article was "The things we do in the high-stakes holistic admissions structure we've set up are insane" more than anything to do with competition math in specific.
The high school math Olympiad team is the fondest memory of mine. It was like solving puzzle games every day. But I guess if you are forced into puzzle games when you don’t like it can be a torture. I also changed my career to tech but I still look fondly back into the days when I could be immersed into solving problems without getting pulled into various meetings.
Same. Not American so perhaps the experience is different across the ocean, but it was truly magical. As an awkward, weird kid that didn't really know what to do with himself it was life changing.
My former team is still alive and going strong, and after uni I went back as a coach. I still hear the same stories about how awesome the group is, and how kids that now are in the same position as I was find it a "safe space", if you will.
I did mathcounts and math team stuff in middle school/HS and it never once occurred to me that the goal was to get into a good college. It just seemed fun.
I wasn't great at it, but I was pretty good considering that I didn't get any training, and I always thought it would have felt amazing to actually get good at those tests and feel like a math wizard. I guess I'm a bit bitter about it; it's a shame the special opportunities go to people who are depressingly minmaxing instead of people who would really love to have them. Although I also know that at some level if I had 'resolved' to get good I could have, so I can't be too resentful.
The author mentions at one point that he was unable to solve a problem because he didn't memorize the formula for the Euler totient function in order to count the number of numbers relatively prime to 9999.
...but its actually an interesting (and not super difficult) exercise in its own right to figure this out even if you don't know the formula. Encourage you all to give it a shot.
SPOILERS: 9999 = 3^2 * 11 * 101, so first subtract out the multiples of 3 (3333 of them), the multiples of 11 (909 of them), the multiples of 101 (99 of them). Note that we've now double-subtracted multiples of 33 (303 of them), multiples of 303 (33 of them), multiples of 1111 (9 of them) so add these back. Finally subtract 1 to not count 9999 itself.
I guess my point is that the purpose of these problems is not to separate out people who know specific tricks from people who don't—its to separate out people who can reason their way through difficult mathematical problems and people who can't.
The difference for you is that you're doing it as a fun exercise. With contest math, you're drilling these formulas and tricks so you can reproduce them quickly on a timed exam. If you know both of the facts listed in the essay then you can knock off this question in a minute or two.
Trying to come up with everything from scratch could take a lot longer and be very frustrating when you've got other problems waiting for you to solve.
The totient formula isn't the hard part of the problem.
The test has a very short time limit (for the difficulty of the problems), and has many gruelingly complicated problems,so if you dont have the formulas down cold, you'll burn out during the contest.
Of course, if you don't care about silly speed-mathing contests, you can enjot the problems at a leisurely pace.
What's especially fascinating is that the core of the problem is a generalization of the totient computation, so understanding the inclusion-exclusion construction of totient is very helpful to the problem, while simply memorizing the formula is a misdirection.
OP missing this point shows that he really was doing this for all the wrong reasons. He should have done FIRST or science bowl instead.
For every distinct prime factor p (so, of 9 is a factor, use 3 not 9), only (p-1)/p of the natural numbers ≤ n are relatively prime to it. Pime. This overcpunts nothing, since prime factors are relatively prime to each other. (Proving this requires some analysis of remaimders / modular arithmetic. But working an example shows the pattern).
This gives a formula, phi(n = Sum (p_i ^ e^i)) = n • Product ((p_i -1)/p))
This also shows how OP (and maybe the coach) missed the point of math team.
My own path through related stuff is unusual, so nothing like the authors. I do however have experience working with and teaching people who were on very similar paths to theirs.
There are a huge number of kids in this authors shoes, ones who are on a treadmill and not entirely sure why. There are also lots of kids who were always the best at something (in this case math) that any of their teachers etc. had ever seen ... until they move across the country and find themselves in a classroom where they are middling at best. And sure, there are rare kids who are so good it catches you off guard.
FWIW I suspect there are more kids made miserable than those who thrive, but I think it's largely unavoidable in a tournament system like "elite" college apps. There are also plenty of kids who had some fun and made some friends along the way without taking it too seriously. I suspect in this case it's often a good outlook in high school, especially for kids who don't really fit in anywhere else.
High school, at least as far as it serves as a sorting mechanism for top students, follows a kind of Parkinson's law: the number of hoops to jump through increases until it reaches the natural limit of how little sleep the top students can handle.
There were rumblings that my high school, which had plenty of AP classes already, was about to introduce a combination AP/IB curriculum, which absolutely terrified us. I and my AP-taking classmates breathed a huge sigh of relief when it was announced that it would be delayed, and the students in the year below us would be the guinea pigs. They would have to run twice as fast just to stay in place.
Indeed. In my time, it wasn't uncommon to have 6 AP classes a semester along with at least one time-intensive extracurricular. Assuming each class is the equivalent of 3 credit hours, it's the equivalent of an above average number of classes in college (15 being the expected amount, 12 being the minimum to be a full-time student, and 18 considered intensive) while playing a competitive sport.
The best part: Even a decade ago, the above was considered neccesary but not sufficient for admission to a top school. Plenty of people with perfect to near-perfect college entrance exams, Intel International Science and Engineering Fair finalists, etc didn't make the cut. Of the few that did, the majority were the lower Ivy's (Dartmouth and Brown).
There is a book called “Seven checkmarks” in Dutch that argues that succes in the Netherlands is strongly correlated to seven checkmarks to have: male, highly educated parents, white, certain type of elite high school, university educated and one more. Having all the marks, having generally underperformed academically and still coming out on top comparatively I feel there might be some truth in it. It would signal a quite stratified society with a “ruler class” inside a society that thinks of itself as classless for the last 60 years at least. (It’s pretty hard to reason about this being while being under scrutiny.)
Why post this? After reading your comment I thought wouldn’t want to live there or raise children there. But the second thought was, wait - that’s meritocracy in action. Imperfect meritocracy as you point out, but it might still be more equitable not than having seven checkmarks and generally faring worse than those born under a different star. My Rawlsian self thinks grit should be rewarded more than birth, even though testing for grit would probably massively increase burnout.
Thinking even further, I don’t think that societies with “high grit” (Korea, US) are generally considered to treat their children and general society very equitable. Still mentally debating if there is a very socialist argument growing inside of me. That book (read it three months ago) does make me think a lot. It was the first time something ‘near-woke’ made me think so hard. The book mentions the reflective point as well - might I only take it that seriously because it was written by someone from the same “class”? Foundational stuff.
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Yes, there are many students (even more these days) that are in the grind for the accolades and college admissions, but math competitions were genuinely my favorite part of high school. They helped hone my problem solving, grit (!), work ethic, social skills, and leadership skills in a way that I continue to see pay off 15 years later. I am forever grateful for my high school teacher who supported me during this time.
It helped that I was actually passionate about it, which I think is the underlying point here. Weird constructs like math competitions that help kids channel their passions? Incredible. Forced hoop-jumping for the purpose of college admissions? Horrible.
> Nights were for other worthless extracurriculars to pad out our applications. ... The worst part was knowing that it was all going to be extruded into a few lines in an application form, that a committee would review for about ninety seconds
This is the tail wagging the dog; work really hard for 4 years in HS so that the next 4 years will be spent at a comparatively more prestigious college. I don't think we should expect high school students to be able to optimize this correctly themselves, and from my experience guidance counselors weren't particularly helpful. College admissions feel like a local maxima that have a lot of unintended side effects but would be difficult to change.
Personally, I didn't do the extracurriculars that would look good for college, but rather the ones I enjoyed; I didn't study very hard either. Didn't get into Stanford or an Ivy, but I look upon my experience in high school/college fondly rather than bitterly. Life seems to have turned out ok too.
It's also interesting to follow the trajectories of people who were in the group that over-optimized for outcomes but have fallen off that path. I think the author was able to mitigate his burnout (I presume he had a good tech career), but I know a fair amount of folks with good pedigrees who haven't, and are still in limbo (unemployed, underemployed, or taking an extended break early in their career/schooling). I know they have the potential to do great things, but it seems the stakes of burnout are much higher today and harder to recover from financially.
My major takeaway from his article was "The things we do in the high-stakes holistic admissions structure we've set up are insane" more than anything to do with competition math in specific.
My former team is still alive and going strong, and after uni I went back as a coach. I still hear the same stories about how awesome the group is, and how kids that now are in the same position as I was find it a "safe space", if you will.
I wasn't great at it, but I was pretty good considering that I didn't get any training, and I always thought it would have felt amazing to actually get good at those tests and feel like a math wizard. I guess I'm a bit bitter about it; it's a shame the special opportunities go to people who are depressingly minmaxing instead of people who would really love to have them. Although I also know that at some level if I had 'resolved' to get good I could have, so I can't be too resentful.
...but its actually an interesting (and not super difficult) exercise in its own right to figure this out even if you don't know the formula. Encourage you all to give it a shot.
SPOILERS: 9999 = 3^2 * 11 * 101, so first subtract out the multiples of 3 (3333 of them), the multiples of 11 (909 of them), the multiples of 101 (99 of them). Note that we've now double-subtracted multiples of 33 (303 of them), multiples of 303 (33 of them), multiples of 1111 (9 of them) so add these back. Finally subtract 1 to not count 9999 itself.
phi(9999) = 9999 - 3333 - 909 - 101 + 33 + 303 + 9 - 1 = 6000
I guess my point is that the purpose of these problems is not to separate out people who know specific tricks from people who don't—its to separate out people who can reason their way through difficult mathematical problems and people who can't.
Trying to come up with everything from scratch could take a lot longer and be very frustrating when you've got other problems waiting for you to solve.
If that is frustrating instead of thrilling to you then you probably shouldn't do these competitions.
https://artofproblemsolving.com/wiki/index.php/2022_AIME_I_P...
The totient formula isn't the hard part of the problem.
The test has a very short time limit (for the difficulty of the problems), and has many gruelingly complicated problems,so if you dont have the formulas down cold, you'll burn out during the contest.
Of course, if you don't care about silly speed-mathing contests, you can enjot the problems at a leisurely pace.
* The contest has 15 problems and a time limit of 3 hours, giving only an average of 12 minutes per problem.
* The average score is 4.83/15, from the official statistics [1].
* The statistics noted that only 5.17% of test takers answered this problem correctly, making it the second-hardest problem out of the 15 on the exam.
[1] "AIME 02/08/2022" https://amc-reg.maa.org/reports/generalreports.aspx
OP missing this point shows that he really was doing this for all the wrong reasons. He should have done FIRST or science bowl instead.
For every distinct prime factor p (so, of 9 is a factor, use 3 not 9), only (p-1)/p of the natural numbers ≤ n are relatively prime to it. Pime. This overcpunts nothing, since prime factors are relatively prime to each other. (Proving this requires some analysis of remaimders / modular arithmetic. But working an example shows the pattern).
This gives a formula, phi(n = Sum (p_i ^ e^i)) = n • Product ((p_i -1)/p))
This also shows how OP (and maybe the coach) missed the point of math team.
There are a huge number of kids in this authors shoes, ones who are on a treadmill and not entirely sure why. There are also lots of kids who were always the best at something (in this case math) that any of their teachers etc. had ever seen ... until they move across the country and find themselves in a classroom where they are middling at best. And sure, there are rare kids who are so good it catches you off guard.
FWIW I suspect there are more kids made miserable than those who thrive, but I think it's largely unavoidable in a tournament system like "elite" college apps. There are also plenty of kids who had some fun and made some friends along the way without taking it too seriously. I suspect in this case it's often a good outlook in high school, especially for kids who don't really fit in anywhere else.
There were rumblings that my high school, which had plenty of AP classes already, was about to introduce a combination AP/IB curriculum, which absolutely terrified us. I and my AP-taking classmates breathed a huge sigh of relief when it was announced that it would be delayed, and the students in the year below us would be the guinea pigs. They would have to run twice as fast just to stay in place.
The best part: Even a decade ago, the above was considered neccesary but not sufficient for admission to a top school. Plenty of people with perfect to near-perfect college entrance exams, Intel International Science and Engineering Fair finalists, etc didn't make the cut. Of the few that did, the majority were the lower Ivy's (Dartmouth and Brown).
Why post this? After reading your comment I thought wouldn’t want to live there or raise children there. But the second thought was, wait - that’s meritocracy in action. Imperfect meritocracy as you point out, but it might still be more equitable not than having seven checkmarks and generally faring worse than those born under a different star. My Rawlsian self thinks grit should be rewarded more than birth, even though testing for grit would probably massively increase burnout.
Thinking even further, I don’t think that societies with “high grit” (Korea, US) are generally considered to treat their children and general society very equitable. Still mentally debating if there is a very socialist argument growing inside of me. That book (read it three months ago) does make me think a lot. It was the first time something ‘near-woke’ made me think so hard. The book mentions the reflective point as well - might I only take it that seriously because it was written by someone from the same “class”? Foundational stuff.