I had the idea of using vectors\/complex numbers to represent everything and then using lin-alg stuff to get area (I don’t want to spoil). But writing this all out seemed ugly and alg-manip heavy, so I never actually wrote stuff out [3]. This is a common occurrence, so much so that I repeatedly tell myself to write down every technique that I considered.<\/p>\n\n\n\n

Now, back to the tree metaphor where the nodes are ideas\/claims\/patterns, and the edges are techniques (imagine the root at the top). This gets to my mental model of how olympiad skill works: it’s a mix of how quickly you can move around [4], faith a.k.a whether you prematurely cut off branches, and then how well you combine ideas from different branches. When I got stuck on IMO 3\/6s, it felt like I had all of the necessary low-level ideas and observations, but just wasn\u2019t able to put it all together. I started saying, as a joke, that I was getting \u201cIQ locked.\u201d Eventually, it stopped being a bit, but there was definitely some kind of phase transition between doing \u201ceasier\u201d problems (I got a 14\/15 on AIME and 5\/6 solves on JMO), and doing hard, late-USAMO problems. I have a theory that the feeling I had toward these IMO problems is the same feeling that most people have towards calculus.<\/p>\n\n\n\n

But, faith might just be a derivative of the \u201chow fast you can move along the tree\u201d factor. Speed can help the assessment and synthesis of branches since you can rediscover things quickly. Additionally, it is not always true that more faith is better. On the Wikipedia page \u201cList of disproved mathematical ideas,\u201d there is an entry saying \u201cErik Christopher Zeeman tried for 7 years to prove that one cannot untie a knot on a 4-sphere. Then one day he decided to try to prove the opposite, and he succeeded in a few hours.\u201d <\/p>\n\n\n\n

<\/figure>\n\n\n\n

I really enjoyed writing down every single one of my ideas, but I no longer do this and instead jot down promising ideas, especially those that reframe the problem in a different way with different degrees of freedom. Math-wise, I think this idea will one day help me decide whether olympiads are really as orthogonal to research mathematics as adults claim. And non-math-wise, I think lowering the bar on what\u2019s a reasonable idea is generally a good thing [5]. Maybe in conspiracy theories, but definitely in choosing your model of the world. [6]<\/p>\n\n\n\n

<\/p>\n\n\n\n

[1] I\u2019m not religious (although I wish I was). It would\u2019ve been nice to have started out with a pre-made moral framework instead of building it out on my own, synthesizing ideas from a multitude of sources. <\/p>\n\n\n\n

[2] An amazing feature which lets you basically do the find my iPhone alert, but it only plays the sound once.<\/p>\n\n\n\n

[3] aka I\u2019m bad at formulas (https:\/\/web.evanchen.cc\/static\/otis-samples\/synopsis.html<\/a>) problems when I don\u2019t know they\u2019re formulas<\/p>\n\n\n\n

[4] https:\/\/terrytao.wordpress.com\/career-advice\/advice-on-mathematics-competitions\/<\/a><\/p>\n\n\n\n

[5] I’m thinking of Simulated Annealing<\/a> here, and this is kind of arguing for exploration over exploitation.<\/p>\n\n\n\n

[6] Which is a funny thing to say because I’m philosophically conservative on institutions, I give them (perhaps inaccurately) the benefit of the doubt, assuming they were designed like that for a reason. <\/p>\n","protected":false},"excerpt":{"rendered":"

This is a strictly nonreligious piece [1]. Instead, it\u2019s about how, to do things, you have to have faith in their feasibility. This is also a gripe about olympiads and how I never pursue the ideas. Pushing hard on an idea, and actually going deep into the “idea tree,” an abstraction I will use to […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","WB4WB4WP_MODE":"","WB4WP_PAGE_SCRIPTS":"","WB4WP_PAGE_STYLES":"","WB4WP_PAGE_FONTS":"","WB4WP_PAGE_HEADER":"","WB4WP_PAGE_FOOTER":""},"categories":[1],"tags":[],"featured_image_src":null,"author_info":{"display_name":"y.ryan.yang","author_link":"https:\/\/ryanyang.us\/?author=1"},"yoast_head":"\n