The hardest problem on the hardest test

So many things I could say about this, but I"ll let Alan Kay speak for me - [LINK] Alan talks about how ideas in Calculus could be manifested from an early age and a richness in understanding built up as they aged up: - at very young ages, kids really respond well through "doing" / the enactive channel. When asked to draw a circle, kids in Papert's group would first emulate what a LOGO turtle would do by rotating their body in a circle (making tiny increments in x and y). - as they got a bit older, the visual / iconic channel was more developed and they could understand the abstraction of a circle on pencil/paper and how the concepts carried over there - closer to early teens, symbols were much easier to grasp and relate to, etc. With this context in mind, there have been some cool efforts to mix the second and third channels I just mentioned to communicate advanced math concepts. Vi Hart and Grant Sanderson's youtube channels come to mind. Here are my favorite videos by Grant: - "What does it feel like to invent math": [LINK] - On the visual intuition behind a hard problem on the Putnam exam - "The hardest problem on the hardest test": [VIDEO] - From vectors to matrices to vector spaces to higher-level ideas like the link between linear algebra and calculus: "Essence of linear algebra" series: [LINK] What I especially like about Grant's videos, is that he often walks through what a mathematician would do, the questions she would ask, etc.

Indeed, I'd recommend the "interesting proof" vignettes by 3Blue1Brown: [VIDEO] These are my favorite videos of his

3Blue1Brown is really great for learning/refreshing math. He has great visualizations. I especially enjoyed this video (not exactly a math tutorial more of a cool explanation): [VIDEO]

I like the 3blue1brown youtube series for explaining mathematical problem solving, like this example explaining a difficult problem from the Putnam: [VIDEO] "And those of you who follow the channel know that rather than just jumping straight to the solution, which in this case will be surprisingly short, when possible I prefer to take the time to walk through how you might stumble upon the solution yourself. That is, make the video more about the problem-solving process than the particular problem used to exemplify it." I bailed out of math around calculus, and these videos helped me at least appreciate what I was missing.

This is an extension of an animation library used by math YouTuber 3blue1brown [0]. He has an excellent channel that I would recommend anything from. One of my favorite videos of his is where he shows the thought process behind solving a level 6 Putnam competition problem and makes it seem almost doable [1]. [0] [LINK] [1] [VIDEO]

I enjoyed watching this similarly insightful video [1] on "The hardest problem on the hardest test" of the Putnam Competition. [1] [VIDEO]

I strongly disagree with this comment. I sincerely doubt 100% of the HN audience always considers those things. You might think some of those things are basic, but that doesn't make them useless. This kind of lack of empathy for new learners can make it incredibly difficult to teach concepts to those who don't have as much knowledge as you may have. See [VIDEO] for a great video by 3blue1brown on this topic.

Related — these videos give a sense of how someone might actually go about thinking through and solving these kinds of problems: - A 3Blue1Brown video on a particularly nice and unexpectedly difficult IMO problem (2011 IMO, Q2): [LINK] -- And another similar one (though technically Putnam, not IMO): [VIDEO] - Timothy Gowers (Fields Medalist and IMO perfect scorer) solving this year’s IMO problems in “real time”: -- Q1: [LINK] -- Q4: [LINK]