The way the problem was solved at first hand by just "recognizing the pattern of (2n) choose n" wouldn't satisfy me at all, where's the proof ? Why does it work ? This isn't maths, this is "pattern recognition".
Maths IS pattern recognition
My Mathematical Regression
121–130 of 159 posts
Re: My Mathematical Regression
#122Re: My Mathematical Regression
#123An intuitive motivation for the solution in the article (2n choose n). For an n*n grid you have to you will take 2n steps, n "over" and n "down". All that matters is the order of the steps. So if you think of there being 2n "slots", you have to pick n to be "over", and the rest are forced to be "down". So it's n choose 2n indeed. You can also think of it another way, without using the formula combinations, and only t…
The way I thought about it: to get to the goal you have to do 20 steps: 10 times right, 10 times down. But the order of these steps does not matter, ever possible arrangement is a solution.
So for counting, you can basically think about it as a list of twenty initially empty spots. You first fill it in with your 10 down steps. The remaining 10 spots will then be the ones for the 10 right steps. So really the only choice you have to make is where to place the 10 down steps.
This question boils down to: in how many different ways can you distribute the 10 down steps over the 20 empty spots? That's 20 choose 10.
Re: My Mathematical Regression
#124> this is just me fantasizing. At work I would just give it to an AI and continue with my day
I'm sure ~4 yrs ago i would have loved the thought of this. It's so boring. My job is so, so boring.
Re: My Mathematical Regression
#125Earlier quoted context omitted.
Good managers are fairly rare, even though every manager probably thinks they are a good manager.
I'm one of the few who absolutely believe they're not one of the rare "good" - hopefully, "not yet". What resources do you recommend to improve ?
radical candor is a good book that explores a simple framework for people management.
Re: My Mathematical Regression
#126This is a pre-AI phenomena. I observe it quite a lot with stuff I did in high school but usually with complex problems. What's generally happening is that you were working with pen and paper through a hard problem. With adult brain, you'd expect just to know the answers, but in reality you're not much smarter than you were at 14, so you need to do the thing properly. Also if you help little kids with homework, you'll…
Yes, AI can write a solution, but cannot visualized a solution. When I was 9, I refused to learn my times tables, and addition, so I was relegated to work with blocks. I loved the blocks... I was able to complete all the problems, come up with my own, and return to class, with some extreme facility. Two years later, comes a challenge in class... make a formula for summing the integers... well everyone started with 1+…
are block referring to Cuisenaire rods?
Re: My Mathematical Regression
#127I think one of the saddest thing is that the kind of person who would recognize, "we can solve this seemingly complicated problem by just applying this formula", would often have trouble even getting recognized in many corporate environments. I managed a guy like that. He was capable of very complex thinking, but he wasn't in love with complexity, he was in love with simplicity. His solutions tended to be of the form…
I imagine this is where the reputation of a good manager comes in and the ability to say to their boss "hey, we should keep this guy... just trust me on this."
Depends on leadership culture. In toxic (aka “competitive”) environments managers are insecure and fear their own staff as potential competition.
Re: My Mathematical Regression
#128The way the problem was solved at first hand by just "recognizing the pattern of (2n) choose n" wouldn't satisfy me at all, where's the proof ? Why does it work ? This isn't maths, this is "pattern recognition".
in math it's often the case that you notice the solution first and only afterwards prove to yourself that it works. pattern matching and intuition play a large role in math!
this is why I'm not a big fan of "show your work": the "work" is however many years it took to build up my intuition, and often any explanation I could type out for my solution would be a retroactive rationalization. it's still useful, sure -especially for catching your errors, but I place it on the opposite end of the open-fake scale than most people.
of course here the proof is simple: 20 right moves, 20 down moves, any order => of 40 total moves choose any 20 indices to be your down moves => 40 choose 20 is your answer. would that teach you how to solve the next problem though? I'm not so sure.
Re: My Mathematical Regression
#129I think one of the saddest thing is that the kind of person who would recognize, "we can solve this seemingly complicated problem by just applying this formula", would often have trouble even getting recognized in many corporate environments. I managed a guy like that. He was capable of very complex thinking, but he wasn't in love with complexity, he was in love with simplicity. His solutions tended to be of the form…
I have the opposite experience with this, and I personally also default to simplest solutions at least as a baseline. However it’s important to distinguish between simple solutions that approximate the problem very well, to simple solutions that work in limited context or with heavy sacrifice in assumptions because those will hurt you in the long run.
Re: My Mathematical Regression
#130An intuitive motivation for the solution in the article (2n choose n). For an n*n grid you have to you will take 2n steps, n "over" and n "down". All that matters is the order of the steps. So if you think of there being 2n "slots", you have to pick n to be "over", and the rest are forced to be "down". So it's n choose 2n indeed. You can also think of it another way, without using the formula combinations, and only t…
Just noticed this, but intriguingly, Catalan numbers are (2n C n)/(n+1), which hints at a connection with trees. Off the cuff, notice that the diagonal has n+1 intersection points, and a path that never passes through the diagonal gives a forest via the isomorphism with ballot sequences [0]. Any sequence that does pass below the diagonal can be "rotated" into one that doesn't, and so there are probably n+1 paths in e…
Funny, but I've recently started a blog to keep my math and physics muscles working and this exact problem is the first post I've written - it is useful in high order perturbation theory in quantum mechanics that I'm planning to describe down the line. Counting these paths is cool problem but generating all of them was more challenging.
Anyway, here is the post https://kpatucha.github.io/posts/Dyck-paths-Raneys-lemma/