HN

Comments (134)

• Menu

  jp57
  I 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, "we can ignore all these things, and just focus on X, and it will provide all the value." He'd notice something and simplify it and the benefit to the company would be measured in multiples of his salary.Every manager who'd ever directly managed him knew what a treasure he was, but it was often hard for us to convince others of the value of his solutions because they were so simple, and people were convinced that hard problems must have complex solutions. (or else they would have solved them, right?)He eventually got bored. He retired and joined a seminary.
• Menu

  mb7733
  An 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 the fact that there are n! permutations of n objects. We can think of this a permutation of 2n items, made up of two groups of n identical items each. Using (2n!) will overcount, due to the fact that each of the "over" steps are identical, and similarly for the "down" group. We have cut down our answer by dividing out all of the repeated sequences. There will be n! redundancies for all the ways we can permute the "over" group and, the same for the "down" group. So this results in (2n!) / (n! * n!), which is exactly equal to 2n choose n. See [1] which explains permutations with repetion this in general. [Note: We pretty much re-derived the formula for combinations!][1] https://brilliant.org/wiki/permutations-with-repetition/
• Menu

  gobdovan
  This 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 see that some problems are quite difficult as well and require you to actually think, even if it's problems for 10 year olds.
• Menu

  GL26
  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".
• Menu

  dhosek
  There’s a bit of hand-waving in the jump to 2n choose n solution, which I suppose is fine, and my ex–math teacher brain really wants to have a proper proof or at least solid reasoning rather than “it follows the pattern” based on three observations.But I am reminded of how during my engagement 24 years ago, my future father-in-law raised an issue of being able to determine whether they were getting the full amount of sandpaper on large rolls that they were paying for. I was able to simplify the question a bit to one that treated the rolls as if they were simple concentric rolls of a specified thickness and from there could turn it into the good old Gaussian sum formula times 2π to get the length. The engineers working for the company came up with the same solution, but instead of using n(n-1)/2 they did the summation with multiple rows in excel.
• Menu

  kccqzy
  I was sad in a different way. I immediately realized that this could be solved by dynamic programming by computing the recurrence F(x,y)=F(x-1,y)+F(x,y-1) with the base case F(0,0)=1 and F(x,y)=0 if x<0 or y<0. The problem is that I immediately jumped to generating functions as a tool to solve this. I defined G(u,v)=\sum_x \sum_y F(x,y) u^x v^y. After maybe ten minutes of manipulation I arrived at the closed form for G(u,v)=1/(1-u-v). At this point I recognized its series expansion and its coefficients are just given by the binomial theorem.I feel sad because I had forgotten the simple and intuitive construction of choosing “go down” and “go right” directions. When a person learns more advanced mathematics, it is often the case that the person just applies such advanced mathematics by rote without realizing that a solution can be found with more elementary mathematics and more creativity. It reminded me of the time in middle school before derivatives were taught, when my teacher reminded me that using derivatives to solve a problem would receive no credit.
• Menu

  jerf
  Something I haven't seen anyone else mention is that this is problem 15. The author probably did the previous 14 problems too. IIRC the problems aren't strictly cumulative but you do generally end up warmed up and with hints about how to approach problem N efficiently from problems [1, N-1].Skills do decay, I can't deny that. But even when I was still in school going back and doing an end-of-year problem for a class I took two years ago would have been harder than it was for me at the time... but it would have been easier than the first time if I warmed up properly with a bit of review and practice first, and I mean, not just three minutes glancing over things but taking some serious time for it.
• the_red_mist
  Tbh your student reasoning is still dangerous... the patterns could have not generalized nicely. see moser's circle problemneeded to justify viewing this as "arranging down vs right movements" as another comment outlines
• erdaniels
  Since I'm between jobs, I've been taking the time to relearn geometry, trig, calc, and linear algebra. I've finished the first two and it's both humiliating and really fun to relearn some math. As an adult, I feel a very different appreciation for how much fundamental math we take for granted. I definitely recommend https://www.khanacademy.org/ and pauls notes (https://tutorial.math.lamar.edu/).
• Menu

  bmenrigh
  The 2n choose n solution isn't at all intuitive to me but thinking about it in terms of 40 steps, 20 of them rightward and 20 of them downward an then looking at all distinct permutations of these 40 steps as (40!) / ((20!)^2) is intuitive to me. Then it becomes obvious that since 20 is half of 40, k and n - k are the same number (20), which coincides with the binomial coefficient n! / k!(n - k)!. But this seems like a lucky coincidence in 2 dimensions and if you extended the problem into 3D you'd do better thinking about permutations.
• jadar
  Yeah, this is relatable. Part of it comes down to "use it or lose it." We atrophy when we don't exercise. But at the same time, it might come back to the OP faster than he thinks if he went into it. Also, I've found that with AI I'm able to ask specific questions where I'm hazy and pick things up faster than if I just had to watch hours of lectures to pick something up. (There's a place for both.)
• Menu

  d_silin
  Nobody forces you to use AI.It has become sort of junk food for the brain. Temptations and ads for it everywhere.
• storus
  I noticed the same with me after a few years working; my solution was to take all hard mathy online MS degrees/grad certs available one after the another. Now I can understand how does the generative AI math work while designing an electric motor for my own robot. Though the latest AI can do it all better than me.
• Menu

  abetusk
  I appreciate where the author is coming from, as we often have much more context because of training that have since atrophied a bit, but in defense of the dynamic programming solution, this generalizes very well.As stated, the choose(2n,n) solution of course works but as soon as you deviate from a square, things can get more complicated. What if it's a rectangle? An arbitrary shape? One with holes? The dynamic programming solution takes all of this in stride (assuming, of course, that the conditions of only going right and down still hold).Pascal's triangle is, after all, a dynamic programming solution. It just so happens that there's a "closed form" solution to their entries.I'm all for clever tricks but I also appreciate much more a solution that generalizes well and gives more insight into a class of problems.
• floppyd
  Noticing a pattern and just extending it without proving why it works is not really a solution. You can prove it without really "understanding" it using induction, but that still would be proof, same as just counting on a computer.
• Menu

  andredurao
  That Project Euler problem was my first encounter with memoization. At the time, it felt like a magical solution, so I ended up solving it using the central column of Pascal's triangle, which was easier for me to understand back then.I also tried a weird idea involving popcount, but it didn't scale. My approach was to represent each possible path with 0s (don't turn) and 1s (turn), testing the same number of 0s and 1s. However, even with popcount running in O(1) with hardware support, the total number of possible paths made the idea impractical :)
• SilverSlash
  I can definitely relate. My brain also wants to take the path of least effort now, even for simple things like adding two numbers which I could do very quickly in my head in my college days.And with AI the path of least (initial) effort seems to be to just ask the model to solve it. It might get it wrong and then I'll prompt it again and again. But each individual prompt is fairly low effort on my part. Whereas coming up with the right solution myself might've taken less time but the initial effort is a lot more.Last year I used to romanticize about building at least 1 thing each month completely by hand without any LLM coding help. The last such project I worked on was 6 months ago so sadly it's not going so well.
• anon
  undefined
• ramon156
  > this is just me fantasizing. At work I would just give it to an AI and continue with my dayI'm sure ~4 yrs ago i would have loved the thought of this. It's so boring. My job is so, so boring.
• hiAndrewQuinn
  The trick to making these problems intuitive is to mentally rewrite them into a "how many permutations of this string are possible?" problem. Consider the 2 * 3 case. ... ... One way you might get there is Right, right, right, down, down Then you can rewrite this as RRRDD You will always need 3 R's, and you will always need 2 D's. So how many unique strings can be made with this?Well let's actually consider the degenerate cases. ABCDE there are 5 places A can go, then 4 left B can go, then 3 left C can go, and so on, until we get 5! = 120 possible permutations of ABCDE. If you replace the B with another A to get AACDE now there are only 60 permutations, because half of the original 120 only differed by where the A and the B were relative to one another. By that same logic, AACCE has only 30 combinations, and AACCC has only 10 (seeing why it's 10 and not 20 is actually the trickiest part imo, it's because there are 3! ways to arrange CDE, but only 1 to arrange CCC).AACCC is isomorphic to RRDDD, which is how we get 10 possible paths to solve the 2*3 grid. We can check this with the binomial theorem: ((2+3) choose 3) = 10.What's nice about this step by step approach is that it generalizes not just to non-square grids, but to multiple dimensions as well! Imagine trying to get from the top of a 3 by 3 by 3 Rubik's cube to the bottom, how do you do that? Well how many ways are there to rewrite AAABBBCCC ? The logic above would suggest 9! / (3! 3! 3!) = 1,680 unique paths. And you can just derive it by starting from the degenerate case and figuring out how to slice things up!
• totetsu
  I have to say the youtube algorhythm has done wonders from my mathmatical ability by directing me to so much numberphile and 3Blue1Brown etc. And rapid AI prototyping of ideas involving complex maths has really helped me build and check intuitions about things that i might never have made if I got stuck in the weeds of trying to understand it all from 0. Like oh if this works in 2 dimensions I wonder how it looks in 4. 4d connect 4 is quite a fun game.
• Menu

  krackers
  "Compute the first few terms and plug into OEIS" is very high on the reward:effort scale
• Menu

  purple-leafy
  Heh, this grid image is all too familiar to me right now.I’m building a grid based game and engine, and I have a game replay format which is not video.I hit a massive wall with compression, trying to compress unit pathing and was trying to solve a similar solution.Given an NxN grid, and the 4 cardinal directions (NSEW) you can move in, plus an extra action that makes you move 2 cells instead of 1, and considering you can move 4 cells per second…What’s the smallest worst-case raw compression artefact you can output for 1 player for a 1 minute game?It’s an extremely fun problem to solve. I tried:- encoding changes into bits eg using 2 bits for direction- movement pattern batching (ie batching 2 moves into 3 bits)- crowd patterns and movement prediction- treating movement as a “projectile” and deriving intermediate statesAnd all sorts of other wild crap that I will write up about on game launch
• vm64
  We taught this problem in my college’s discrete math course. The intuition we gave is that it’s exactly equivalent to the number of ways to rearrange a string of 20 Rs and 20 Ds (corresponding to a right and down move)
• a_c
  Don't worry, we can regress our programming skill after 6 months of management, management of agents I mean
• utopiah
  Comment as a song https://mariedavidson.bandcamp.com/album/work-it-soulwax-rem...There is no easy way out, you have to rest but you simply can't stop. Your body will rot, your mind too.PS: song isn't an ode to the grind culture or how to slave away in an office, as lyrics say "you’ve got to work for yourself - Love yourself, feed yourself".
• cocoto
  Even if you don’t know or remember the basics of combinatorics you can solve the problem with basic dynamic programming : start with the unit grid and then expend it.
• vatsachak
  This is why you still work on open source and do coursework for fun. The same reason you go for a jog
• blmarket
  I think it's due to calibration - practical work related problems are never that simple & easy so naturally people might start from DP formula.D[A,B] := number of ways to navigate from grid sized AxB = D[A-1,B]+D[A,B-1]and the aha moment is realising this is just a binomial coefficient.
• Menu

  Mainan_Tagonist
  The more i think about math these days, the more i see it as a muscle one must constantly train to achieve its potential.Give it too long a rest and you have to go back at full blast for weeks on end to hope to ever achieve past performance.I am very bad at math and have always been in awe of those who can do it well.
• dominicrose
  Even for a math teacher the math isn't the hard part of the job.
• Menu

  jldugger
  Project Euler is like this a lot -- it's presented as a coding challenge but half the problems can be optimized to return 5 # because math
• markus_zhang
  I could be wrong but I feel this solution is not proved to be correct, but could be nevertheless the answer.On the other side, my Math ability definitely goes down to Calculus and I definitely forgot most about geometry.
• aesthesia
  I've had the same experience looking back at solutions to old problem sets and wondering how I ever came up with them.
• Menu

  noosphr
  All problems in project Euler have closed form solutions.The computer programming part of it is just a quick way to develop candidate solutions.
• meindnoch
  Manhattan distance is 2n steps. Of these, exactly n are to the right, the rest is downwards. A 2n step path is completely determined by choosing which n steps will be to the right. Hence 2n choose n.This is high school math.
• Sharlin
  If you want to keep your math brain working, and to find use for any and all the math you learned in school, look no further than graphics programming.
• soseng
  This is also me. I was a double CS and Math major in university and one of my favorite classes as a young lad was combinatorics and probability.. 25 years ago..It's true, if you don't activate this area of your brain often, it's easier to brute force the solution and reach for the easy mechanical calculation. I can feel this when I'm refactoring code. Today, I just have Claude do it for me with a few instructions. Each day, I feel a tiny bit more ignorant about the actual framework's APIs, its abstractions, and its rules. But I still would rather do other things with my time.As for the problem, luckily for me, this one was easy to derive if you remember factorials, permutations, and remember to account for duplicate patterns
• johndough
  Another approach that often works for these kinds of problems and does not require much intelligence:Work out the first few cases by hand (1,2,6,20 in our case) and then look up the sequence on "The On-Line Encyclopedia of Integer Sequences" (OEIS):https://oeis.org/search?q=1%2C2%2C6%2C20&language=english&go...
• aeve890
  Ha. When I found that problem I draw the grids and paths from the example, left for a coffee and when came back I just look at the drawings at an angle and thought "well this is just Pascal's triangle". And the solution was obvious.
• Menu

  ogogmad
  me@localhost:~> bc d=1; for(i=21; i < 41; i++){d *= i;}; print d; print "\n"; 335367096786357081410764800000 n = 1; for(i = 1; i < 21; i++){n *= i;}; print n; print "\n"; 2432902008176640000 d/n; 137846528820 I couldn't start Python for some reason, so I went 1337 and used BC, which comes preinstalled in every Unix-like OS. BC has a surprising advantage here since 40!/20! cannot be represented as a 64-bit integer since its value exceeds 2^64. That said, BC's stdlib does not provide the factorial function* - so I had to resort to using for-loops instead.* - What it does contain is sine, cosine, exponential, log, arctan, and Bessel J (?!?!?!?!)
• rabbitlord
  More like mathematical depression.
• Imustaskforhelp
  I am currently just going to college after high school and I was preparing for the JEE exam and I remember such a question was within the textbooks of maths itself.At first glance of the question, I had imagined it to be hard but then I read through the solution and other comments to recognize that I had in fact done such a question previously and I had solved it independently during the class if I remember correctly or such classes of problems.I also agree with the AI and spreadsheets part of thing for what its worth but I can only tell more when I get into job but I have heard such things from my senior brothers.I feel like there has to be a right balance of complexity though, and for what its worth I think that there are so many other things that one optimizes later on in life with tangential benefits as well with real knowledge about real life use-cases and edge-cases and so much more! I feel like it would be hard to replace with AI as much as (some) people (mostly Marketing) want it to feel so.I do hope that people don't atrophy their skills though and to solve some coding questions or make projects perhaps as well without LLM by hand if given/having the time. Not everything probably has to be done by the fastest or the most accelerated way as you wouldn't know the destination as it would be found along the way itself. I suppose just like life, so stay safe and have a nice day.
• jasonmp85
  [dead]
• Menu

  shshsjsj
  this is like the most basic leetcode question. Some comments here sound like they need to brush up the basics