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.
> 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.
I did some of that a few times in my life. But I also realised that a large part of the value I brought was not necessarily in coming up with the solution, but in convincing the rest of the company---and in training up enough of the rest of the team to understand and maintain the system.
For example at Goldman, I used an integer linear programming solver to re-shuffle how we assigned compute capacity in different data centres to various departments and how to compute fail-over plans ahead of time.
The actual modelling and implementing barely took any time at all; I used an off-the-shelf open source solver. But I spend multiple weeks teaching the team enough about linear programming so that they can eg change the model when business requirements change.
This is basically my professional career. I've stayed 5+ years at the same place now and it's gotten to the point where people ask for me to build an optimization before it's even clear they have a problem amenable to optimization. Yay! My work has a good rep. But also I'm called upon entirely too frequently to explain work I did years ago or to answer questions like which constraint "comes first". Honestly it's getting very tiring. I think a switch to a new company would help but I wonder what will happen to the optimization work at my current company after I leave. It's weird that such a useful academic field still feels like black magic to much of corporate America. I guess it's just sophisticated enough.
Maybe, though just applying a commercial or open-source off-the-shelf solver is a lot easier than most of what they ask you to figure out in your average programming interview. Probably about as hard as doing some SQL queries, which we even entrust semi-technical people with.
Of course, when your first naive modelling attempt doesn't work (eg can't be solved in any reasonable amount of time), there's a bag of more complicated tricks and some real ingenious engineering you can do; but again, the lower rungs of that aren't all that dissimilar from what your local database guru does when she tells you to add more indices.
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."
I think thats a good start probably haha. I am not a manager, and have not had a "good" one in a very long time, so I have no resources unfortunately. That isn't to say mine are bad, they let me be autonomous as an IC and I have a lot of flexibility. I am coming more from a human angle, and not a "these are the 5 things you need to do to be a good manager" angle. In my company, most people are too busy with day to day activities to really focus on developing people. When I parted with my last one, he thought we had only been together for 2 years...it had been 5. I think a lot of managers just don't really listen, they nod their head and give some advice and then 4 weeks later we're just having the same conversation again.
Everyone's situation is different, so without context this probably just reads as pouting or possibly me being incompetent haha.
If the company runs on reputation it’s only a matter of time until a consultant comes in and processes are established to move to a more efficient metric based management style.
Yeah, it's tough either way. Some managers might be biased and keep praising some of their reports that don't actually provide good value for the company. For an outside observer that has no intimate knowledge to the work, how could this be differentiated from the manager having a backbone to support a report that does truly great work but there is no good metric to prove it?
If his monetary value to the company was as said why would any other metric like complexity even remotely matter or need convincing assuming the main goal of the company was to make money.
Money would matter even more than the interpersonal stuff in most cases but on top of it even the managers treasured him so there should've been even less of an issue of communicating value.
Getting bored is totally understandable though given his calibre but that's a separate issue from how the company evaluates performance.
> If his monetary value to the company was as said why would any other metric like complexity even remotely matter
Here is why: I turned off a feature flag in our feature flagging service which saved company 10% infra cost, do you think I can be promoted to Staff+ and lead 50 engineers?
Promotions and/or recognitions in corporate environments works differently.
I don't agree with it, but this is how it works: If what you did feels simple, anyone else can do it as well, why should we promote you for finding such silly mistake or improvement.
Well, I was on the tech side which (at least at Goldman) has comparatively higher powered titles at less reports compared to the real bankers.
I suspect it's because the tech side doesn't run the model where ambitious young people without a clear idea of what to do go work for an investment bank for three years as sort-of 'finishing school'. So there are less juniors to herd for the techies.
Models of extra compensation works elsewhere, like commissions and base salaries. I would think we could come up with something for engineers and ops in return for saving the company money that doesn't result in exploitative behavior.
There is just no collective bargaining power to put it into effect.
Maybe. It looking easy isn't the point. If you have the knowledge and skills such that doing so is a semi-repeatable endeavor, especially in a world where your colleagues apparently missed that 10% savings lying on the floor, is that not (part of) the point of a promotion? [0]
> lead 50 engineers
That's a totally separate ballgame. Nothing in your example says anything about leadership ability. Maybe you have those skills, and maybe you don't, but technical acumen is separate from leadership.
> If his monetary value to the company was as said why would any other metric like complexity even remotely matter or need convincing assuming the main goal of the company was to make money.
I'll just be the Nth commenter to say it, but corporations, especially larger ones, are anything but efficient. I don't know if it ever was true, except maybe for companies focused on producing high volumes of highly standardized/specific products in a competitive environment. That's not to say that efficiency isn't desirable or beneficial in general, but as soon as it becomes difficult to put a value tag on the work being done (which unfortunately gets harder in more services oriented corporations), competing for clever ideas just rewards less than competing for the boss's attention. There's no justice or fairness in that.
Companies can be efficient and inefficient at the same time. Efficient at some things, and inefficient at other things.
For things with direct bottom line impact and fierce competition, many companies can suddenly become very efficient. But in a big company, that's often the exception rather than the rule.
> even the managers treasured him so there should've been even less of an issue of communicating value.
I’m not a fan of doing politicking, bjut after much courses on writing and communication, I strongly believe that such simple solutions could have been presented in a way to justify rewards.
There are people that do nothing worthwhile and can find words to justify themselves. If someone brings value, you can find words to earn him recognition.
This is what i hate about modern corporate culture (or human culture in general perhaps).
I dont want to expend effort politiking. I dont want to expend effort blowing my own trumpet. The value of my work is self-evident, but requires an equally intelligent person to understand.
And most people do not understand, and thus, fail to recognize the value.
I don't think this has anything to do with _modern_ corporate culture. There was never a golden age where corporate culture was better across the board.
Sometimes they do understand, but often your nice work may highlights someone's else egregious errors. And the political landscape may not work in your favor. So private praise (if you're lucky) and public silence.
Oh, I've worked in more than a dozen of software companies. When it comes to planning activities and setting goals, I've rarely seen a lot of sense.
I mean, we are in the industry where it used to be a standard practice, not so long ago, to deliver daily reports about one's activities while "planking", or throwing a beach ball to another person doing some silly acrobatics...
It should come as no surprise that there's no rigorous assessment protocol for these kinds of things anywhere. Retrospectively, I will admit, that enormous amount of effort and resources are wasted due to bad planning. But it's still not done.
I can imagine that with the field becoming more competitive, eventually, the industry specialists will come together and try to address the problem, but so far and for so long the resources just kept flowing in, the huge waste wasn't really a problem.
>> He eventually got bored. He retired and joined a seminary.
Wow, got bored and joined a seminary - Do you know how does he feel there? A genuine question - Did he expect to get excited and challenged in a seminary?
A good quote. I didn’t mean the question as “atheist vs believer argument”, but rather as an environment where ideas are accepted and flourish. Because I understood the parent comment: “He eventually got bored.” as "He got bored by the corporate environment not getting his ideas and thus went to a better organization in that regard."
So now, how was he, the one with brilliant and unorthodox ideas, got accepted in an institution like seminary? How he felt having arguments there, etc. …
I think a better way to think about it is that humans in general just make so many mistakes and imperfect judgements. Smart people just do them slightly less.
Smart people also typically have biases as well which affects their judgement, their basis are just more complicated than normal people.
Like obviously being smart is an advantage for any decision, but saying if someone smart does something it must be smart makes no sense in a world where smart people usually can’t agree on what is right.
it's a pretty good guess that when a smart person does something like this, it's something he considered long and hard and applied his full smartness to, and therefore it is probably the smart move. Right? Let's not split hairs.
well, they are known for being places to discuss theology all day. Theology isnt that far from philosophy, it's certainly interesting enough to talk about
It tells a lot about the state of software "engineering" if it's difficult to convince people of the value of the very best ones (the "lazy and smart officers" in the old story). No good engineer can ever be in love with complexity, that's like an automatic disqualification.
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.
How is that sad? A disruptive competitor will come along and eat their lunch. Creative destruction is the natural order of things. Ideally the badly managed organization will eventually be liquidated, thus freeing up resources for more productive activities.
okay, another POV is that hardly any real problems are math problems, but we seem to call solving math problems "problem solving" when that's really not true at all. like the easy part is the math. the hard part is persuading other people.
A great many real world problems have, at their crux, a mathematical problem. A mathematician paired with a subject matter expert can be powerful indeed when each sees deeply into the others' blind spots. But, I've heard statements like yours a lot over the years: assuming mathematicians aren't fully aware of the limitations of what you think math is.
Its all a math problem, and if it isn't a math problem, its a database problem.
I come across a lot of problems, and since I always try to reduce it to a math problem... I intuitively come up with both the solution to the problem, but solutions to other problems. But, if it's not a math problem, of course, its a database problem.
I would safely assume that there are no limitations of what mathematicians can do, with one important exception: Andrew, for whom I argued about the mis-uses of Infinity. Andrew is, well, rather famous.
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!]
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.
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 each "path class" on average.
Conversely, this would suggest that all paths contained in just one upper or lower triangle of the square can be counted by the Catalan numbers. Indeed, a 2x2 square has just 2 such paths and (2n C n)/(n+1) = 6/3 = 2.
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.
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.
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+2+... I starred with blocks, 1+n, 2+n-1... I had the complete formula in minutes...
That was the very last class for which I was with my peers of that grade... I was put in a HP High Potential class, with a high school algebra book, and although was a bit lonely, was in my element.
The point is- the recognition of the problem, can save huge amounts of time, where as AI can only brute force it, or use a pretrained solution.
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.
There is nothing wrong in using generating functions. A very handy and powerful tool. I wish I was better at it than I am.
It is a common experience in mathematical problem solving that the first solution leads to more insight which illuminates a shorter slap-my-forehead solution -- bruised forehead.
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.
You can go down or right at any point. To go in bottom-right corner, you need n down steps and n right steps. In how many ways can you arrange n things on type A and n things of type B? In C(2n, n). The problem is about modeling, once you model it correctly, you get the definition of combinations.
C(2n, n) isn’t about permutations, but choosing subsets of a set without order, but as I’ve thought about it, the model here is that there are 2n moves to be made and you need to choose n of them to be down (or right, depending on your preference).
Which suggests another approach to the solution; you have 20Ls and 20Rs which can appear in any order so it’s a permutation with repetition problem and thus 40!/(20!20!)
That approach has the advantage that it’s easily adapted to non square rectangular grids (n+m)!/(n!m!)
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".
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.
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.
2n choose n is just: you must move East 20 times and South 20 times. Hence any solution looks like a permutation of 20 Es and 20 Ss. Now, only look at the indices for the Es. There are 20 of them. Out of 40.
40 indices, pick 20. Those are East moves, the rest are South moves.
This interpretation also generalizes nicely to 3d. You have 60 moves, 20 of which will be up (60 choose 20), then 20 are east (40 choose 20), then 20 are south (20 choose 20).
FWIW the generalization of binomial coefficient which allows you to express an n-dimensional solution is called a multinomial coefficient [0]. So in a 3d 20x20x20 box we would have (60 multichoose 20, 20, 20) paths.
Also, the wiki article doesn't mention this but the growth rate of (n multichoose k1, k2, ..., km) as we increase n but fix the ratios p1 = k1 / n, ..., pm = km / n is precisely the Shannon entropy of the categorical distribution with probabilities p1, ..., pm . The wiki article for entropy [1] states the result for the binomial coefficient, which can be written as (n choose k) = (n multichoose k, (n - k)) .
Actually a lot of basic information-theoretic results about entropy and related quantities (e.g. the properties of the Boltzmann distribution/softmax function) can be derived from similar discrete counting problems after taking a large-n limit. I don't have links at the ready but I might edit this comment if I remember places which explain this stuff.
That was in the first calendar quarter this year. In the second calendar quarter CEOs started saying AI is too fucking expensive and let's stop doing it.
maybe your C suite. ours just joined the hype train. We got an email a few weeks ago saying all new code must be ai generated by 2027 and introducing internal metrics for ai usage by team
I’m currently in the middle of this process. I’m finding refusing to be working remarkably well, though I am also in a country with worker protections. Join a union.
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 (?!?!?!?!)
After the i-th iteration of the for loop, ans will contain n!/((n-i)!i!) which is exactly \binom{n}{i}, an integer.
Technically "ans" can grow above the final result in my example, but even that could be fixed if one really wants (e.g. i must divide either ans or n-i, you play a bit with divmod to figure out which division you do first.)
- Python's native integer handling, which already has no size limit.
- PLUS part of the Decimal module in Python's stdlib: BC's floats are DECIMAL by default, not binary.
- PLUS an implementation of Bessel's J function, while neglecting Bessel's K.
- Some features for base conversion using `ibase` and `obase`. So, I suppose you can output numbers to base 60. [EDIT: Correction from earlier: ibase is allowed to be at most 16, while POSIX allows for the maximum value of obase to be at least 99, which therefore does allow for formatting output to base 60.]
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.
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 :)
The binomial coefficient is derived from Pascal’s triangle, which is a fact I remember but cannot explain, based on some hazy memories of 10th grade math class.
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.
The rectangle case is just choose(w+h, h), with w, h being the width and height, respectively. (Or choose(w+h, w), which is the same.)
Depending on how well you know binomial coefficients, this can also lead you to the dynamic programming solution. After all, for the rectangle case this is nothing but construction of Pascal's triangle by summing two adjacent numbers in a row to get the number below them. If you recognize this, the solution for paths on grids with holes falls out almost immediately.
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!
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.
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 states
And all sorts of other wild crap that I will write up about on game launch
What a lot of games do is run a strictly deterministic simulation in lockstep. Then you don't save the path of every unit, you save one move command for the whole group. Then the game replays inputs, and the pathing algorithm should give the same result if there are no desyncs.
Yes you are definitely onto something! Love to see more people talking about deterministic games.
My game is strictly deterministic, so I get bot movement for free - but the player has agency so I need to capture their deviations
That’s the tricky part! Right now I do capture input (actually just deviations) and can replay whole games, but I think I’m at the limits in terms of compression - talking bytes here not KB
I will post back with real numbers tonight, but naive approach did not compress well at all (KB easily).
But of the “smart” approaches (pre compression):
- 5 move motif over 2 bytes - best
- 2 move motif - insanely small
- 2 move motif with non linear tick delta even better
- “naive” 1 byte cardinal directions (worst)
- less-naive - byte relative direction changes (middle of the pack)
—-
But post compression with brotli the naive approach was second best and the less-naive approach was first (2-10% better than naive), I was so bummed that the better ones didn’t compress as well (about 10% worse than second best on average)
Came here to say this-- I've solved a lot of otherwise seemingly quite hard problems by numerically/brute-force finding the first few terms then asking OEIS then convincing myself it it's indeed the solution.
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.
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.)
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)
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.
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.
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".
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.
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 sequence numbers of problems on Project Euler only represent the order of publication. They aren't deliberately connected except for a few problems that have one or two follow-ups.
Most of the first 100 problems can be solved without any understanding of the problem, should you so desire.
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/).
Because the daily world has much less math in it than we care to admit. You don't need calculus to understand which elevator button needs to be pushed.
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.
At least the site itself is honest about it, describing the challenges as "mathematical/computer programming problems". I always found it weird when people recommended them as ways to learn a new programming language because, perhaps because of my own math background, I ended up solving most of them with math and so little programming. Where the programs only used the basics of any language so that nothing novel would be learned by rewriting except for how loops look in two different languages (or don't, in the case of some FP languages), or how arrays look, etc.
Professional mathematicians and computer scientists use AI as well. Maybe that's not very reassuring from the standpoint of skills regression in modern adult work culture, but at least it demonstrates that skills and tool usage aren't mutually exclusive.
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.
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.
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
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.
> 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.
I did some of that a few times in my life. But I also realised that a large part of the value I brought was not necessarily in coming up with the solution, but in convincing the rest of the company---and in training up enough of the rest of the team to understand and maintain the system.
For example at Goldman, I used an integer linear programming solver to re-shuffle how we assigned compute capacity in different data centres to various departments and how to compute fail-over plans ahead of time.
The actual modelling and implementing barely took any time at all; I used an off-the-shelf open source solver. But I spend multiple weeks teaching the team enough about linear programming so that they can eg change the model when business requirements change.
This is basically my professional career. I've stayed 5+ years at the same place now and it's gotten to the point where people ask for me to build an optimization before it's even clear they have a problem amenable to optimization. Yay! My work has a good rep. But also I'm called upon entirely too frequently to explain work I did years ago or to answer questions like which constraint "comes first". Honestly it's getting very tiring. I think a switch to a new company would help but I wonder what will happen to the optimization work at my current company after I leave. It's weird that such a useful academic field still feels like black magic to much of corporate America. I guess it's just sophisticated enough.
> I guess it's just sophisticated enough.
Maybe, though just applying a commercial or open-source off-the-shelf solver is a lot easier than most of what they ask you to figure out in your average programming interview. Probably about as hard as doing some SQL queries, which we even entrust semi-technical people with.
Of course, when your first naive modelling attempt doesn't work (eg can't be solved in any reasonable amount of time), there's a bag of more complicated tricks and some real ingenious engineering you can do; but again, the lower rungs of that aren't all that dissimilar from what your local database guru does when she tells you to add more indices.
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."
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 ?
I think thats a good start probably haha. I am not a manager, and have not had a "good" one in a very long time, so I have no resources unfortunately. That isn't to say mine are bad, they let me be autonomous as an IC and I have a lot of flexibility. I am coming more from a human angle, and not a "these are the 5 things you need to do to be a good manager" angle. In my company, most people are too busy with day to day activities to really focus on developing people. When I parted with my last one, he thought we had only been together for 2 years...it had been 5. I think a lot of managers just don't really listen, they nod their head and give some advice and then 4 weeks later we're just having the same conversation again.
Everyone's situation is different, so without context this probably just reads as pouting or possibly me being incompetent haha.
radical candor is a good book that explores a simple framework for people management.
A massively abused book that led many to justify being assholes. You don’t have to read it, here’s the Cliff’s Notes:
Care personally, challenge directly.
If the company runs on reputation it’s only a matter of time until a consultant comes in and processes are established to move to a more efficient metric based management style.
Yeah, it's tough either way. Some managers might be biased and keep praising some of their reports that don't actually provide good value for the company. For an outside observer that has no intimate knowledge to the work, how could this be differentiated from the manager having a backbone to support a report that does truly great work but there is no good metric to prove it?
Depends on leadership culture. In toxic (aka “competitive”) environments managers are insecure and fear their own staff as potential competition.
If his monetary value to the company was as said why would any other metric like complexity even remotely matter or need convincing assuming the main goal of the company was to make money.
Money would matter even more than the interpersonal stuff in most cases but on top of it even the managers treasured him so there should've been even less of an issue of communicating value.
Getting bored is totally understandable though given his calibre but that's a separate issue from how the company evaluates performance.
> If his monetary value to the company was as said why would any other metric like complexity even remotely matter
Here is why: I turned off a feature flag in our feature flagging service which saved company 10% infra cost, do you think I can be promoted to Staff+ and lead 50 engineers?
Promotions and/or recognitions in corporate environments works differently.
I don't agree with it, but this is how it works: If what you did feels simple, anyone else can do it as well, why should we promote you for finding such silly mistake or improvement.
Same with executives and upper management.
One cannot be a director/VP of 3 people. They need an empire…
One can be a VP of 0 people.
Look into what the title means at banks!
I was an Executive Director / Vice President at Goldman with 0 reports.
Banks are the inversion where a Director is higher than a VP! An ED w/o reports is actually moderately impressive though!
Well, I was on the tech side which (at least at Goldman) has comparatively higher powered titles at less reports compared to the real bankers.
I suspect it's because the tech side doesn't run the model where ambitious young people without a clear idea of what to do go work for an investment bank for three years as sort-of 'finishing school'. So there are less juniors to herd for the techies.
Models of extra compensation works elsewhere, like commissions and base salaries. I would think we could come up with something for engineers and ops in return for saving the company money that doesn't result in exploitative behavior.
There is just no collective bargaining power to put it into effect.
> staff+
Maybe. It looking easy isn't the point. If you have the knowledge and skills such that doing so is a semi-repeatable endeavor, especially in a world where your colleagues apparently missed that 10% savings lying on the floor, is that not (part of) the point of a promotion? [0]
> lead 50 engineers
That's a totally separate ballgame. Nothing in your example says anything about leadership ability. Maybe you have those skills, and maybe you don't, but technical acumen is separate from leadership.
[0] https://mosaicstrategy.us/2016/10/10/know-where-to-tap/
> If his monetary value to the company was as said why would any other metric like complexity even remotely matter or need convincing assuming the main goal of the company was to make money.
I'll just be the Nth commenter to say it, but corporations, especially larger ones, are anything but efficient. I don't know if it ever was true, except maybe for companies focused on producing high volumes of highly standardized/specific products in a competitive environment. That's not to say that efficiency isn't desirable or beneficial in general, but as soon as it becomes difficult to put a value tag on the work being done (which unfortunately gets harder in more services oriented corporations), competing for clever ideas just rewards less than competing for the boss's attention. There's no justice or fairness in that.
Companies can be efficient and inefficient at the same time. Efficient at some things, and inefficient at other things.
For things with direct bottom line impact and fierce competition, many companies can suddenly become very efficient. But in a big company, that's often the exception rather than the rule.
> even the managers treasured him so there should've been even less of an issue of communicating value.
I’m not a fan of doing politicking, bjut after much courses on writing and communication, I strongly believe that such simple solutions could have been presented in a way to justify rewards.
There are people that do nothing worthwhile and can find words to justify themselves. If someone brings value, you can find words to earn him recognition.
This is what i hate about modern corporate culture (or human culture in general perhaps).
I dont want to expend effort politiking. I dont want to expend effort blowing my own trumpet. The value of my work is self-evident, but requires an equally intelligent person to understand.
And most people do not understand, and thus, fail to recognize the value.
I don't think this has anything to do with _modern_ corporate culture. There was never a golden age where corporate culture was better across the board.
Sometimes they do understand, but often your nice work may highlights someone's else egregious errors. And the political landscape may not work in your favor. So private praise (if you're lucky) and public silence.
Oh, I've worked in more than a dozen of software companies. When it comes to planning activities and setting goals, I've rarely seen a lot of sense.
I mean, we are in the industry where it used to be a standard practice, not so long ago, to deliver daily reports about one's activities while "planking", or throwing a beach ball to another person doing some silly acrobatics...
It should come as no surprise that there's no rigorous assessment protocol for these kinds of things anywhere. Retrospectively, I will admit, that enormous amount of effort and resources are wasted due to bad planning. But it's still not done.
I can imagine that with the field becoming more competitive, eventually, the industry specialists will come together and try to address the problem, but so far and for so long the resources just kept flowing in, the huge waste wasn't really a problem.
>> He eventually got bored. He retired and joined a seminary.
Wow, got bored and joined a seminary - Do you know how does he feel there? A genuine question - Did he expect to get excited and challenged in a seminary?
"The first gulp from the glass of natural sciences will turn you into an atheist, but at the bottom of the glass, God is waiting for you."
A good quote. I didn’t mean the question as “atheist vs believer argument”, but rather as an environment where ideas are accepted and flourish. Because I understood the parent comment: “He eventually got bored.” as "He got bored by the corporate environment not getting his ideas and thus went to a better organization in that regard."
So now, how was he, the one with brilliant and unorthodox ideas, got accepted in an institution like seminary? How he felt having arguments there, etc. …
If a smart guy joins a seminary then joining a seminary is probably smart.
I think a better way to think about it is that humans in general just make so many mistakes and imperfect judgements. Smart people just do them slightly less. Smart people also typically have biases as well which affects their judgement, their basis are just more complicated than normal people. Like obviously being smart is an advantage for any decision, but saying if someone smart does something it must be smart makes no sense in a world where smart people usually can’t agree on what is right.
it's a pretty good guess that when a smart person does something like this, it's something he considered long and hard and applied his full smartness to, and therefore it is probably the smart move. Right? Let's not split hairs.
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Compare and contrast https://en.wikipedia.org/wiki/Ted_Kaczynski
I don't get it, is the point you're trying to make that Ted Kaczynski wasn't smart?!
Just the opposite: he was very smart; but most people don't consider his overall life choices to be something you would want to copy.
Ah, got it! Thanks for the clarification.
Sorry, I was perhaps being too indirect.
well, they are known for being places to discuss theology all day. Theology isnt that far from philosophy, it's certainly interesting enough to talk about
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It tells a lot about the state of software "engineering" if it's difficult to convince people of the value of the very best ones (the "lazy and smart officers" in the old story). No good engineer can ever be in love with complexity, that's like an automatic disqualification.
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.
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How is that sad? A disruptive competitor will come along and eat their lunch. Creative destruction is the natural order of things. Ideally the badly managed organization will eventually be liquidated, thus freeing up resources for more productive activities.
Unless they are, in addition to bad at management, really good at regulatory capture.
E.g. Experian, Transunion and their ilk are unlikely to be eaten as lunch soon.
Apologies to the good engineers and managers in those orgs!
reminds me of this other post https://news.ycombinator.com/item?id=48498385
great story.
okay, another POV is that hardly any real problems are math problems, but we seem to call solving math problems "problem solving" when that's really not true at all. like the easy part is the math. the hard part is persuading other people.
A great many real world problems have, at their crux, a mathematical problem. A mathematician paired with a subject matter expert can be powerful indeed when each sees deeply into the others' blind spots. But, I've heard statements like yours a lot over the years: assuming mathematicians aren't fully aware of the limitations of what you think math is.
Its all a math problem, and if it isn't a math problem, its a database problem. I come across a lot of problems, and since I always try to reduce it to a math problem... I intuitively come up with both the solution to the problem, but solutions to other problems. But, if it's not a math problem, of course, its a database problem.
I would safely assume that there are no limitations of what mathematicians can do, with one important exception: Andrew, for whom I argued about the mis-uses of Infinity. Andrew is, well, rather famous.
The hard part is often finding a way to turn your business problem into a math problem in the first place.
Database problems are math problems too.
Perfection is achieved, not when there is nothing left to add, but nothing to take away. ― Antoine de Saint-Exupéry
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/
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.
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 each "path class" on average.
Conversely, this would suggest that all paths contained in just one upper or lower triangle of the square can be counted by the Catalan numbers. Indeed, a 2x2 square has just 2 such paths and (2n C n)/(n+1) = 6/3 = 2.
[0]:https://blog.wilsonb.com/posts/2026-02-27-easy-random-trees....
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/
This was one of my favorite Project Euler problems, but getting to the mathematical solution admittedly took me a couple of weeks.
Perhaps because I was pigeon-holing this as a programming optimization problem.
I wrote about it too! [0]
[0] https://nmn.gl/blog/vibe-coding-gambling
or just rotate it by 45 degrees and pretend it's a pascal's triangle.
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.
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+2+... I starred with blocks, 1+n, 2+n-1... I had the complete formula in minutes...
That was the very last class for which I was with my peers of that grade... I was put in a HP High Potential class, with a high school algebra book, and although was a bit lonely, was in my element.
The point is- the recognition of the problem, can save huge amounts of time, where as AI can only brute force it, or use a pretrained solution.
are block referring to Cuisenaire rods?
Eh, AI is getting better at creating little visualisations, too.
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.
There is nothing wrong in using generating functions. A very handy and powerful tool. I wish I was better at it than I am.
It is a common experience in mathematical problem solving that the first solution leads to more insight which illuminates a shorter slap-my-forehead solution -- bruised forehead.
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.
You can go down or right at any point. To go in bottom-right corner, you need n down steps and n right steps. In how many ways can you arrange n things on type A and n things of type B? In C(2n, n). The problem is about modeling, once you model it correctly, you get the definition of combinations.
C(2n, n) isn’t about permutations, but choosing subsets of a set without order, but as I’ve thought about it, the model here is that there are 2n moves to be made and you need to choose n of them to be down (or right, depending on your preference).
You always either go left or down so total 40 steps, choose 20 to be down (or 20 to be right)
Which suggests another approach to the solution; you have 20Ls and 20Rs which can appear in any order so it’s a permutation with repetition problem and thus 40!/(20!20!)
That approach has the advantage that it’s easily adapted to non square rectangular grids (n+m)!/(n!m!)
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".
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.
Maths IS pattern recognition
Followed by a proof via induction.
Tbh your student reasoning is still dangerous... the patterns could have not generalized nicely. see moser's circle problem
needed to justify viewing this as "arranging down vs right movements" as another comment outlines
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.
2n choose n is just: you must move East 20 times and South 20 times. Hence any solution looks like a permutation of 20 Es and 20 Ss. Now, only look at the indices for the Es. There are 20 of them. Out of 40.
40 indices, pick 20. Those are East moves, the rest are South moves.
This interpretation also generalizes nicely to 3d. You have 60 moves, 20 of which will be up (60 choose 20), then 20 are east (40 choose 20), then 20 are south (20 choose 20).
That's a nice way to put it.
FWIW the generalization of binomial coefficient which allows you to express an n-dimensional solution is called a multinomial coefficient [0]. So in a 3d 20x20x20 box we would have (60 multichoose 20, 20, 20) paths.
Also, the wiki article doesn't mention this but the growth rate of (n multichoose k1, k2, ..., km) as we increase n but fix the ratios p1 = k1 / n, ..., pm = km / n is precisely the Shannon entropy of the categorical distribution with probabilities p1, ..., pm . The wiki article for entropy [1] states the result for the binomial coefficient, which can be written as (n choose k) = (n multichoose k, (n - k)) .
Actually a lot of basic information-theoretic results about entropy and related quantities (e.g. the properties of the Boltzmann distribution/softmax function) can be derived from similar discrete counting problems after taking a large-n limit. I don't have links at the ready but I might edit this comment if I remember places which explain this stuff.
[0] https://en.wikipedia.org/wiki/Multinomial_theorem#Number_of_...
[1] https://en.wikipedia.org/wiki/Entropy_(information_theory)#A...
Nobody forces you to use AI.
It has become sort of junk food for the brain. Temptations and ads for it everywhere.
If you have performance based metrics about your AI usage then you are essentially being forced to use AI (or become unemployed)
Plenty of people are experiencing this nowadays
The idea that no one is being forced to use AI is nonsense
That was in the first calendar quarter this year. In the second calendar quarter CEOs started saying AI is too fucking expensive and let's stop doing it.
You’re now being forced to optimize your AI cost usage and produce the same results as before you cared about it.
maybe your C suite. ours just joined the hype train. We got an email a few weeks ago saying all new code must be ai generated by 2027 and introducing internal metrics for ai usage by team
I’m currently in the middle of this process. I’m finding refusing to be working remarkably well, though I am also in a country with worker protections. Join a union.
I'm becoming convinced that mandatory AI use is a humiliation ritual, akin to neighboring tribes forcing Hebrews to eat pork.
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There is no metric a reasonably intelligent person can't sabotage or subvert...
...for example, you can write a script to burn tokens and write the code yourself.
e.g. <https://github.com/dtnewman/burn-baby-burn>
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* - What it does contain is sine, cosine, exponential, log, arctan, and Bessel J (?!?!?!?!)
You don't need space for 40!/20!, for example:
The same idea can be trivially tweaked to compute any binomial coefficient without ever storing an integer greater than the final result.Good point. But what if `i` does not divide `ans` evenly? I suppose you could use floats and then round.
It always divides it evenly, that's why it works.
After the i-th iteration of the for loop, ans will contain n!/((n-i)!i!) which is exactly \binom{n}{i}, an integer.
Technically "ans" can grow above the final result in my example, but even that could be fixed if one really wants (e.g. i must divide either ans or n-i, you play a bit with divmod to figure out which division you do first.)
Just noting that Python natively handles integers larger than the machine word size since version 2.5, so this would have worked in Python as well.
I think BC's features are equivalent to:
- Python's native integer handling, which already has no size limit.
- PLUS part of the Decimal module in Python's stdlib: BC's floats are DECIMAL by default, not binary.
- PLUS an implementation of Bessel's J function, while neglecting Bessel's K.
- Some features for base conversion using `ibase` and `obase`. So, I suppose you can output numbers to base 60. [EDIT: Correction from earlier: ibase is allowed to be at most 16, while POSIX allows for the maximum value of obase to be at least 99, which therefore does allow for formatting output to base 60.]
And does it quite fast too:
https://www.wilfred.me.uk/blog/2014/10/20/the-fastest-bigint...
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.
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 :)
The binomial coefficient is derived from Pascal’s triangle, which is a fact I remember but cannot explain, based on some hazy memories of 10th grade math class.
You can take the two variables to be outcomes of a binomial event:
- at zero events: we have one outcome, nothing
- at one event: we have two outcomes, 1X and 1Y
- at two events, each of those two outcomes have two outcomes, but one X and one Y outcome both become XY, so we have 1X^2 2XY 1Y^2.
You can continue combining this way, adding an X and Y to each term (for the two possible outcomes) — and then grouping like terms.
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.
The rectangle case is just choose(w+h, h), with w, h being the width and height, respectively. (Or choose(w+h, w), which is the same.)
Depending on how well you know binomial coefficients, this can also lead you to the dynamic programming solution. After all, for the rectangle case this is nothing but construction of Pascal's triangle by summing two adjacent numbers in a row to get the number below them. If you recognize this, the solution for paths on grids with holes falls out almost immediately.
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 Then you can rewrite this as 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.
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 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, has only 30 combinations, and 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
? 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!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.
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 states
And all sorts of other wild crap that I will write up about on game launch
What a lot of games do is run a strictly deterministic simulation in lockstep. Then you don't save the path of every unit, you save one move command for the whole group. Then the game replays inputs, and the pathing algorithm should give the same result if there are no desyncs.
Yes you are definitely onto something! Love to see more people talking about deterministic games.
My game is strictly deterministic, so I get bot movement for free - but the player has agency so I need to capture their deviations
That’s the tricky part! Right now I do capture input (actually just deviations) and can replay whole games, but I think I’m at the limits in terms of compression - talking bytes here not KB
How much did your clever approach save over a naive approach + file compression?
I will post back with real numbers tonight, but naive approach did not compress well at all (KB easily).
But of the “smart” approaches (pre compression):
- 5 move motif over 2 bytes - best
- 2 move motif - insanely small
- 2 move motif with non linear tick delta even better
- “naive” 1 byte cardinal directions (worst)
- less-naive - byte relative direction changes (middle of the pack)
—-
But post compression with brotli the naive approach was second best and the less-naive approach was first (2-10% better than naive), I was so bummed that the better ones didn’t compress as well (about 10% worse than second best on average)
"Compute the first few terms and plug into OEIS" is very high on the reward:effort scale
Came here to say this-- I've solved a lot of otherwise seemingly quite hard problems by numerically/brute-force finding the first few terms then asking OEIS then convincing myself it it's indeed the solution.
not really solving the problem though. would be easier to directly search for project euler xx solution.
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.
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.)
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)
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.
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.
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".
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.
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 sequence numbers of problems on Project Euler only represent the order of publication. They aren't deliberately connected except for a few problems that have one or two follow-ups.
Most of the first 100 problems can be solved without any understanding of the problem, should you so desire.
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/).
This is why you still work on open source and do coursework for fun. The same reason you go for a jog
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.
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.
> 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.
Because the daily world has much less math in it than we care to admit. You don't need calculus to understand which elevator button needs to be pushed.
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.
Project Euler is like this a lot -- it's presented as a coding challenge but half the problems can be optimized to
At least the site itself is honest about it, describing the challenges as "mathematical/computer programming problems". I always found it weird when people recommended them as ways to learn a new programming language because, perhaps because of my own math background, I ended up solving most of them with math and so little programming. Where the programs only used the basics of any language so that nothing novel would be learned by rewriting except for how loops look in two different languages (or don't, in the case of some FP languages), or how arrays look, etc.
Professional mathematicians and computer scientists use AI as well. Maybe that's not very reassuring from the standpoint of skills regression in modern adult work culture, but at least it demonstrates that skills and tool usage aren't mutually exclusive.
Don't worry, we can regress our programming skill after 6 months of management, management of agents I mean
I've had the same experience looking back at solutions to old problem sets and wondering how I ever came up with them.
Even for a math teacher the math isn't the hard part of the job.
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.
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.
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.
A lot of life is "use it or lose it," due to limited resources I imagine.
All problems in project Euler have closed form solutions.
The computer programming part of it is just a quick way to develop candidate solutions.
https://projecteuler.net/problem=76 does this have a closed form solution? vast majority of the problems don't seem to have a closed form solution
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
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...
More like mathematical depression.
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this is like the most basic leetcode question. Some comments here sound like they need to brush up the basics
thanks for keeping us honest. what would we ever do without this insightful comment of yours?