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.
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.
> 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.
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.
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."
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!]
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.
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).
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.
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.
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.
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!
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 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.
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
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.
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.
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)
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.
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 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.
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.
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.]
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.
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.
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.
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.
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.
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.
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/
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.
40 indices, pick 20. Those are East moves, the rest are South moves.
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.
needed to justify viewing this as "arranging down vs right movements" as another comment outlines
It has become sort of junk food for the brain. Temptations and ads for it everywhere.
Plenty of people are experiencing this nowadays
The idea that no one is being forced to use AI is nonsense
...for example, you can write a script to burn tokens and write the code yourself.
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 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 :)
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.
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.
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
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
This is high school math.
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.
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".
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
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.
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...
* - 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.)
https://www.wilfred.me.uk/blog/2014/10/20/the-fastest-bigint...
- 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.]