6 ms·

A math problem generator

145 points by mindcrime 4y ago

lupire 4y ago
The author is talented, but these are mundane exercises, not problems. They are all trivial mechanical computations best done by a calculator, that do nothing to promote understanding and insight. A lot of work went into this, to generate trivial exercises to train rote memorization, like flashcards.
- BeetleB 4y ago
  Almost every physicist would disagree :-) While mathematicians hate this kind of stuff, it is the bread and butter of physicists. While it is true most of them will not compute the cross product of two concrete vectors by hand, most will compute them symbolically, and the skills to do both are the same.
  - el_sinchi 4y ago
    yet, he's right. these are "exercises", not "problems" they don't involve creative thinking. look up the difference on any text about "problem solving"
- vouaobrasil 4y ago
  I have a degree in math and I actually enjoy these types of problems. Keeping sharp at mental arithmetic is really fun and it prompts the brain to come up with shortcuts that can lead to more interesting math. Like, is 51 prime? (No, 5+1 = 6). It's not really rote memorization but getting better at mental arithmetic, which is actually meditative in a way.
  - melony 4y ago
    Why do you sum the digits? Goldbach?
    - arbitrandomuser 4y ago
      It's divisible by 3 , because 6
      - bobthepanda 4y ago
        To say this in more detail, it’s observed that any number with digits adding up to multiples of 3 is divisible by 3 https://math.stackexchange.com/questions/341202/how-to-prove-the-divisibility-rule-for-3-casting-out-threes https://math.stackexchange.com/questions/341202/how-to-prove...
        shagie 4y ago
        https://everything2.com/title/m_turner%2527s+postulate https://everything2.com/title/m_turner%2527s+postulate For any integer base B greater than 2, the multiples of the number B-1, when represented in base B will always have the sum of the digits be a multiple of B-1. If B-1 is a square number (4,9,16,...) then the square root of B-1 will also have this property. (the following node has a proof)
        schoen 4y ago
        > If B-1 is a square number (4,9,16,...) then the square root of B-1 will also have this property. In fact, if d is any divisor of B-1 (including a trivial divisor), then d will have this property. (The author of the proof on the page you linked to is aware of this, but the author of the original conjecture isn't. For example, the reply notes that the digital root test for divisibility in hexadecimal works for 3 or 5, which are divisors of 16-1=15, regardless of the fact that 15 isn't a square. It also works for 1 or 15, which are also divisors of 15.)
    - vouaobrasil 4y ago
      It is because a number is base ten is like this: a + 10b + 100c + 1000d + .... But 10 leaves a remainder of 1 on division by 3 so this number divided by 3 is the same as a + b + c + d + ...
- tibbon 4y ago
  Curiously, ChatGPT falls on its face with about half of these that I checked. It was confidently wrong about them too, but tried its logic again when I corrected it. YMMV. So if your calculator is a huge neural net, maybe the problems aren't trivial!
  - whatyesaid 4y ago
    Try ask ChatGPT what 12931239 + 712380 is. It can only solve 30 + 46 because it actually memorized/'saw' the latter, but that's nothing new.
  - sesm 4y ago
    But Wolfram Alpha will do them all easily.
- lukew3 4y ago
  True, perhaps this would be better called an exercise generator. However, when studying this in school, the teachers always called them problems, hence why the title was chosen. The project is aims to help students K-12 and possibly undergrads whose curriculum is aimed to get them comfortable with solving these computations without aid of a calculator, or at least without programs that solve it for them. This is not supposed to be the primary means of teaching, but rather a tool that teachers can use to provide infinite exercises for their students to practice on, with the benefit of students not being able to copy answers directly out of a textbook without understanding how it works.
mindcrime 4y ago
Personally I don't care about the quibbling over semantics as far as "problem" vs "exercise". Fine, call it a "math exercise generator". That's something I personally wanted and was very happy to find this, and I expect it's something a lot of other people (especially people doing self-study) will find useful. I think it would also be fun to look at contributing to this, and see about maybe adding some generators for additional probl... er, exercises. In particular, adding more stuff in the Calculus section might be an interesting challenge, and could make this even more useful.
- nutanc 4y ago
  Agree. This can be very useful for an online learning tool. Will see how I can leverage this. Thanks for this.
tuukkah 4y ago
Are there generators for programming exercises?
- ruune 4y ago
  Codewars [0] and sites like it [1] maybe [0] https://www.codewars.com/ https://www.codewars.com/ [1] https://alternativeto.net/software/codewars/ https://alternativeto.net/software/codewars/ https://www.reddit.com/r/learnprogramming/comments/43upct/31_coders_games_and_puzzle_sites/ https://www.reddit.com/r/learnprogramming/comments/43upct/31...
lukew3 4y ago
Hi, I'm the project author. It's cool to see so many people suddenly take interest in this project. I haven't touched it for a while, but seeing such support has restored my desire to work on it. I would love to hear your thoughts and would be happy to respond to any questions!
- ytygg775 4y ago
  I applaud your effords. But unfortunately, this perpetuates the notion that this is what math is. It's mundane calculations, best done by a machine. Nobody even underatands why it's meaningful to multiply matrices like that. How come that high school math is full of this stuff, like doing manual matrix multiplications over and over, but not a single mathematical proof? Society would really benefit from people training logical reasoning, which is what a mathematical proof is. Rather that than a manual matrix multiplication. Heck, even most of my SWE colleagues can't even write down a real proof.
  - lukew3 4y ago
    Thanks. I don't believe that math is simply learning how to perform calculations. However, I also don't think that I'm in the best position to teach students how to understand why they do these calculations. It is entirely up to teachers and those who create curriculum to teach students why these calculations work. This project simply gives teachers another tool to help students solidify their what they have learned.
    - _the_inflator 4y ago
      I love your Repo. It is fun to solve these tasks and to have an abundance of these is even more compelling. I love Math and this is sort of casual gaming.
    - ranting-moth 4y ago
      Don't let the naysayers grind you down. Your project is very useful for what it is! You can't please everybody but in this case you're doing a pretty good job.
xchkr1337 4y ago
Why is there an exercise called "complex division" even though it has nothing to do with complex division?
- lukew3 4y ago
  Thanks for pointing this out. This is a naming error, where complex is used as an antonym of simple. I think a better name would be "rational result division" or even "fraction to decimal". This is being corrected now.
derelicta 4y ago
Absolutely love the concept.
c7b 4y ago
I guess it's my problem, but having been blown away by AI so often recently, I was sort of expecting more here. I suppose an AI that could generate full text problem statements together with sample solutions for basic university level math courses like probability or linear algebra isn't too far out of reach at this point anymore. There could also be some actual market opportunities in that, both with educators and students who want to prep for tests.
- Herbstluft 4y ago
  > isn't too far out of reach If you are referring to the recent ChatGPT developments, it can't even solve these very basic exercises.
  - c7b 4y ago
    I was more thinking of Github Copilot, but ChatGPT is similarly good at programming. Programming is arguably even less error-forgiving than providing a written answer to math problems (I was more thinking of problems that ask for an analytical rather than numerical answers, which is also what most university-level math courses ask for). I wouldn't have expected those models to perform nearly as well at programming as they do, so I've updated my prior. Of course you'd need to find a math corpus of a similar size to github's repositories first. Overleaf has a bit of a similar position to github in STEM research, so that would be an interesting place to start looking (if it's legally possible). It's probably still too small, but would be interesting to see what a GPT version trained on the Overleaf corpus could do.
petesergeant 4y ago
Extend it to produce (and sell!) sample GMAT papers
i13e 4y ago
I think the coolest thing about this tool that I haven't seen mentioned yet is randomizing homework and tests for students on pen and paper without an expensive web platform. This completely eliminates cheating in a math class. Would some sample AI generated text for problems be nice? Sure. But this is the bulk of the work already done IMO.
vitorsr 4y ago
Those are not problems. See [1], [2], [3] and many others [4]. [1] https://bookstore.ams.org/content?PageName=prb https://bookstore.ams.org/content?PageName=prb [2] https://www.maa.org/press/ebooks/problem-books https://www.maa.org/press/ebooks/problem-books [3] https://www.springer.com/series/714 https://www.springer.com/series/714 [4] https://zbmath.org/?q=cc%3A00A07 https://zbmath.org/?q=cc%3A00A07
- SquareShoe 4y ago
  Wolfram has a similar product that they call a “problem generator” [1] [1]https://www.wolframalpha.com/problem-generator/ https://www.wolframalpha.com/problem-generator/

145 points by mindcrime 4y ago

The author is talented, but these are mundane exercises, not problems. They are all trivial mechanical computations best done by a calculator, that do nothing to promote understanding and insight. A lot of work went into this, to generate trivial exercises to train rote memorization, like flashcards.

BeetleB 4y ago
Almost every physicist would disagree :-) While mathematicians hate this kind of stuff, it is the bread and butter of physicists. While it is true most of them will not compute the cross product of two concrete vectors by hand, most will compute them symbolically, and the skills to do both are the same.
- el_sinchi 4y ago
  yet, he's right. these are "exercises", not "problems" they don't involve creative thinking. look up the difference on any text about "problem solving"
vouaobrasil 4y ago
I have a degree in math and I actually enjoy these types of problems. Keeping sharp at mental arithmetic is really fun and it prompts the brain to come up with shortcuts that can lead to more interesting math. Like, is 51 prime? (No, 5+1 = 6). It's not really rote memorization but getting better at mental arithmetic, which is actually meditative in a way.
- melony 4y ago
  Why do you sum the digits? Goldbach?
  - arbitrandomuser 4y ago
    It's divisible by 3 , because 6
    - bobthepanda 4y ago
      To say this in more detail, it’s observed that any number with digits adding up to multiples of 3 is divisible by 3 https://math.stackexchange.com/questions/341202/how-to-prove-the-divisibility-rule-for-3-casting-out-threes https://math.stackexchange.com/questions/341202/how-to-prove...
      - shagie 4y ago
        https://everything2.com/title/m_turner%2527s+postulate https://everything2.com/title/m_turner%2527s+postulate For any integer base B greater than 2, the multiples of the number B-1, when represented in base B will always have the sum of the digits be a multiple of B-1. If B-1 is a square number (4,9,16,...) then the square root of B-1 will also have this property. (the following node has a proof)
        schoen 4y ago
        > If B-1 is a square number (4,9,16,...) then the square root of B-1 will also have this property. In fact, if d is any divisor of B-1 (including a trivial divisor), then d will have this property. (The author of the proof on the page you linked to is aware of this, but the author of the original conjecture isn't. For example, the reply notes that the digital root test for divisibility in hexadecimal works for 3 or 5, which are divisors of 16-1=15, regardless of the fact that 15 isn't a square. It also works for 1 or 15, which are also divisors of 15.)
  - vouaobrasil 4y ago
    It is because a number is base ten is like this: a + 10b + 100c + 1000d + .... But 10 leaves a remainder of 1 on division by 3 so this number divided by 3 is the same as a + b + c + d + ...
tibbon 4y ago
Curiously, ChatGPT falls on its face with about half of these that I checked. It was confidently wrong about them too, but tried its logic again when I corrected it. YMMV. So if your calculator is a huge neural net, maybe the problems aren't trivial!
- whatyesaid 4y ago
  Try ask ChatGPT what 12931239 + 712380 is. It can only solve 30 + 46 because it actually memorized/'saw' the latter, but that's nothing new.
- sesm 4y ago
  But Wolfram Alpha will do them all easily.
lukew3 4y ago
True, perhaps this would be better called an exercise generator. However, when studying this in school, the teachers always called them problems, hence why the title was chosen. The project is aims to help students K-12 and possibly undergrads whose curriculum is aimed to get them comfortable with solving these computations without aid of a calculator, or at least without programs that solve it for them. This is not supposed to be the primary means of teaching, but rather a tool that teachers can use to provide infinite exercises for their students to practice on, with the benefit of students not being able to copy answers directly out of a textbook without understanding how it works.

mindcrime 4y ago

Personally I don't care about the quibbling over semantics as far as "problem" vs "exercise". Fine, call it a "math exercise generator". That's something I personally wanted and was very happy to find this, and I expect it's something a lot of other people (especially people doing self-study) will find useful. I think it would also be fun to look at contributing to this, and see about maybe adding some generators for additional probl... er, exercises. In particular, adding more stuff in the Calculus section might be an interesting challenge, and could make this even more useful.

nutanc 4y ago
Agree. This can be very useful for an online learning tool. Will see how I can leverage this. Thanks for this.

tuukkah 4y ago

Are there generators for programming exercises?

ruune 4y ago
Codewars [0] and sites like it [1] maybe [0] https://www.codewars.com/ https://www.codewars.com/ [1] https://alternativeto.net/software/codewars/ https://alternativeto.net/software/codewars/ https://www.reddit.com/r/learnprogramming/comments/43upct/31_coders_games_and_puzzle_sites/ https://www.reddit.com/r/learnprogramming/comments/43upct/31...

lukew3 4y ago

Hi, I'm the project author. It's cool to see so many people suddenly take interest in this project. I haven't touched it for a while, but seeing such support has restored my desire to work on it. I would love to hear your thoughts and would be happy to respond to any questions!

ytygg775 4y ago
I applaud your effords. But unfortunately, this perpetuates the notion that this is what math is. It's mundane calculations, best done by a machine. Nobody even underatands why it's meaningful to multiply matrices like that. How come that high school math is full of this stuff, like doing manual matrix multiplications over and over, but not a single mathematical proof? Society would really benefit from people training logical reasoning, which is what a mathematical proof is. Rather that than a manual matrix multiplication. Heck, even most of my SWE colleagues can't even write down a real proof.
- lukew3 4y ago
  Thanks. I don't believe that math is simply learning how to perform calculations. However, I also don't think that I'm in the best position to teach students how to understand why they do these calculations. It is entirely up to teachers and those who create curriculum to teach students why these calculations work. This project simply gives teachers another tool to help students solidify their what they have learned.
  - _the_inflator 4y ago
    I love your Repo. It is fun to solve these tasks and to have an abundance of these is even more compelling. I love Math and this is sort of casual gaming.
  - ranting-moth 4y ago
    Don't let the naysayers grind you down. Your project is very useful for what it is! You can't please everybody but in this case you're doing a pretty good job.

xchkr1337 4y ago

Why is there an exercise called "complex division" even though it has nothing to do with complex division?

lukew3 4y ago
Thanks for pointing this out. This is a naming error, where complex is used as an antonym of simple. I think a better name would be "rational result division" or even "fraction to decimal". This is being corrected now.

derelicta 4y ago

Absolutely love the concept.

c7b 4y ago

I guess it's my problem, but having been blown away by AI so often recently, I was sort of expecting more here. I suppose an AI that could generate full text problem statements together with sample solutions for basic university level math courses like probability or linear algebra isn't too far out of reach at this point anymore. There could also be some actual market opportunities in that, both with educators and students who want to prep for tests.

Herbstluft 4y ago
> isn't too far out of reach If you are referring to the recent ChatGPT developments, it can't even solve these very basic exercises.
- c7b 4y ago
  I was more thinking of Github Copilot, but ChatGPT is similarly good at programming. Programming is arguably even less error-forgiving than providing a written answer to math problems (I was more thinking of problems that ask for an analytical rather than numerical answers, which is also what most university-level math courses ask for). I wouldn't have expected those models to perform nearly as well at programming as they do, so I've updated my prior. Of course you'd need to find a math corpus of a similar size to github's repositories first. Overleaf has a bit of a similar position to github in STEM research, so that would be an interesting place to start looking (if it's legally possible). It's probably still too small, but would be interesting to see what a GPT version trained on the Overleaf corpus could do.

petesergeant 4y ago

Extend it to produce (and sell!) sample GMAT papers

i13e 4y ago

I think the coolest thing about this tool that I haven't seen mentioned yet is randomizing homework and tests for students on pen and paper without an expensive web platform. This completely eliminates cheating in a math class. Would some sample AI generated text for problems be nice? Sure. But this is the bulk of the work already done IMO.

vitorsr 4y ago

Those are not problems. See [1], [2], [3] and many others [4]. [1] https://bookstore.ams.org/content?PageName=prb https://bookstore.ams.org/content?PageName=prb [2] https://www.maa.org/press/ebooks/problem-books https://www.maa.org/press/ebooks/problem-books [3] https://www.springer.com/series/714 https://www.springer.com/series/714 [4] https://zbmath.org/?q=cc%3A00A07 https://zbmath.org/?q=cc%3A00A07

SquareShoe 4y ago
Wolfram has a similar product that they call a “problem generator” [1] [1]https://www.wolframalpha.com/problem-generator/ https://www.wolframalpha.com/problem-generator/