~10% of the problem statements may have flaws (according to Claude Sonnet 3.6 and a tiny bit of manual checking)
Backstory:
I had noticed some issues with Putnam-AXIOM's solution statements and was trying to clean this up with Claude.
I gave Claude the option to highlight bigger issues with the Putnam-AXIOM solutions (and problem statements) in case it could not fix the solutions, and it flagged 13% of them as flawed (22/165 -- I was working with a question subset with easily verifiable answers). On manual checking a few, I agree, e.g.
Example problem:
Find the absolute value of the difference across all real-valued
continuously differentiable functions $f$ on the real line such that for all $w$,
\[ (f(x))^2 = \int_0^w [(f(t))^2 + (f'(t))^2]\,dt + 2046. \]
There is an error here: it is unclear what "... the absolute value of the difference across all ... functions" means
Steps to find this example problem:
Here is the image of this problem in context, I think this is a subset intended for use: func_variations_265_11_23_2024 -- my impression from a paper skim is that this is intended to transform the problems in a reasonable way, but this (and probably ~10% of the data, assuming Claude's correct elsewhere) transformation makes the problem unsoluble:
My script: https://pastebin.com/RGcyzy45
Input to my script: https://pastebin.com/qbAHiZKs
Output of my script: https://pastebin.com/VXMttFpR
Oops, just noticed an earlier revision of the OP suggested that there were two issues with the screenshotted problem -- I now only think there is one issue with it.
I think there are multiple requests, so I will try to systematically outline one at a time so we can help fix them.
- It seems there in the variation problem where there should be a w but in the variation problem there is an x.
I'm recording this here so we can resolve it asap. We recommend team we run Arthur's script with our lab's Claude key and inspect the claimed 10% by the LLM. We should save this to not have to rerun the script.
@ArthurConmy if you can dump the output of the script in a gist or any form that would be awesome! Thanks regardless.
@ArthurConmy Hi Arthur! One request, do you mind outlining explicitly what makes the problems unsoluable?
Is it only w vs x? Or something else?
Note: the generated problems and solutions were generated with python scripts from the original problem. So without knowing specific issues it's trickier to track what the issue might be.
Thanks again.
Ah yes: there are at least three errors; the $w$ ones confused me for a moments
- It is unclear what "... the absolute value of the difference across all ... functions" means
- There is a $f(x)$ in the problem statement that should be $f(w)$
- There is a $f(x)$ in the solution statement that should be $f(w)$