  Current LLM training positions mathematical reasoning as a core capability.
With publicly available sources fully tapped, there is unmet demand for diverse
and challenging math questions. Relying solely on human experts is both
time-consuming and costly, while LLM-generated questions often lack the
requisite diversity and difficulty. We present a design framework that combines
the strengths of LLMs with a human-in-the-loop approach to generate a diverse
array of challenging math questions. We leverage LLM metacognition skills
[Didolkar et al., 2024] of a strong LLM to extract core "skills" from existing
math datasets. These skills serve as the basis for generating novel and
difficult questions by prompting the LLM with random pairs of core skills. The
use of two different skills within each question makes finding such questions
an "out of distribution" task for both LLMs and humans. Our pipeline employs
LLMs to iteratively generate and refine questions and solutions through
multiturn prompting. Human annotators then verify and further refine the
questions, with their efficiency enhanced via further LLM interactions.
Applying this pipeline on skills extracted from the MATH dataset [Hendrycks et
al., 2021] resulted in MATH$^2$ - a dataset of higher-quality math questions,
as evidenced by: (a) Lower performance of all models on MATH$^2$ than on MATH
(b) Higher performance on MATH when using MATH$^2$ questions as in-context
examples. Although focused on mathematics, our methodology seems applicable to
other domains requiring structured reasoning, and potentially as a component of
scalable oversight. Also of interest is a striking relationship observed
between models' performance on the new dataset: the success rate on MATH$^2$ is
the square on MATH, suggesting that successfully solving the question in
MATH$^2$ requires a nontrivial combination of two distinct math skills.
