For which $n$ are there $n$ points in ${\mathbb{R}}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

These are now usually called **crescent configurations**: $n$ points in the plane in general position (no 3 collinear, no 4 concyclic) such that the (\binom n2) pairwise distances take only $n-1$ distinct values, and the multiplicities of those values are exactly (1,2,\dots,n-1) (in some order). ([arXiv][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.