The Ring Loading Problem

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open.

Problem
Problem 61 in the AlphaEvolve repository: The Ring Loading Problem
Credit
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
Checked
Construction can be checked directlyPreprint (arXiv, 2025)
Timeline
Paper submitted 2025-11-03
Sources
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
PARTIALImproved bound · new construction

The problem

Let CC be the infimum of all reals α\alpha for which the following statement holds: for all positive integers mm and nonnegative reals u1,…,umu_1, \ldots, u_m and v1,…,vmv_1, \ldots, v_m with ui+vi≤1u_i + v_i \leq 1, there exist z1,…,zmz_1, \ldots, z_m such that for every kk, we have zk∈{vk,−uk}z_k \in \{v_k, -u_k\}, and

∣∑i=1kzi−∑i=k+1mzi∣≤α.\left|\sum_{i=1}^k z_i - \sum_{i=k+1}^m z_i\right|\leq \alpha.

Obtain upper and lower bounds on CC that are as strong as possible.

Statement quoted from problem 61 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.41 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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