The Ring Loading Problem

The problem

Let $C$ be the infimum of all reals $\alpha$ for which the following statement holds: for all positive integers $m$ and nonnegative reals $u_1, \ldots, u_m$ and $v_1, \ldots, v_m$ with $u_i + v_i \leq 1$, there exist $z_1, \ldots, z_m$ such that for every $k$, we have $z_k \in \{v_k, -u_k\}$, and

$$\left|\sum_{i=1}^k z_i - \sum_{i=k+1}^m z_i\right|\leq \alpha.$$

Obtain upper and lower bounds on $C$ 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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