Three people want to share a few indivisible items, and each of them values the items differently. When is a division fair?

One demanding answer is called pairwise maximin share fairness. Take any two of the people. Imagine they pooled their items, and one of them split the pool into two piles, knowing she would be left with the worse pile. A division passes the test if everyone already has at least what that split could guarantee, against each of the other two.

In September twenty twenty-six, researchers showed that such a division can fail to exist. For three people, the examples used nine items. That left an obvious question: is nine the smallest number that can go wrong?

This Evidence Press candidate says yes. With eight items or fewer, a fair division in this sense always exists, provided each person's values simply add up across items.

The proof turns a hypothetical eight-item failure into ten large logical puzzles, and shows that none of them has a solution. A separate program rebuilds every rule of every puzzle from the definitions, and two separate proof checkers confirm each refutation. Smaller cases, with six and seven items, are proved again along the way.

This remains an unrefereed candidate. The checks were run by the producer, not yet by an unaffiliated team, and the argument has not been formalised in a proof assistant.

The paper, code and certificates accompany Pairwise maximin share allocations for three agents exist up to eight goods, released on the tenth of October twenty twenty-six. This briefing uses an AI-generated voice and is not additional mathematical evidence.