Let q(n) be the maximum number of queens that can be placed on an n× n chessboard such that every queen attacks at most one other queen. Determine q(n) for every positive integer n. The cited preprint claims that q(n)=n for n≤ 5 and q(n)=lfloor frac4n3rfloor for every n≥ 6. It also claims that, under the stricter condition that every queen attacks exactly one other queen, the maximum number of queens is 2 lfloorfrac2n3rfloor for every positive integer n.

Queens on an n×n board, each attacking at most one other

Let $q(n)$ be the maximum number of queens that can be placed on an $n\times n$ chessboard such that every queen attacks at most one other queen. Determine $q(n)$ for every positive integer $n$ . The cited preprint claims that $q(n)=n$ for $n\leq 5$ and $q(n)=\left\lfloor\frac{4n}{3}\right\rfloor$ for every $n\geq 6$ . It also claims that, under the stricter condition that every queen attacks exactly one other queen, the maximum number of queens is $2\left\lfloor\frac{2n}{3}\right\rfloor$ for every positive integer $n$ .

References

Primary source

arXiv

Additional references

Closing the gap and settling the problem of queens on an n×n board, each attacking at most one other — arXiv

combinatorics recreational-mathematics

4 days ago

Progress summary

Refreshed 4 days ago

Claimed solved

An unrefereed preprint claims to determine the largest possible placement for every board size, including the version where every queen has exactly one attacker.

The problem asks for the maximum number of queens that can be placed on a square board so that each queen attacks at most one other queen. The reported result also addresses the stricter version in which every queen attacks exactly one other.

August 2026 claimed resolution

A preprint claims that the conjectured $4n/3$ threshold is the exact maximum for every board size and that the exactly-one-attacker variant is likewise settled. The claim is presented as closing both extremal placement problems, but the preprint is unrefereed.

Current status (as of August 2026): A preprint claims the $4n/3$ bound and both variants are settled for every board size, but the result remains unverified.

Sources

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