# P11732: When is a cubic-lattice ball Manhattan-optimal?

- ID: `P11732`
- Reference: `when-is-a-cubic-lattice-ball-manhattan-optimal`
- Page: https://theoremdb.org/statements/P11732
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/when-is-a-cubic-lattice-ball-manhattan-optimal?representation=markdown
- Record maturity: Reviewed problem

## The problem

For \(n\ge 0\), let \(B_n=\{(x,y,z)\in\mathbb{Z}^3:x^2+y^2+z^2\le n\}\), let \(S(n)\) be the sum of Manhattan distances over all unordered pairs in \(B_n\), and let \(D(N)\) be the minimum such sum among all \(N\)-point subsets of \(\mathbb{Z}^3\). Determine all further \(n\) for which \(S(n)=D(|B_n|)\), beyond the reported equality radii \(0,1,2,3,6,11,21,30,\ldots\).

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `when-is-a-cubic-lattice-ball-manhattan-optimal`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.