# P3648: Largest first Betti number of a connected twenty-voxel solid

- ID: `P3648`
- Reference: `twenty-voxel-maximum-first-betti`
- Page: https://theoremdb.org/statements/P3648
- 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/twenty-voxel-maximum-first-betti?representation=markdown
- Record maturity: Reviewed problem

## The problem

Among face-connected unions \(X\) of twenty unit cubes chosen from a \(5\times5\times5\) cubical grid, determine the largest possible value of \(\dim_{\mathbb F_2}H_1(X;\mathbb F_2)\).

### Problem setup

- **Definition.** Two cubes are face-adjacent when they share a square face, and face-connected means connected in that adjacency graph.
- **Remark.** The union is treated as a cubical complex including all faces of every selected cube.
- **Definition.** The first Betti number is the dimension of H_1 over the field with two elements.

### What counts as a solution

- Give a twenty-cube construction attaining the optimum and a complete canonical-enumeration or integer-programming certificate for the upper bound, with independently checkable cubical boundary ranks.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Background and intake notes

A planar construction already gives four independent tunnels. Three-dimensional branching could improve it, so planar perimeter arguments alone do not certify optimality.

- Canonical augmentation by adding a face-adjacent cube gives a complete search through free polycubes, followed by rejection of shapes that do not fit the box.
- Betti numbers should be computed from exact boundary matrices over F_2. Counting visible holes in projections can miss linked or enclosed three-dimensional features.

- Recorded example: At z=0, take every cube (x,y,0) with 0 <= x,y <= 4 except (0,0,0), (1,1,0), (1,3,0), (3,1,0), and (3,3,0). This uses twenty cubes.

### Computational notes

- The displayed union is face-connected and deformation retracts onto a planar region with four holes, so its Betti vector is (1,4,0).
- The raw labeled search has C(125,20)=20276875628250225138100 subsets. Connectivity and cube symmetries must be imposed before homology evaluation.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `twenty-voxel-maximum-first-betti`, 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.