# P3696: Hamiltonian cycles of the four-by-four-by-four grid

- ID: `P3696`
- Reference: `hamilton-cycles-four-cubical-grid-count`
- Page: https://theoremdb.org/statements/P3696
- 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/hamilton-cycles-four-cubical-grid-count?representation=markdown
- Record maturity: Reviewed problem

## The problem

How many undirected Hamiltonian cycles does the grid graph \(P_4\square P_4\square P_4\) have, counting cycles up to cyclic choice of starting vertex and reversal but retaining spatially symmetric copies?

### Problem setup

- **Remark.** Vertices are triples in {0,1,2,3}^3 and edges join triples differing by 1 in one coordinate.
- **Definition.** A Hamiltonian cycle visits all 64 vertices exactly once.

### What counts as a solution

- Give the exact integer count with a connectivity-transfer transcript or modular residues and a proven reconstruction bound.

## Status

**Resolution status:** Open.

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

## Research packet

### Background and intake notes

This is the compact self-avoiding polygon count for the smallest genuinely three-dimensional even cube beyond the trivial order two case.

- A connectivity-state transfer across 4 by 4 layers tracks vertex degrees and pairs of open path ends. Premature closed components must be rejected.
- Counts from directed rooted cycles must be divided by 128 under the stated convention. Spatial automorphisms remain distinct.

- Recorded example: A Hamiltonian cycle exists by splicing serpentine cycles across the four layers.

### Computational notes

- The graph has 64 vertices and 144 edges. Every target cycle uses 64 edges and gives 128 rooted directed encodings.
- A layer transfer needs only four 16-vertex interfaces, while a direct subset dynamic program would have 64 times 2^64 endpoint states.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `hamilton-cycles-four-cubical-grid-count`, 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.