# P3660: Longest Rule 110 cycle on a thirty-cell ring

- ID: `P3660`
- Reference: `rule-110-ring-thirty-max-cycle`
- Page: https://theoremdb.org/statements/P3660
- 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/rule-110-ring-thirty-max-cycle?representation=markdown
- Record maturity: Reviewed problem

## The problem

Under elementary cellular automaton Rule \(110\) with periodic boundary conditions, what is the maximum cycle length among the \(2^{30}\) binary states on a ring of thirty cells?

### Remarks

- **Remark.** At each synchronous step, a cell's new bit is read from Rule 110 using its left, center, and right bits.
- **Remark.** Transient steps before entering a cycle do not count toward cycle length.

### What counts as a solution

- Give a state on a longest cycle and a replayable exhaustive certificate that every functional-graph component has been classified.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Background and intake notes

Cycle lengths fluctuate sharply with ring size, so extrapolation from nearby sizes is unreliable. The completed prefix is useful chiefly as an implementation check.

- The global update is a functional graph on 2^30 states. A complete answer can mark components in packed arrays and retain one representative plus a hash for every longest cycle.
- Rotation and reflection must be handled carefully. Rotation commutes with the update, while reflection changes the local rule and is not a symmetry of Rule 110.

- Recorded example: The maximum cycle lengths for ring sizes 17, 18, 19, and 20 are 578, 81, 285, and 240.

### Computational notes

- Exact enumeration of every state gave maximum cycle lengths 2,1,9,14,16,7,25,110,18,351,91,295,32,578,81,285,240 for ring sizes 4 through 20.
- The order-thirty scan has 1073741824 states. A packed two-bit visitation array uses 256 MiB before cycle representatives and checksums.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `rule-110-ring-thirty-max-cycle`, 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.