# Icosian game:

### Hamiltonian Cycles

#### (Claude Cycles)

<br>

**Hobson Lane**

Python User Group

*March 26, 2026*

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## searching graphs for your next move

#### ...or token

#### ...or atom

#### ...or chemical reaction

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## Graphs == Networks

- Chess & Go moves (Alpha Go Zero)
- Biochemistry, SMILES, ORD (Alpha Fold)
- Neural network *circuits*
- LLMs (*degree* 50k)

[![/home/hobs/code/hobs/mkslides/slides/Knight's_tour_anim_2_wikipedia.gif](/Knight's_tour_anim_2_wikipedia.gif)](/Knight's_tour_anim_2_wikipedia.gif)

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## _Claude Cycles_

by Don Knuth

[![claude_cycles_knuth.png](/claude_cycles_knuth.png)](/claude_cycles_knuth.png)

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![](/icosian-game-01-wrap-around-arc-stubs.plotly.png)

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![](/icosian-game-02wrap-around-arcs-with-cones-and-thin-lines.plotly.png)

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![](/icosian-game-03-wrap-around-arcs-with-cones-thick-lines-dotdash.plotly.png)

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![](/icosian-game-04-wrap-around-arcs-without-cones.plotly.png)
(/icosian-game.drawio.html)

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## Hamiltonian cycles

https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cycles.pdf

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## Icosohedron != Dodecohedron

Dice with greek letters on them

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## ORD

- Open Reaction Database
- 2M chemical reactions in a graph

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## SMILES

- Molecule representation as emojies ;-)
- Physics of bonds => grammar of language

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## Knuth Postulate Python

```python
>>> m = 3
... dimensions = 3
... nodes = []
... edges = []
```

Can you count the nodes and edges in your head? ...

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## Generate graph

```python
... for i in range(m):
...     for j in range(m):
...         for k in range(m):
...             src = (i, j, k)
...             nodes.append(src)
...             for d in range(dimensions):
...                 dest = list(src)
...                 dest[d] = (dest[d] + 1) % m
...                 edges.append((src, tuple(dest)))
```

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## Number of nodes?

```python
>>> len(nodes)
27
```

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## Number of edges?

```
>>> len(edges)
81
```

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## iPython tip

```python
>>> hist -o -p -f icosian_game.hist.ipy
```

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## Claude

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❯ Consider the digraph with $$m$^3$ vertices $(i)(j)(k)$ for $0 \leq (i), (j), (k) < $m$$, and three arcs from each vertex, namely to $(i)^+(j)(k)$,                  
$(i)(j)^+_k$, and $(i)(j)(k)^+$, where $(i)^+ = ((i)+1) mod $m$$. Try to find a general decomposition of the arcs into three directed $$m$^3$-cycles, for all $$m$ > 2$.

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● This is a beautiful graph theory problem! Let me first clarify the problem statement with proper LaTeX formatting:

Problem: Consider the digraph with $m^3$ vertices $ijk$ for $0 \leq i, j, k < m$, and three arcs from each vertex to $i^+jk$, $ij^+k$, and $ijk^+$, where $i^+ = (i+1) \bmod m$.
  Find a decomposition of all arcs into three directed Hamiltonian cycles (each visiting all $m^3$ vertices exactly once) for all $m > 2$.

---

Structure Analysis

This graph is the Cayley graph Cay($\mathbb{Z}_m^3$, ${e_1, e_2, e_3}$) where:
  - Vertices are elements of $\mathbb{Z}_m^3$ (triples $(i,j,k)$)
  - The three arc types correspond to adding generators: $(1,0,0)$, $(0,1,0)$, $(0,0,1)$

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Key properties:
  - Total arcs: $3m^3$
  - Each vertex has out-degree 3 and in-degree 3
  - We need 3 arc-disjoint Hamiltonian cycles (each using $m^3$ arcs)

Proposed Construction

Here's a general construction using a balanced Gray code approach:

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Cycle 1: "Scanline pattern"

```
For k = 0 to m-1:
    For j = 0 to m-1:
        Traverse i from 0 to m-1 using type-1 arcs (increment i)
        Then use type-2 arc to increment j (except at j=m-1)
  Use type-3 arc to increment k (except at k=m-1)
```

This creates a path visiting all vertices. The cycle closes when returning from $(m-1,m-1,m-1)$ to $(0,0,0)$.

---

Cycles 2 and 3:

By symmetry, we can construct these by permuting the coordinate roles:

- Cycle 2: Same pattern but with coordinates permuted as $(j,k,i)$
  - Cycle 3: Same pattern but with coordinates permuted as $(k,i,j)$

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## AI makes promises

Would you like me to:
  1. Write Python code to verify this construction for specific values of $m$?
  2. Provide a formal proof of correctness?
  3. Visualize the cycles for a small example like $m=3$?

✻ Cogitated for 42s

❯ 1. Write the Python code to verify this construction for m = 3.

### AI can't remember

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## AI has no long-term memory

✻ Conversation compacted (ctrl+o for history)

⎿  API Error: Claude's response exceeded the 32000 output token maximum. To configure this behavior, set the CLAUDE_CODE_MAX_OUTPUT_TOKENS environment variable.

✻ Baked for 51m 11s

#### *!!! Hour of churn !!!*

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## Filip "fixed" this with:

- Starting each task with empty context
- Continuously updating `plan.md`

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## Claude can't see

No dotted lines or arrows for wrap-around arcs are visible in torus_digraph.html despite being mentioned in the title of the plot. Please add the missing wrap-around arcs or arrows to
  the plotly visualization.

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## Pedantic and mathy

[![icosian-game-01-wrap-around-arc-stubs.plotly.png](/icosian-game-01-wrap-around-arc-stubs.plotly.png)](/icosian-game-01-wrap-around-arc-stubs.plotly.png)

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## Claude can prompt your thinking

"""**Caley graph**... set of generators... combinatorial and geometric group theory"""

#### Generators are

- Algorithms
- Deterministic *generative models*
- Generative AI??
- AI *agent policy*
- Game play *strategy*

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## "Torus

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[![exocija-cycles.png](/icosian-game.drawio.png)](slides/icosian-game.drawio)

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![](/icosian-game-01-wrap-around-arc-stubs.plotly.png)

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![](/icosian-game-02wrap-around-arcs-with-cones-and-thin-lines.plotly.png)

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![](/icosian-game-03-wrap-around-arcs-with-cones-thick-lines-dotdash.plotly.png)

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![](/icosian-game-04-wrap-around-arcs-without-cones.plotly.png)
(/icosian-game.drawio.html)

---

## Solution

[![exocija-cycles.png](/exocija-cycles.png)](/exocija-cycles.png)

---

## Litellm

## links

- LiteLLM 1.82.7 & 1.82.8
- **Rotate all secrets, tokens, and credentials** everywhere on affected devices
- Search for '~/.config/sysmon/sysmon.py' and related systemd services
- Search for '/tmp/pglog' and '/tmp/.pg_state'
- Review Kubernetes clusters: unauthorized pods in 'kube-system' namespace
- Monitor outbound traffic to known attacker domains (`opensnitch`)