Tower of hanoi.md

# Tower of Hanoi

**Transfer N disks between three pegs following size constraints**

---

## Overview

The Tower of Hanoi is a classic recursive puzzle consisting of three pegs (labeled A, B, and C) and N disks of different sizes, numbered from 1 (smallest) to N (largest). This puzzle is famous in computer science for demonstrating recursion and exponential time complexity.

### Difficulty Rating: ⭐⭐⭐⭐⭐ (Very Hard - Exponential)

---

## 📊 Statistics

| Metric | Value |
|--------|-------|
| **Total Puzzles** | 60 |
| **Total Moves** | 12,216 |
| **Training Puzzles (N=1-7)** | 42 |
| **Test Puzzles (N=8-10)** | 18 |
| **Difficulty Parameter** | N (number of disks) |
| **Number of Pegs** | 3 (A, B, C) |
| **Solution Length** | L(N) = **2^N - 1** (exponential!) |
| **Transition Locality** | O(N) - must check top disk constraints |

---

## 🎯 Puzzle Rules

### Objective
Transfer all N disks from a designated **start peg** to a **target end peg** while maintaining size ordering (largest at bottom, smallest at top) throughout all intermediate states.

### Constraints
1. **Single Disk Movement**: Only one disk may be moved at a time
2. **Top Disk Access**: Only the topmost disk from any peg can be selected for movement
3. **Size Ordering Constraint**: A larger disk may **never** be placed on top of a smaller disk

### Why Tower of Hanoi is Extremely Challenging

Tower of Hanoi is the **hardest** puzzle in the RecurrReason benchmark:

1. **Exponential Solution Length**: L(N) = 2^N - 1
   - N=3: 7 moves
   - N=7: 127 moves
   - N=10: **1,023 moves!**

2. **Recursive Structure**: Optimal solution requires decomposing problem recursively:
   - Move top N-1 disks to auxiliary peg
   - Move largest disk to target peg
   - Move N-1 disks from auxiliary to target peg

3. **Compounding Errors**: With per-step error rate ε, success probability is:
   ```
   P(success) ≈ (1-ε)^(2^N - 1) → 0 as N grows
   ```

---

## 📋 State Representation

States are represented as **lists of three lists**, where each list represents one peg (A, B, C) containing disks ordered from **top to bottom**.

### Format
```python
[[1, 2, 3], [], []]
```

This represents:
- **Peg A**: Disks 1 (top), 2, 3 (bottom)
- **Peg B**: Empty
- **Peg C**: Empty

**Important**: Disks are numbered 1 (smallest) to N (largest).

### Move Representation
```python
[1, 'A', 'B']
```

This represents: **Move disk 1 from peg A to peg B**

Format: `[disk_number, source_peg, destination_peg]`

---

## 🖼️ Example Puzzle

![Tower of Hanoi Example](https://github.com/gowravmannem/Recurrent-Reasoning-on-Puzzles/blob/main/assets/tower_of_hanoi.png?raw=true)

### Example Trajectory (N=2)

**Initial State**: `[[], [], [1, 2]]` (both disks on peg C)  
**Goal State**: `[[], [1, 2], []]` (both disks on peg B)  
**Start Peg**: C  
**Goal Peg**: B  
**Optimal Solution Length**: 3 moves (2^2 - 1 = 3)

**Step-by-step solution:**

| Step | Current State | Next State | Move | Description |
|------|--------------|-----------|------|-------------|
| 0 | `[[], [], [1, 2]]` | `[[1], [], [2]]` | `[1, 'C', 'A']` | Move disk 1 from C to A |
| 1 | `[[1], [], [2]]` | `[[1], [2], []]` | `[2, 'C', 'B']` | Move disk 2 from C to B |
| 2 | `[[1], [2], []]` | `[[], [1, 2], []]` | `[1, 'A', 'B']` | Move disk 1 from A to B |
| 3 | `[[], [1, 2], []]` | `[[], [1, 2], []]` | `['_', '_', '_']` | Goal reached! |

### Recursive Pattern

The recursive pattern for N disks:
```
function HANOI(n, source, target, auxiliary):
    if n == 1:
        move disk 1 from source to target
    else:
        HANOI(n-1, source, auxiliary, target)  # Move n-1 to aux
        move disk n from source to target       # Move largest
        HANOI(n-1, auxiliary, target, source)  # Move n-1 to target
```

---

## 📁 CSV Column Descriptions

### Columns

| Column | Type | Description |
|--------|------|-------------|
| `N` | int | Number of disks (difficulty parameter) |
| `start_state` | string | Initial configuration of all three pegs |
| `goal_state` | string | Target configuration to achieve |
| `start_peg` | string | Starting peg ('A', 'B', or 'C') |
| `goal_peg` | string | Target peg ('A', 'B', or 'C') |
| `current_state` | string | State before this move |
| `next_state` | string | State after applying this move |
| `move` | string | Action taken: `[disk, source_peg, dest_peg]` |
| `num_moves` | int | Total moves in optimal solution (2^N - 1) |

### Data Format

Each row represents one **move** in a solution trajectory.

**Example CSV rows:**
```csv
N,start_state,goal_state,start_peg,goal_peg,current_state,next_state,move,num_moves
2,"[[],[],[1,2]]","[[],[1,2],[]]",C,B,"[[],[],[1,2]]","[[1],[],[2]]","[1,'C','A']",3
2,"[[],[],[1,2]]","[[],[1,2],[]]",C,B,"[[1],[],[2]]","[[1],[2],[]]","[2,'C','B']",3
2,"[[],[],[1,2]]","[[],[1,2],[]]",C,B,"[[1],[2],[]]","[[],[1,2],[]]","[1,'A','B']",3
2,"[[],[],[1,2]]","[[],[1,2],[]]",C,B,"[[],[1,2],[]]","[[],[1,2],[]]","['_','_','_']",3
```

---

## 💡 Usage Tips

### For Model Training

⚠️ **Warning**: Tower of Hanoi is **difficult** for current sequence models.

Suggested approaches:
1. **Add explicit subgoal markers**: Annotate when recursive subproblems start/end
2. **Hierarchical representations**: Encode recursive structure explicitly
3. **Search augmentation**: Use beam search or MCTS during decoding
4. **Curriculum learning**: Start with N=1, slowly increase (but likely still fails at N≥3)

### For Evaluation

```python
from datasets import load_dataset

# Load Tower of Hanoi
dataset = load_dataset("gmannem/RecurrReason", "tower_of_hanoi")

# WARNING: Expect very low success rates!
# Models typically solve only N=1

def evaluate_hanoi(model, example):
    """
    Evaluation with strict constraints.
    
    A single size-ordering violation = immediate failure.
    """
    current = example['start_state']
    goal = example['goal_state']
    steps = 0
    max_steps = 2 * example['num_moves']  # 2 × (2^N - 1)
    
    while steps < max_steps:
        next_state = model.predict(current, goal)
        
        # Check size ordering (CRITICAL!)
        if violates_size_constraint(next_state):
            return "INVALID_MOVE", steps
            
        if next_state == goal:
            return "SUCCESS", steps
            
        current = next_state
        steps += 1
    
    return "TIMEOUT", steps

def violates_size_constraint(state):
    """Check if any peg has larger disk on top of smaller."""
    for peg in state:
        for i in range(len(peg) - 1):
            if peg[i] > peg[i+1]:  # Larger disk on top!
                return True
    return False
```

---

## 🔬 Research Directions

Tower of Hanoi poses fundamental challenges for sequence models:

1. **Hierarchical Planning**: How to encode recursive subgoals?

2. **Search Integration**: Can we augment models with A* or MCTS?

3. **Neuro-Symbolic Approaches**: Combine neural prediction with symbolic constraint checking

4. **Explicit Memory**: External memory to track subproblem state

5. **Length Generalization**: Current models cannot extrapolate from short to long sequences on this task

**Key Insight**: Success on Tower of Hanoi likely requires **search** or **explicit hierarchical representations**, not just larger models.

---


## 📚 References

**Main Paper:**
```bibtex
@inproceedings{mannem2026recurrent,
  title={Recurrent Reasoning on Symbolic Puzzles with Sequence Models},
  author={Gowrav Mannem and Chowdhury Marzia Mahjabin and Jason Chen and Shivank Garg and Kevin Zhu},
  booktitle={ICLR 2026 Workshop on Logical Reasoning of Large Language Models},
  year={2026}
}
```

**Classic Reference:**
```bibtex
@article{lucas1883tower,
  title={Récréations mathématiques},
  author={Lucas, Édouard},
  journal={Gauthier-Villars},
  year={1883}
}
```

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