Block World
Rearrange blocks in stacks from initial to goal configuration
Overview
Block World is a classical planning puzzle that tests sequential reasoning and dependency analysis. The puzzle involves uniquely labeled blocks (A, B, C, etc.) arranged in K stacks, where the objective is to rearrange blocks from an initial configuration to a specified goal configuration.
Difficulty Rating: ⭐⭐ (Moderate)
📊 Statistics
| Metric | Value |
|---|---|
| Total Puzzles | 849 |
| Total Moves | 5,827 |
| Training Puzzles (N=1-7) | 549 |
| Test Puzzles (N=8-10) | 300 |
| Difficulty Parameter | N (number of blocks) |
| Number of Stacks | K = 3 |
| Solution Length | L(N) = O(N) (linear) |
| Transition Locality | O(1) - only check top of stacks |
🎯 Puzzle Rules
Objective
Move blocks from the initial configuration to the goal configuration following movement constraints.
Constraints
- Top Block Movement: Only the topmost block from any stack can be moved
- Valid Placement: A block can only be placed:
- On an empty stack position, OR
- On top of another block
Why Block World is Interesting
Block World is the most learnable puzzle in the RecurrReason benchmark for three key reasons:
- O(1) Transition Locality: Verifying whether a move is legal requires checking only the top of two stacks (source and destination), independent of total problem size
- Linear Solution Length: L(N) = O(N), meaning solution length grows linearly with difficulty
- Dense Training Signal: All 549 training puzzles share the same movement grammar, providing consistent learning opportunities
📋 State Representation
States are represented as lists of K lists, where each inner list represents one stack containing blocks ordered from bottom to top.
Format
[['B'], [], ['A', 'C']]
This represents:
- Stack 0: Block B (bottom)
- Stack 1: Empty
- Stack 2: Block A (bottom), Block C (top)
Move Representation
['C', 2, 0]
This represents: Move block C from stack 2 to stack 0
Format: [block_name, source_stack_index, destination_stack_index]
🖼️ Example Puzzle
Example Trajectory
Initial State: [['B'], [], ['A', 'C']]
Goal State: [['B'], ['A'], ['C']]
Optimal Solution Length: 3 moves
Step-by-step solution:
| Step | Current State | Next State | Move | Description |
|---|---|---|---|---|
| 0 | [['B'], [], ['A', 'C']] |
[['B', 'C'], [], ['A']] |
['C', 2, 0] |
Move C from stack 2 to stack 0 |
| 1 | [['B', 'C'], [], ['A']] |
[['B', 'C'], ['A'], []] |
['A', 2, 1] |
Move A from stack 2 to stack 1 |
| 2 | [['B', 'C'], ['A'], []] |
[['B'], ['A'], ['C']] |
['C', 0, 2] |
Move C from stack 0 to stack 2 |
| 3 | [['B'], ['A'], ['C']] |
[['B'], ['A'], ['C']] |
['_', '_', '_'] |
Goal reached! |
Note: The final row with ['_', '_', '_'] indicates puzzle completion (sentinel move).
📁 CSV Column Descriptions
Columns
| Column | Type | Description |
|---|---|---|
N |
int | Number of blocks in the puzzle (difficulty parameter) |
K |
int | Number of stacks (always 3 in this dataset) |
start_state |
string | Initial configuration of all stacks |
goal_state |
string | Target configuration to achieve |
current_state |
string | State before this move |
next_state |
string | State after applying this move |
move |
string | Action taken: [block, source_stack, dest_stack] |
num_moves |
int | Total number of moves in the optimal solution |
Data Format
Each row represents one move in a solution trajectory. A complete puzzle solution consists of multiple rows (one per move) plus a final row indicating completion.
Example CSV rows:
N,K,start_state,goal_state,current_state,next_state,move,num_moves
3,3,"[['B'],[],['A','C']]","[['B'],['A'],['C']]","[['B'],[],['A','C']]","[['B','C'],[],['A']]","['C',2,0]",3
3,3,"[['B'],[],['A','C']]","[['B'],['A'],['C']]","[['B','C'],[],['A']]","[['B','C'],['A'],[]]","['A',2,1]",3
3,3,"[['B'],[],['A','C']]","[['B'],['A'],['C']]","[['B','C'],['A'],[]]","[['B'],['A'],['C']]","['C',0,2]",3
3,3,"[['B'],[],['A','C']]","[['B'],['A'],['C']]","[['B'],['A'],['C']]","[['B'],['A'],['C']]","['_','_','_']",3
💡 Usage Tips
For Model Training
- Start with Block World: It's the most learnable puzzle—ideal for validating your training pipeline
- Use full trajectories: Train on complete state→action→next_state sequences
- Implement early stopping: Monitor validation accuracy, stop when plateaus
- Try curriculum learning: Train progressively from N=1 → N=7
For Evaluation
from datasets import load_dataset
# Load Block World
dataset = load_dataset("gmannem/RecurrReason", "block_world")
# Evaluation metrics to track:
# 1. Success Rate: % of puzzles solved correctly
# 2. Move Validity: % of generated moves that are legal
# 3. Optimality Gap: (|solution| - |optimal|) / |optimal|
# 4. Termination: Reaches goal within 2×optimal steps?
def evaluate_trajectory(model, example):
"""
Autoregressive rollout evaluation.
Start from initial state, generate next states until:
- Goal reached (success!)
- Invalid move generated (constraint violation)
- Loop detected (repeated state)
- Horizon exceeded (2× optimal length)
"""
current = example['start_state']
goal = example['goal_state']
visited = set()
steps = 0
max_steps = 2 * example['num_moves']
while steps < max_steps:
# Generate next state
next_state = model.predict(current, goal)
# Check termination conditions
if next_state == goal:
return "SUCCESS", steps
if next_state in visited:
return "LOOP", steps
if not is_valid_state(next_state):
return "INVALID", steps
visited.add(next_state)
current = next_state
steps += 1
return "TIMEOUT", steps
🔬 Research Directions
Based on our findings, promising research directions include:
Architecture Search: Why does bidirectional attention help so much? Can we design minimal architectural changes for goal-conditioning?
Length Generalization: Block World shows gradual OOD degradation—can we improve N=8-10 performance further?
Transfer Learning: Does Block World skill transfer to other local-structure planning tasks?
Hybrid Approaches: Combining neural models with explicit state-space search
📚 References
Main Paper:
@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}
}
Original Puzzle Introduction:
@article{shojaee2025illusion,
title={The illusion of thinking: Understanding the strengths and limitations of reasoning models via the lens of problem complexity},
author={Shojaee, Parshin and Mirzadeh, Iman and Alizadeh, Keivan and Horton, Maxwell and Bengio, Samy and Farajtabar, Mehrdad},
journal={arXiv preprint arXiv:2506.06941},
year={2025}
}