# Self-Supervision is All You Need for Solving Rubik's Cube

arXiv: https://arxiv.org/abs/2106.03157 · HTML: https://mlsanity.com/paper/2106.03157/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 41 references that are in the map.
- First posted on arXiv 6 Jun 2021.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Reinforcement Learning for Combinatorial Optimization: A Survey](https://mlsanity.com/paper/2003.03600/) (2020)
- [A Constructive Prediction of the Generalization Error Across Scales](https://mlsanity.com/paper/1909.12673/) (2019)
- [Understanding the Difficulty of Training Transformers](https://mlsanity.com/paper/2004.08249/) (2020)
- [Learning the Travelling Salesperson Problem Requires Rethinking Generalization](https://mlsanity.com/paper/2006.07054/) (2020)
- [Learning Combinatorial Optimization on Graphs: A Survey with Applications to Networking](https://mlsanity.com/paper/2005.11081/) (2020)
- [Solving the Rubik's Cube Optimally is NP-complete](https://mlsanity.com/paper/1706.06708/) (2017)
- [A historical note on the 3/2-approximation algorithm for the metric traveling salesman problem](https://mlsanity.com/paper/2004.02437/) (2020)

## Built on by (most cited first, 4 of 4)

- [CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs](https://mlsanity.com/paper/2502.18663/) (2025)
- [Learning Shortest Paths with Generative Flow Networks](https://mlsanity.com/paper/2603.01786/) (2026)
- [CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)](https://mlsanity.com/paper/2509.19162/) (2025)
- [Learning Model Successors](https://mlsanity.com/paper/2502.00197/) (2025)

## Abstract (arXiv)

Existing combinatorial search methods are often complex and require some level of expertise. This work introduces a simple and efficient deep learning method for solving combinatorial problems with a predefined goal, represented by Rubik's Cube. We demonstrate that, for such problems, training a deep neural network on random scrambles branching from the goal state is sufficient to achieve near-optimal solutions. When tested on Rubik's Cube, 15 Puzzle, and 7×7 Lights Out, our method outperformed the previous state-of-the-art method DeepCubeA, improving the trade-off between solution optimality and computational cost, despite significantly less training data. Furthermore, we investigate the scaling law of our Rubik's Cube solver with respect to model size and training data volume.