Image restoration techniques, spanning from the convolution to the transformer paradigm, have demonstrated robust spatial representation capabilities to deliver high-quality performance.Yet, many of these methods, such as convolution and the Feed Forward Network (FFN) structure of transformers, primarily leverage the basic first-order channel interactions and have not maximized the potential benefits of higher-order modeling. To address this limitation, our research dives into understanding relationships within the channel dimension and introduces a simple yet efficient, high-order channel-wise operator tailored for image restoration. Instead of merely mimicking high-order spatial interaction, our approach offers several added benefits: Efficiency: It adheres to the zero-FLOP and zero-parameter principle, using a spatial-shifting mechanism across channel-wise groups. Simplicity: It turns the favorable channel interaction and aggregation capabilities into element-wise multiplications and convolution units with $1 \times 1$ kernel. Our new formulation expands the first-order channel-wise interactions seen in previous works to arbitrary high orders, generating a hierarchical receptive field akin to a Rubik's cube through the combined action of shifting and interactions. Furthermore, our proposed Rubik's cube convolution is a flexible operator that can be incorporated into existing image restoration networks, serving as a drop-in replacement for the standard convolution unit with fewer parameters overhead. We conducted experiments across various low-level vision tasks, including image denoising, low-light image enhancement, guided image super-resolution, and image de-blurring. The results consistently demonstrate that our Rubik's cube operator enhances performance across all tasks.

Progress in understanding expert performance is limited by the scarcity of quantitative data on long-term knowledge acquisition and deployment. Here we use the Rubik's Cube as a cognitive model system existing at the intersection of puzzle solving, skill learning, expert knowledge, cultural transmission, and group theory. By studying competitive cube communities, we find evidence for universality in the collective learning of the Rubik's Cube in both sighted and blindfolded conditions: expert performance follows exponential progress curves whose parameters reflect the delayed acquisition of algorithms that shorten solution paths. Blindfold solves form a distinct problem class from sighted solves and are constrained not only by expert knowledge but also by the skill improvements required to overcome short-term memory bottlenecks, a constraint shared with blindfold chess. Cognitive artifacts such as the Rubik's Cube help solvers navigate an otherwise enormous mathematical state space. In doing so, they sustain collective intelligence by integrating communal knowledge stores with individual expertise and skill, illustrating how expertise can, in practice, continue to deepen over the course of a single lifetime.

Games and puzzles play important pedagogical roles in STEM learning. New AI algorithms that can solve complex problems offer opportunities for scaffolded instruction in puzzle solving. This paper presents the ALLURE system, which uses an AI algorithm (Deep CubeA) to guide students in solving a common first step of the Rubik's Cube (i.e., the white cross). Using data from a pilot study we present preliminary findings about students' behaviors in the system, how these behaviors are associated with STEM skills - including spatial reasoning, critical thinking and algorithmic thinking. We discuss how data from ALLURE can be used in future educational data mining to understand how students benefit from AI assistance and collaboration when solving complex problems.

Section 2 provides the implementation details of Rubik's cube convolution within the image restora-4 Section 3 provides the evaluation of our proposed Rubik's cube convolution on the classification task. Section 4 provides more quantitative and qualitative results. Specifically, the input feature is separated into five groups, where the last four are shifted into four direction and the first is unchanged. Rubik's cube convolution operation into the baseline will achieve the performance improvement, "Acc-1" and "Acc-5" indicate the top-1 and As illustrated in Figure 3 and 4, integrating our Rubik's cube In contrast, the baseline combined with our Rubik's cube convolution operator achieves details

In classical AI, perception relies on learning state-based representations, while planning, which can be thought of as temporal reasoning over action sequences, is typically achieved through search. We study whether such reasoning can instead emerge from representations that capture both perceptual and temporal structure. We show that standard temporal contrastive learning, despite its popularity, often fails to capture temporal structure due to its reliance on spurious features. To address this, we introduce Combinatorial Representations for Temporal Reasoning (CRTR), a method that uses a negative sampling scheme to provably remove these spurious features and facilitate temporal reasoning. CRTR achieves strong results on domains with complex temporal structure, such as Sokoban and Rubik's Cube. In particular, for the Rubik's Cube, CRTR learns representations that generalize across all initial states and allow it to solve the puzzle using fewer search steps than BestFS, though with longer solutions. To our knowledge, this is the first method that efficiently solves arbitrary Cube states using only learned representations, without relying on an external search algorithm.

Heuristic functions are central to the performance of search algorithms such as A-star, where admissibility - the property of never overestimating the true shortest-path cost - guarantees solution optimality. Recent deep learning approaches often disregard admissibility and provide limited guarantees on generalization beyond the training data. This paper addresses both of these limitations. First, we pose heuristic learning as a constrained optimization problem and introduce Cross-Entropy Admissibility (CEA), a loss function that enforces admissibility during training. On the Rubik's Cube domain, this method yields near-admissible heuristics with significantly stronger guidance than compressed pattern database (PDB) heuristics. Theoretically, we study the sample complexity of learning heuristics. By leveraging PDB abstractions and the structural properties of graphs such as the Rubik's Cube, we tighten the bound on the number of training samples needed for A-star to generalize. Replacing a general hypothesis class with a ReLU neural network gives bounds that depend primarily on the network's width and depth, rather than on graph size. Using the same network, we also provide the first generalization guarantees for goal-dependent heuristics.

The core mechanical system is built around three stepper motors for physical manipulation, a microcontroller for hardware control, a camera and YOLO detection model for real-time cube state detection. A significant software component is the development of a user-friendly graphical user interface (GUI) designed in Unity. The initial state after detection from real-time YOLOv8 model (Precision 0.98443, Recall 0.98419, Box Loss 0.42051, Class Loss 0.2611) is virtualized on GUI. To get the solution, the system employs the Kociemba's algorithm while physical manipulation with a single degree of freedom is done by combination of stepper motors' interaction with the cube achieving the average solving time of ~2.2 minutes.

The giant formula for simultaneously enlarging the resolution, depth and width provides us a Rubik's cube for neural networks. So that we can find networks with high efficiency and excellent performance by twisting the three dimensions.

This paper presents Auto-RubikAI, a modular autonomous planning framework that integrates a symbolic Knowledge Base (KB), a vision-language model (VLM), and a large language model (LLM) to solve structured manipulation tasks exemplified by Rubik's Cube restoration. Unlike traditional robot systems based on predefined scripts, or modern approaches relying on pretrained networks and large-scale demonstration data, Auto-RubikAI enables interpretable, multi-step task execution with minimal data requirements and no prior demonstrations. The proposed system employs a KB module to solve group-theoretic restoration steps, overcoming LLMs' limitations in symbolic reasoning. A VLM parses RGB-D input to construct a semantic 3D scene representation, while the LLM generates structured robotic control code via prompt chaining. This tri-module architecture enables robust performance under spatial uncertainty. We deploy Auto-RubikAI in both simulation and real-world settings using a 7-DOF robotic arm, demonstrating effective Sim-to-Real adaptation without retraining. Experiments show a 79% end-to-end task success rate across randomized configurations. Compared to CFOP, DeepCubeA, and Two-Phase baselines, our KB-enhanced method reduces average solution steps while maintaining interpretability and safety. Auto-RubikAI provides a cost-efficient, modular foundation for embodied task planning in smart manufacturing, robotics education, and autonomous execution scenarios. Code, prompts, and hardware modules will be released upon publication.

Breakthroughs, discoveries, and DIY tips sent every weekday. Mitsubishi's bragging rights for designing the world's fastest Rubik's cube-solving robot have officially been stolen by a team of undergrads in Indiana. Earlier this month, Purdue University announced four collaborators in its Elmore Family School of Electrical and Computer Engineering (ECE) successfully designed and built a bot that not only set the new Guinness World Record--it absolutely demolished the multinational company's previous time. Meet Purdubik's Cube: a machine capable of completing a randomly shuffled Rubik's cube in just 0.103 seconds. At 1-2 times faster than the blink of a human eye, the feat is difficult to see, much less comprehend.