A.1 MCTS-kSubS algorithm In Algorithm 4 we present a general MCTS solver based on AlphaZero. Solver repeatedly queries the planner for a list of actions and executes them one by one. Baseline planner returns only a single action at a time, whereas MCTS-kSubS gives around kactions - to reach the desired subgoal (number of actions depends on a subgoal distance, which not always equals k in practice). MCTS-kSubS operates on a high-level subgoal graph: nodes are subgoals proposed by the generator (see Algorithm 3) and edges - lists of actions informing how to move from one subgoal to another (computed by the low-level conditional policy in Algorithm 2). The graph structure is represented by treevariable. For every subgoal, it keeps up to C3 best nearby subgoals (according to generator scores) along with a mentioned list of actions and sum of rewards to obtain while moving from the parent to the child subgoal. Most of MCTS implementation is shared between MCTS-kSubS and AlphaZero baseline, as we can treat the behavioral-cloning policy as a subgoal generator with k = 1. MCTS-kSubS and the baseline are encapsulated in GEN_CHILDREN function (Algorithms 5 and 6).

Technology Robots Brothers build a robot to solve Rubik's cubes in record-setting time The robot completed the puzzle in just 45.3 seconds, breaking its own record of 55 seconds made just moments earlier. The Revenger set a world record. Breakthroughs, discoveries, and DIY tips sent six days a week. A pair of brothers in the U.K. have officially broken the Guinness World Record for the fastest time solving a four-by-four Rubik's Cube with a robot. Their DIY machine, which the brothers call The Revenger, completed the puzzle in only 45.3 seconds.

In this paper, we present a novel learning framework for finding shortest paths in graphs utilizing Generative Flow Networks (GFlowNets). First, we examine theoretical properties of GFlowNets in non-acyclic environments in relation to shortest paths. We prove that, if the total flow is minimized, forward and backward policies traverse the environment graph exclusively along shortest paths between the initial and terminal states. Building on this result, we show that the pathfinding problem in an arbitrary graph can be solved by training a non-acyclic GFlowNet with flow regularization. We experimentally demonstrate the performance of our method in pathfinding in permutation environments and in solving Rubik's Cubes. For the latter problem, our approach shows competitive results with state-of-the-art machine learning approaches designed specifically for this task in terms of the solution length, while requiring smaller search budget at test-time.

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.