루빅큐브를 통한 평생 전문가 학습의 보편성

Universality in Collective Intelligence on the Rubik’s Cube David Krakauer1*, Gülce Kardeş1,2 and Joshua A. Grochow2,3 1*Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, 87501, NM, USA. 2Department of Computer Science, University of Colorado Boulder, 1111 Engineering Dr., ECOT 717, 430 UCB, Boulder, 80309, CO, USA. 3Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, 80309, CO, USA. *Corresponding author(s). E-mail(s): dk@santafe.edu; Contributing authors: gulcekardes@gmail.com; jgrochow@colorado.edu; Abstract Progress in understanding expert performance is limited by the scarcity of quanti- tative 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 condi- tions: expert performance follows exponential progress curves whose parameters reflect the delayed acquisition of algorithms that shorten solution paths. Blind- fold 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 over- come 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 intelli- gence 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. Keywords: Collective Intelligence, Rubik’s Cube, Cognitive Artifacts, Combinatorial Search, Learning Curves 1 arXiv:2511.18609v1 [cs.AI] 23 Nov 2025 1 Introduction The Rubik’s Cube, invented by Erno Rubik in 1974, is a popular puzzle and the phys- ical embodiment of a mathematical structure, the Cayley graph of a permutation group generated by face turns [1]. Solving the puzzle involves a sequence of moves forming a path in its state graph from a scrambled state to the solved state, where each edge is a legal move. In the solved state, each of the six faces of the n×n×n cube is monochromatic. The longest path through this graph from a maximally jumbled start to the solution (i.e., the diameter of the graph) is given by its “God’s Number” [2] (Gn), which for the 3-cube is 20: every position is solvable in at most 20 face turns (counting both quarter-turns and half-turns as single moves), and some positions require exactly 20. The mapping of a popular puzzle directly onto a formal mathematical struc- ture offers numerous advantages, whereby formal concepts like entropy, difficulty, search, and optimality, as well as cognitive concepts such as skill and expertise, can be formally connected through a cognitive artifact [3, 4]. We describe the simplifi- cation (demonstrable reduction in dimension) of combinatorial search problems by physical artifacts as the Principle of Materiality, and show how the application of memorized algorithms to the Rubik’s Cube reveals fundamental characteristics of collectively intelligent systems [5]. The Rubik’s Cube has provided the basis for an ongoing series of tournaments in which highly skilled competitors seek to minimize both the time and the number of steps required to solve sighted and blindfolded n-cubes, where n denotes the cube’s linear dimension [6]. In this paper, we ana- lyze record-breaking solutions for all sighted cubes from n = [3,7] spanning up to 19 years of competitive play and blindfolded competition with nb = [3,5] spanning 17 years. These are recorded by the World Cube Association—a volunteer-organized association that has overseen thousands of competitions and millions of recorded solves under refereed conditions (https://www.worldcubeassociation.org). 1.1 Algorithms Efficient human solutions of the Rubik’s Cube rely on memorized “algorithms”, pat- tern–action rules that map a perceived global or local cube configuration to a specific sequence of face turns [6, 7]. The shared repertoire of these algorithms in the speed- cubing community has grown steadily over the past several decades. From the 2000s onward, CFOP (the Fridrich method) became the dominant method in speed-solving competitions. In its fully algorithmic form, CFOP requires on the order of a hun- dred memorized algorithms for the first two layers and the last layer, although many solvers rely on reduced two-look variants with only a few dozen core cases. Addi- tional methods subsequently gained popularity, including Roux, Petrus, and ZZ [8], which are well known within the speedcubing community. Together with specialized last-layer sets such as ZBLL, which alone comprises hundreds of distinct cases, these methods have expanded the shared

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