Related papers: Educational Challenges of Rubik's Cube
Spatial reasoning and collaboration are essential for childhood development, yet blind and visually impaired (BVI) children often lack access to tools that foster these skills. Tactile maps and assistive technologies primarily focus on…
This article illustrates pedagogy through training in the handling of abstractions. Mental arithmetic is not limited to numerical calculation; one can mentally calculate primitives and simplify analytical expressions. Even if there is…
We present an exactly solvable random-subcube model inspired by the structure of hard constraint satisfaction and optimization problems. Our model reproduces the structure of the solution space of the random k-satisfiability and k-coloring…
Large language models (LLMs) excel at many supervised tasks but often struggle with structured reasoning in unfamiliar settings. This discrepancy suggests that standard fine-tuning pipelines may instill narrow, domain-specific heuristics…
A cubic surface in $P^3$ is known to contain 27 lines, out of which one can form 36 Schlafli double - sixes i.e., collections $l_1,...,l_6, l'_1,..., l'_6\}$ of 12 lines such that each $l_i$ meets only $l'_j, j\neq i$ and does not meet…
Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…
Reinforcement learning algorithms rely on carefully engineering environment rewards that are extrinsic to the agent. However, annotating each environment with hand-designed, dense rewards is not scalable, motivating the need for developing…
We provide ongoing results from the development of a personalized learning system integrated into a serious game. Given limited instructor resources, the use of computerized systems to help tutor students offers a way to provide higher…
It is a $300$ year old counterintuitive observation of Prince Rupert of Rhine that in cube a straight tunnel can be cut, through which a second congruent cube can be passed. Hundred years later P. Nieuwland generalized Rupert's problem and…
Effective visual representation learning is crucial for reinforcement learning (RL) agents to extract task-relevant information from raw sensory inputs and generalize across diverse environments. However, existing RL benchmarks lack the…
The Dirac belt trick is often employed in physics classrooms to show that a $2\pi$ rotation is not topologically equivalent to the absence of rotation whereas a $4\pi$ rotation is, mirroring a key property of quaternions and their…
Rigid origami has shown potential in large diversity of practical applications. However, current rigid origami crease pattern design mostly relies on known tessellations. This strongly limits the diversity and novelty of patterns that can…
In this paper we give a upper bound of 40 on Rubik's cube group. The previously known upper bound has been 42 since 1995. In order to prove our claim we use computational methods. The program used is GAP computer algebra. Further more we…
This paper studies two families of constraints for two-dimensional and multidimensional arrays. The first family requires that a multidimensional array will not contain a cube of zeros of some fixed size and the second constraint imposes…
Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful…
Quantum mechanics is a notoriously abstract subject, and therefore challenging to teach at pre-college and introductory college levels. Here we introduce the Bloch Cube, a hands-on educational tool which can illustrate key quantum concepts…
Particle Dobble is an open-access, gamified learning tool designed to address persistent misconceptions in particle physics education by symbolically representing elements of the Standard Model. Aimed at upper secondary and introductory…
Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to…
Let $R$ be a polynomial ring over a field and $I$ an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the…
The application of Generative Pre-trained Transformer (GPT-2) to learn text-archived game notation provides a model environment for exploring sparse reward gameplay. The transformer architecture proves amenable to training on solved text…