Rubik's Cube Research Articles

Machine discovery of effective admissible heuristics

Admissible heuristics are an important class of heuristics worth discovering: they guarantee shortest path solutions in search algorithms such as A* and they guarantee less expensively produced, but boundedly longer solutions in search algorithms such as dynamic weighting. Unfortunately, effective (accurate and cheap to compute) admissible heuristics can take years for people to discover. Several researchers have suggested that certain transformations of a problem can be used to generate admissible heuristics. This article defines a more general class of transformations, called abstractions, that are guaranteed to generate only admissible heuristics. It also describes and evaluates an implemented program (Absolver II) that uses a means-ends analysis search control strategy to discover abstracted problems that result in effective admissible heuristics. Absolver II discovered several well-known and a few novel admissible heuristics, including the first known effective one for Rubik's Cube, thus concretely demonstrating that effective admissible heuristics can be tractably discovered by a machine.

Machine discovery of effective admissible heuristics

Admissible heuristics are an important class of heuristics worth discovering: they guarantee shortest path solutions in search algorithms such as A * and they guarantee less expensively produced, but boundedly longer solutions in search algorithms such as dynamic weighting. Unfortunately, effective (accurate and cheap to compute) admissible heuristics can take years for people to discover. Several researchers have suggested that certain transformations of a problem can be used to generate admissible heuristics. This article defines a more general class of transformations, called abstractions, that are guaranteed to generate only admissible heuristics. It also describes and evaluates an implemented program (Absolver II) that uses a means-ends analysis search control strategy to discover abstracted problems that result in effective admissible heuristics. Absolver II discovered several well-known and a few novel admissible heuristics, including the first known effective one for Rubik's Cube, thus concretely demonstrating that effective admissible heuristics can be tractably discovered by a machine.

Amuse (TM): A practitioner's device for maintaining a copywriter's creativity

A Rubik cube has been transformed to provide a trigger for creative ideas. The inventor dreamed up and tested the idea (Amuse, TM) while working as a copywriter for an advertising agency. He describes its workings, and links it to a brief, theoretical account of creativity.

“RUBIK'S CUBE” AS A TRANSPOSITION DEVICE

This paper points out that the well-known “Rubik's Cube” can be used as a device for implementing a transposition cipher which is simple to operate and which should afford a high level of security. A particular system for using the cube is proposed. The system is secure against brute force attacks; since it permits a different scrambled ordering of letters for each letter block enciphered, it is also secure against multiple anagramming.

Rubik's Tesseract

can't build a physical model and experiment with it. The results described below were discovered with the aid of a simulation of the tesseract on a Macintosh computer. In this simulation the computer displays a representation of the tesseract on the screen, and the user uses a pointing device (a mouse) to ask the computer to twist sides of the tesseract. To understand the graphic representation of the tesseract used in this simulation, it might be helpful to consider first the easier problem of representing the ordinary three-dimensional Rubik's cube in a way that two-dimensional people could understand. One way to make a two-dimensional representation of the Rubik's cube would be to imagine unfolding the surface of the cube in the familiar way illustrated in FIGURE 1. (In all of the figures in this paper, the of the sides of the cube are represented as black-and-white patterns. We will continue to refer to them, however, as colors.) Unfortunately, this representation would not be very useful to a two-dimensional person trying to solve the cube. The problem is that this representation doesn't show clearly which are attached to different sides of the same cubie. (The 27 small cubes that make up a Rubik's cube are usually called cubies.) For example, the three colors attached to the cubie in the front bottom right comer of the cube in FIGURE 1 are stripes, gray, and dots, but it takes some thought to figure this out from the two-dimensional representation in FIGURE 1.

The Pancreatic Islet as Rubik's Cube: Is Phospholipid Hydrolysis a piece of the Puzzle?

At least three types of phospholipase exist in the beta-cells of the pancreatic islet. Data regarding their physiological activation are incomplete but suggest that glucose (or its metabolite glyceraldehyde) either activates or potentiates the activation of several phospholipases. At least seven phospholipid hydrolysis by-products (diacylglycerol, myo-inositol 1,4,5-trisphosphate, lysophospholipids, arachidonic acid and its cyclooxygenase- and lipoxygenase-derived metabolites, phosphatidate) have been demonstrated to have effects compatible with their postulated roles as mediators or modulators of islet function. Presumptive mechanisms of action have been tentatively identified for these metabolites. However, key studies in the puzzle are missing, and current methodologies have important limitations. Shortcomings include the paucity of measurements of the mass of metabolites; the frequent use of static incubations rather than perfusions; a lack of complete time- and agonist concentration-dependence curves; the equation of metabolite accumulation with rates of metabolite generation (which ignores metabolite removal as a key variable); the use of nonspecific, insensitive, or ambiguous phospholipase assays; and the need for more studies directly correlating lipid metabolism and insulin secretion in physiologically functioning preparations. Like Rubik's Cube, the pancreatic islet is a dynamic puzzle comprised of many interrelated components requiring proper alignment and integration. Phospholipid turnover is one "panel" in the islet; however, an obligate role for phospholipase activation in glucose-induced insulin secretion is not yet rigorously established, despite tantalizing, inferential evidence. It may be that glucose serves principally to potentiate the phospholipase and secretory responses to other signals that act by initiating phospholipid hydrolysis.

The pancreatic islet as Rubik's Cube. Is phospholipid hydrolysis a piece of the puzzle?

The pancreatic islet as Rubik's Cube. Is phospholipid hydrolysis a piece of the puzzle?

The geometrical representation of path planning problems

The path planning problem for arbitrary devices is first and foremost a geometrical problem. For the field of control theory, advanced mathematical techniques have been developed to describe and use geometrical structure. In this paper we use the notions of the flow of vector fields and geodesics in metric spaces to formalize and unify path planning problems. A path planning algorithm based on flow propagation is briefly discussed. Applications of the theory to motion planning for a robot arm, a maneuvering car, and Rubik's Cube are given. These very different problems (holomic, non-holomic and discrete, respectively) are all solved by the same unified procedure.

Integrating perception with problem solving

Problem solving is a process to convert the world from an initial state to a goal state. This paper describes an approach that interleaves perception with action in problem solving. We presume that an intelligent agent starts this process with uncertain knowledge about the initial state of the world, a specified goal state, a given set of permissible actions to alter the world state, and a given set of sensors to observe the partially understood world state. The task is to generate and execute a plan to reach a state in which the agent knows that it has reached the goal.Given a problem described by all feasible sequences of actions and perceptions applicable to coverting any given state to the goal state, we demonstrate how to construct plans that can guide both action and perception by achieving a sequence of subgoals and show it is equivalent to decomposing the representation of the problem into components corresponding to achieving subgoals. We first show that feasible sequences of actions can be decomposed into components, independent of perception, for instance through a principal series (in the case of a group, a subnormal series). This leads to subgoals with which certain observables can be associated. On the other hand, the perceptive problem can be decomposed by switching on sensors in series and performing actions that make the perceived state indistinguishable from the goal state. Finally, we show how perception can be intergrated with problem solving; we use perception to guide actions; hence, to reduce the overall cost of problem solving. The ideas are illustrated with problems taken from subgroups of Rubik's Cube; we are currently extending them to arbitrary semi-groups.

The Eternal Triangle—A History of a Counting Problem

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsMogens Esrom LarsenMogens Esrom Larsen was born in 1942. He graduated from the University of Copenhagen in 1965 where he remained, except for a recess at M.I.T. during 1969–70, and is now associate professor of mathematics. His interests cover most of mathematics from several complex variables, to differential equations, to finite groups, e.g., Rubik's cubes, and from pre-Euclidean history of mathematics to the teaching and applications of modern mathematics, including numerical analysis. In addition, he writes a monthly problem column in a popular Scandinavian science journal where this triangle problem was presented. Fond of games, he also founded the Copenhagen Go Club in 1972.

Kepler's Spheres and Rubik's Cube

Kepler's Spheres and Rubik's Cube

Kepler's Spheres and Rubik's Cube

How many spheres of radius r may simultaneously be tangent to a fixed sphere of the same radius? This question goes back at least to the year 1694, when it was considered by Isaac Newton and his contemporary David Gregory. Johann Kepler had already shown in 1611 [8] that twelve outer spheres could be arranged around a central sphere, and now Gregory claimed that a thirteenth could be added. Newton disagreed, but neither man proved his claim, and it was not until 1874 that a proof was found, substantiating Newton's conjecture. (R. Hoppe's original proof is described in [2]; more recently, proofs have been found by Giinter [7] and by Schiitte and van der Waerden [10]. Perhaps the most elegant proof known is the one given by John Leech [9]. For a history of the problem, see [6].) It's worth noting that the 4-dimensional version of this problem is still unsolved; no one knows how to arrange more than twenty-four hyperspheres around a central hypersphere, but neither has it been shown that a twenty-fifth cannot be added. In higher dimensions the situation is even less well understood, with the remarkable exception of dimensions 8 and 24, for which the maximum contact numbers are precisely known [1]. The source of difficulty in the original Gregory-Newton problem (and the reason it took 180 years to be solved) is that there is almost room for a thirteenth sphere; the twelve spheres of Kepler can be pushed and pulled in all sorts of ways, and it's credible that some sort of fiddling could create a space big enough to accommodate an extra sphere. This note will deal with a related question, in the spirit of Erno Rubik: if we label the twelve spheres and roll them over the surface of the inner sphere at will, what permutations are achievable? The answer is surprising, and the proof requires only the rudiments of analytic geometry and group theory. One fringe benefit of this enterprise is that it leads to a natural coordinatization of the vertices of the regular icosahedron. An equivalent formulation of the problem is gotten by considering only the centers of the spheres: we imagine twelve vertices, free to move on a sphere of fixed radius R = 2r about a fixed origin but subject to the constraint that no two vertices may ever be closer together than R. We will find it convenient to switch back and forth between the two formulations. For definiteness, put r = 1, 1 = 1. To begin, we must prescribe an initial configuration for the outer spheres. One possibility that springs to mind is to arrange the twelve vertices to form a regular icosahedron. As we'll see, any two distinct vertices of the icosahedron inscribed in the unit sphere are at least

Selecting and compensating mentor teachers: A win‐win scenario

What had gone wrong? They had studied hard, taken the prescribed courses with the finest professors, pored over the material on the psychology of learning and the tenets of effective pedagogy, deepened their dedication. Each was ready-but not for this. This was a real class with real children, real challenges-a human Rubik's cube. What these teachers were missing (this knowledge dawned slowly) was the help of those who had stood before where they stood now. They needed mentors. The idea that teachers just might be the experts on teaching was anathema to those who ruled the bureaucratic structures that entangled schools when I started teaching in the 1960s. What would be taught and how it would be taught was dictated from on high-from centralized district offices staffed by people far removed from the realities of classroom practice, by superintendents accountable to state policymakers and wed to state policy (ECS, 1986; Wirt & Kirst, 1988; Wise, 1979). Not much has changed. In very few schools are today's teachers released from the two-by-fourby-six structural straightjacket that traps both

The rotation group of Rubik's cube

In this project, the group theory program "Cayley" is used to solve some problems associated with Rubik's Cube, including the original problem of restoring the initial state of the cube.

Noise and verbal or spatial solutions of Rubik's cube

Subjects were required to solve simple problems using a Rubik's cube, posed either spatially (a pictorial representation of the problem) or verbally (written instructions). Results showed that subjects classified on a pretest as spatially able performed better on the Rubik's cube than those classified as verbally able. In addition, verbal performance was better under 95 dBA than 70 dBA and spatial performance better under 70 dB A than under 95 dB A white noise. The results are considered in the light of the contradictory literature on the effects of noise on human memory, with particular reference to the adoption of problem-solving strategies in noise.

Computing short generator sequences

The diameter of a group is the length of the longest product of generators required to reach a group element. We show that the diameter of a permutation group of degree n generated by cycles of bounded degree is O ( n 2 ). This bound is the best possible. Additionally, such short products can be found in polynomial time. The techniques presented here may be applied to many permutation-group puzzles such as Alexander's Star, the Hungarian Rings, Rubik's Cube, and some of their generalizations.

The Subgroup Algorithm for Generating Uniform Random Variables

We suggest a simple algorithm for Monte Carlo generation of uniformly distributed variables on a compact group. Example include random permutations, Rubik's cube positions, orthogonal, unitary, and symplectic matrices, and elements ofGLnover a finite field. the algorithm reduces to the “standard” fast algorithm when there is one, but many new example are included.

A Common Sense Approach to Cost/Rate Analysis

Abstract : The premise of this paper is that cost/production rate analysis is an individualistic issue that demands a determinative, non-statistical treatment, and that cost/rate statistics are certainly not now, and may never be, appropriate to measure the cost/rate effect. Cost/rate analysis, one of the two basic analyses that constitute production cost estimating, is receiving more and more high-level attention. Unfortunately, there is widespread misinterpretation, misuse, and superficial treatment of cost/rate methodologies. Some of this arises from analysts selecting from the methodologies available without recognizing the limited range of applicability or the fact that some are underdeveloped or unverified theories. Realizing this and faced with his own specific need for a valid methodology, the author developed a new determinative, nonstatistical equation and approach to cost/rate analysis. The problem (that obstructs and almost defies a valid methodology) is that the production cost formula is a virtual Rubik's Cube of interdependent and unpredictable factors (e.g., the degree of labor union resistance to labor force cutbacks, resulting from a falling production rate). In addition, the number of factors is perhaps limitless and the values of some seem indeterminable.

On the mobility of bodies in ℝn

We may define a mobility problem to be one of the type: ‘Given a (fully or partlys pecified) collection of bodies in ℝn, assumed not to interpenetrate, can they be moved in some (fully or partly specified) fashion?’ Early examples of mobility problems nclude ‘Chinese puzzles’ and Sam Loyd's ‘15 puzzle’ (to which could be added the more recent Rubik's Cube!) Less combinatorial examples include the ‘piano mover’ problem (see, e.g. [2, 3, 9]) and the several recent papers on collision avoidance, such as [1, 5, 6, 7, 8].

A Bug's Shortest Path on a Cube

This can be readily obtained by identifying each shortest path either as a selection of m objects from m + n objects or as a permutation of m + n objects consisting of exactly m of the symbols E (east) and n of the symbols N (north). Now one naturally wonders about the 3-dimensional analogue of this problem. Imagine for example, that a bug situated at one corner of an n X n X n Rubik's cube wants to reach the opposite corner of the opposite face in such a way that its path consists of line segments of integer length each of which is parallel to one of the axes. What is the number of such shortest paths? If the bug can chew its way through the interior of the cube, then the same argument used in the 2-dimensional version immediately gives the answer (3 n) !/( n !)3. But if the bug can only crawl on the surface of the cube and because the surface is slippery, must crawl along the edges or the grooves between the small cubes, to determine the number, f (n), of such shortest paths turns out to be a much harder problem than the 2-dimensional version. Though it is obvious that f(1) = 6, to compute the value of f(2) by brute force might be quite a challenging job. (Any one with 2 x 2 x 2 Rubik's cube is encouraged to give it a try.) To determine the value of f(3), with or without the use of an ordinary Rubik's cube, seems to be a formidable task. For the interest of the readers, we will not reveal these two values until the end of the article. To answer this question, we consider the more general problem in which the cube is replaced by a rectangular box. Let Q be the I X m X n rectangular prism in space with vertices at A = (0, 0, 0), B = (1, 0, 0), C=(l,m,O), D=(O,m,0), E=(0,m,n), F=(0,0,n), G=(l,0,n) and H=(l,m,n). See FIGURE 1, where 1, m, and n are positive integers. A bug wants to go from A to H on the surface of Q in such a way that its path consists of line segments of integer lengths each of which is parallel to one of the axes. What is the number f (1, m, n) of different shortest paths? (It is clear that the length of such a path is I + m + n.) This is question (1) asked in [1].