A scientist, a musician, and a normal student discovering their aspirations. I can usually be found at my discord server. Head over if you want to receive updates when I post, or have a question, or even just simply want to be part of a group.
See below for my unadulterated thoughts, or head to archives for a list of all of them.
I would like to extend a token of thanks to several people, without whom this website would not exist:
Jocelyn Baker, for providing all the resources that allowed me to build this;
Morgan Arnold, for reigniting my desire to build a website; and
Rashid Al-Abri, for being a wonderful head of Hack Club and reccommending the use of Visual Studio Code.
So I have been working on a project in the past few months. It was mostly a coding project, but due to its nature being related to schoolwork, deadlines prevented me from actually completing the project to my standards.
This will most likely be a three part series, with the first post (this one) explaining our results, the second post going deeper into the process, and potentially a third post to follow up after a summer’s worth of improvements.
This paper was co-authored by Callum Lehingrat and myself. Thank you very much for joining me on this endeavour.
The Efficiency of the Rubik's Cube
A Numerical Analysis of Rubik's Cube Speed-Solving Methods, with a Focus on CFOP & Roux
Abstract
CFOP and Roux are the two most popular 3x3 Rubik’s Cube speed-solving methods among solvers today. In our report, we seek to answer the question: with respect to varying metrics, which method is faster Our project consists of creating a large sample size of solves by creating a computer program to solve a multitude of cubes, whilst taking additional factors into account such as average move speed, which includes the average number of moves required to solve the cube, look time, finger-tricking, cube repositioning, and algorithm-specific time deductions. After we collected data from the computer using the different methods to solve one million cubes with identical scrambles, we discovered that with respect to most metrics, Roux outperforms CFOP.
Before moving on, we recommend consulting the supplemental information section of our paper at the bottom of this page, as it contains definitions of unavoidable cubing jargon used in the rest of this report.
Introduction
As newer, faster-turning models of the cube continue to be released and as solvers further optimize their methods, the world record for the fastest 3x3 solve continues to decrease [8]. Fridrich’s method, better known as CFOP, consists of solvers using an abundance of algorithms to solve the first two layers of the cube simultaneously [6], and then solving the final layer using only two algorithms [6]. On the other hand, the Roux method focuses on solving the corners of the cube first [5], and then working inwards to solve the cube by repositioning the middle pieces [5]. Both methods have their differences. On average, CFOP has a higher move count than Roux [4]. Furthermore, the Roux method makes use of moving the middle layer at a high frequency [5], whereas CFOP rarely requires such movements [6]. Furthermore, with respect to setting new world records, CFOP is the overwhelming winner [8]. Despite their differences, both methods allow solvers to employ a variety of finger-tricks, allowing for quick algorithm execution. Finally, both methods require the solver to reposition their hands several times over the course of a solve, and to look multiple times to reassess the situation.
We hypothesize that CFOP is faster than Roux, due to its prevalence in competitive solving.
Methods
Metric
Traditionally, the efficiency of an algorithm or a method is often measured in turn metrics such as HTM and STM. However, these two metrics both have basic assumptions: HTM assumes that every single face move will take the same amount of time, and middle slice turns will take double that; meanwhile, STM assumes all the moves to be equal. As we have asserted above, not all moves and sequence of moves are created equal. As such, we have identified four different factors that will affect a metric’s accuracy as a time-unit.
Look time: The approximate amount of time (in number of moves) taken to inspect the cube and identify moves.
Cube rotations: Since rotating a cube is not turning any faces, it is not included in any metrics. However, it still takes time to rotate a cube.
Regrips: Some moves on the cube cannot be performed (as easily) without a regrip, which is essentially removing your hand from the cube and then placing it down differently.
Deduction factor: When alternating moves on the same two faces, through the flexible design of speed cubes and techniques such as “corner-cutting” and “finger-tricking”, the effective time it takes is less than the actual number of moves, since they are often performed in one fluid motion.
By incorporating these four factors in different proportions, we have created two corrections to be made to existing metrics. The Adjustment Factor (AHTM and ASTM seen below) adds two moves every time inspection and rotation is needed, and adds one move for every regrip; the No-Regrips Factor (NRHTM and NRSTM seen below), on the other hand, adds one move each for inspection and rotation, and ignores regrips, since at the highest level of speed-cubing, the number of finger-tricks available completely removes the need for regrips.
Code
We emulated the scrambled state of a cube and solved it while keeping track of the metrics mentioned above. This process is repeated a million times to obtain the distributions for the two methods shown below. The link to the GitHub repository can be found here.
Results
Figure 1. The speeds of CFOP and Roux methods over 1 million solves, measured in half turn metric.
Figure 2. The speeds of CFOP and Roux methods over 1 million solves, measured in slice turn metric.
Figure 3. The speeds of CFOP and Roux methods over 1 million solves, measured in half turn metric with adjustment factor.
Figure 4. The speeds of CFOP and Roux methods over 1 million solves, measured in slice turn metric with adjustment factor.
Figure 5. The speeds of CFOP and Roux methods over 1 million solves, measured in half turn metric, omitting regrips.
Figure 6. The speeds of CFOP and Roux methods over 1 million solves, measured in slice turn metric, omitting regrips.
As presented above, since CFOP uses less middle slice turns than Roux, under the assumption that the two methods are equal in speed, it is expected that CFOP will generate lower values in all HTMs, and Roux will generate lower values in all STMs.
While the 3 STMs predictably favoured Roux, out of the 3 half turn metrics, only HTM and NRHTM favoured CFOP, while AHTM indicated Roux to be faster. Due to the high number of samples, the p-values evaluated from the two-tailed t-test for all six metrics is effectively 0, and we can visually identify the quicker method from the overlapping distributions.
Table 1. The means and standard deviations in number of moves of CFOP and Roux, measured in the six metrics. T-score is calculated using a paired Student’s T test.
HTM
STM
AHTM
ASTM
NRHTM
NRSTM
CFOP mean
68.905347
67.962297
79.414499
78.471449
66.602106
65.659056
Roux mean
73.609764
64.264263
77.544101
68.198600
68.809291
59.463790
CFOP SD
6.581088
6.904131
6.531167
6.877734
5.661963
5.993907
Roux SD
7.116545
6.163162
7.203812
6.189490
6.428597
5.422036
T-score
485
399
192
1110
257
766
Discussion
The accepted average STM values for CFOP and Roux is 60 and 48 respectively [9] [10], which significantly deviates away from the experimentally obtained values of 68 and 64. This can be explained by the use of adjust U-face techniques [11], which rotate a single face instead of the entire cube to match specific cases; this takes up moves, as opposed to cube rotations, which do not take up moves, but are not widely used in actual speed-cubing, as a single move takes less time than a cube rotation.
The other source of error would be due to the not perfectly optimized intuitive steps. As we have taken an algorithmic approach to simulate these steps, it is likely that it is not the best way to do so. However, since Roux, having two intuitive steps, should be less optimized than that of CFOP, which only has one intuitive step, it is logical to conclude that Roux is quicker than CFOP.
Although the data shown suggests that the Roux method is faster than CFOP, we must keep in mind that different variations can be introduced to both methods, further increasing their efficiency and speed. Realistically, the number of variations we can use is limited by the number of cases and algorithms the cuber can memorize and perform efficiently. At the highest level of speed-cubing, cubers will seldom only use the basic forms of these methods, but rather take the approach of that method, and combine steps from different methods to decrease their solve times.
In addition, there is a historical factor involved. CFOP was first proposed in 1981 [9], but Roux was introduced in 2003 [10]. As a result, CFOP is much more researched, with more variations and algorithms discovered. This leads to a higher popularity among speed-cubers, since a lot more information surrounding CFOP is readily available.
Furthermore, although our data supports the idea that the basic form of Roux is faster than that of CFOP, we cannot recommend CFOP cubers to change over to Roux. This is because at the highest level, variations for both methods will share many algorithms. Therefore, it is unlikely that any one method will have a definite advantage over the other. However, for beginners going into speed-cubing, looking to choose a method, we can suggest Roux as a quicker method than CFOP.
Concluding Remarks
Roux is shown to be quicker than CFOP in 4 metrics out of a total of 6 measured and should be recommended to beginners. However, since speed comes with instinct, not thought, we cannot recommend speed-cubing veterans to change methods after years of practicing and training.
Jargon & Supplemental Information
CFOP (Fridrich’s Method): A speed-solving method whereby the cube is solved in 4 steps. A cross is created on one side, and then an abundance of different algorithms is used to solve the first two layers of the cube simultaneously. The final layer is then solved in two steps, first by correctly orienting those blocks, and then permuting them into the correct position [6].
Roux method: A speed-solving method whereby solvers first focus on solving the corners of the cube through the building of two sets of 1x2x3 blocks, then solve the four remaining corners, and then completing the last six edges [5].
Corner-cutting: The phenomenon in which when a side is not completely aligned, another perpendicular side can still be turned due to the flexible design of cubes. This results in a distinctive “snap” when the unaligned side clicks into place.
Finger-Tricking: An unofficial term employed by competitive cubers whereby cubers solve cubes much more quickly by the ability to turn sides with a single finger movement, rather than the movement of the entire hand [2]. This allows cubers to turn multiple sides simultaneously.
HTM (Half-Turn Metric): A metric for the 3x3x3 Rubik’s Cube where any turn of any face, by any angle, counts as 1 turn [3].
QTM (Quarter-Turn Metric): A metric for the 3x3x3 Rubik’s Cube where any turn of any face by 90 degrees counts as one turn. It differs from the half-turn-metric because half-turns (turning the same face twice) count as two moves instead of one [3].
STM (Slice-Turn-Metric): A metric for the 3x3x3 Rubik’s Cube where any turn of any layer, by any angle, counts as one turn. It differs from the half-turn metric because middle-layer turns count as one move instead of two [3].
If you believe in a globe Earth, please finish reading this before making any conclusions.
Introduction
The Standard Horizontal Inertial Theory (SHIT) is a reformulation of multiple current flat-earth theories to become more scientifically sound and believable, and less religiously based. Its main goal is to correct the common misconception that the Earth is spherical, and to rectify the idea that a flat-earth theory must include edges where people can fall off of.
SHIT is an extension of my previous interpretation of the flat-earth theory, the Continuous Unlimited New-world Theory (CUNT), which has multiple erroneous statements that did not account for certain natural phenomena. SHIT attempts to cover what CUNT has missed, and is hopefully compatible with our current understanding of physics and astronomy.
Postulates
The Earth is an infinite, 2-dimensional plane.
1.1. The Earth carries repetitive features. It is mirrored across polar regions (i.e. the direction of the poles shift), and translated across meridians (i.e. East/West is consistent).
The Earth locally accelerates through space at 9.8 m/s².
2.1. The Earth creates a gravitational field of strength 9.8 m/s² that affects all objects.
2.2. The Earth collects matter from the universe, as a result of the acceleration, which forms the atmosphere.
2.3. By Newton’s Third Law, space can also be said to be accelerating towards earth at 9.8 m/s². However, since laws of motion are translationally invariant, and there is no reference point in space, we cannot determine the speed or position of the Earth.
2.4. Since the Earth carries no speed, relativity does not apply to the motion of the Earth.
2.5. The lack of a measurement for the speed of the Earth implies that the Earth has been accelerating since the beginning of time (if there is one).
Aether is a fundamental, underlying feature of the universe.
3.1. Aether is not considered an object, and is therefore not affected by the gravitational field.
3.3. Aether is not collected by the Earth, and therefore is displaced by the atmosphere to be distributed in the universe.
The presence of aether, or lack thereof, affects our perception of the Earth.
4.1. The density of aether increases with the distance from the observer, due to warped space.
4.2. As the density of aether increases, we perceive space to be warped.
4.3. The density of aether reaches a critical point at the horizon, which we cannot locally see beyond. The area that the horizon encapsulates is termed the vision circle.
4.4. The vision circle can be extended to the space above the Earth. There can only be a limited amount of space that can be visible to an observer at one time. That region of space is termed the vision cone.
Aether forms the majority of outer space.
5.1. The night sky is unchanging. However, the motion of the aether allows us to perceive objects in space as moving.
5.2. Planets outside of Earth are not real. However, the ability to explore a simulation of another world is possible, and can be explained by a theory of simulation that conforms with SHIT.
Evidence
So, you might be asking. Why would we want to abandon thousands of years’ worth of physics for such a theory? The truth is, physics was never right; it never even attempted to be right. Physics was developed to undermine the power of the Roman Catholic Church, and although it does not contradict the existence of deities, it is not immediately apparent to those who are unfamiliar with modern physics.
On the contrary, every single postulate here is constructed from simple observations, and should be apparent to both the religious and the atheists.
In postulate 1, we observe that the Earth is locally flat, but yet we can travel both eastwards and westwards to the same location. We can easily predict that the Earth is therefore an infinitely repeating plane, just like the ones we see in mathematics textbooks.
In postulate 2, we observe that the Earth is experiencing constant acceleration. By applying the principles of conventional physics, we are forced to conclude that it contradicts itself, sadly. We must therefore introduce something that explains the phenomenon, just like how we improve on theories when previous ones do not make sense.
The result is the aether, which is by itself not a new concept. By outlining its function and rules in the next 3 postulates, SHIT is able to explain natural phenomenon without resorting to an absurd conclusion that the Earth is somehow curved.
Ongoing Development
For a theory to be accepted, it must be simple enough to be understood. Like classical mechanics, SHIT is governed by simple, concise statements that can be understood universally, and are mutually compatible.
However, by this principle, we have to conclude that modern physics is unfortunately, not accepted, due to its mathematical complexity.
SHIT is open to revision and additions, as we have only just begun to rediscover the lost knowledge of a flat Earth. Feel free to contact me at my discord server to discuss and correct this theory.
Happy New Year’s to those to follow this blog. I hope you are well-rested, and prepared for a new year of challenges.
YouTube
I have made a new YouTube channel, hopefully posting my own performances and compositions. Nothing much has been planned for it, but please subscribe for more updates!
New Year’s Resolutions
I usually don’t make goals like these, since nobody ever follows them; but after an especially tough 2020, it might not be too bad of an idea to make myself some vague guidelines for the next year.
Put a reasonable effort into school.
Make good music.
Be happy; don’t be depressed (I know it doesn’t work that way, but so far, forgetting about being sad keeps me going).
For Humanity
I would like to take some time to recognize the troubles that the planet has been suffering in the past year. Maybe I am privileged and unable to understand the challenges of the poor, maybe I will never experience the discriminations of the minorites, but I hope in the future and in the present, we can work in toleration of each other, for peace, for progress, and for humanity.
I swear, this is my final post about the Conjecture. I have had enough topology for this year.
This is just a replication of my final written report that I have submitted, sprinkled with the animations that I have made for the presentation.
The Century-Old Problem
A Brief Overview of the Poincaré Conjecture and Perelman's Solution
The Conjecture
The Poincaré Conjecture, first posed by Henri Poincaré in 1904, is a problem in topology that asks whether a compact 3-manifold that allows every simple closed curve within the manifold to be deformed continuously to a point is homeomorphic to the 3-sphere (Milnor, 2004). To understand the conjecture, first we must define several terms used in the description.
A topological manifold in the nth dimension is said to “locally look like” a Euclidean space in the same dimension (Lee, 2012, p. 1). In essence, a 2-manifold “looks like” a 2-dimensional plane locally; for example, when examined at a close distance, the surface of the Earth seems flat, and therefore it is a 2-manifold. A 3-manifold, by analogy, should “look like” 3-dimensional space locally; for example, outer space can be considered a 3-manifold.
Compactness implies a manifold is closed and bounded (Lee, 2011, p. 85). A piece of paper with negligible thickness is not compact, because there are edges; however, the surface of a sphere is considered compact, since there are no edges.
A closed curve deforming into a point can be thought as wrapping a rubber band around the surface of sphere. If we tighten that rubber band, it will glide along the surface, reducing the area inside it, and eventually tightening to a point. However, if we loop that same rubber band around a doughnut, passing through the hole, we cannot tighten that rubber band to a point without destroying the doughnut.
Homeomorphism is a continuous process that maps an object to another (Lee, 2011, p. 28). If a manifold can be stretched and deformed into another manifold without punching extra holes in it, they are homeomorphic to each other. In a classic example, since both a coffee mug and a doughnut have exactly one hole, they are homeomorphic to each other.
Finally, a 3-sphere is a hypersphere. A circle, or a 1-sphere, is constructed using all the points on a plane a fixed distance away from a single point. A sphere, or a 2-sphere, is constructed using all the points in space a fixed distance away from a single point. A 3-sphere, by analogy, is constructed using all the points in 4-dimensional space a fixed distance away from a single point. Note that in topology, an n-sphere refers to a hollow sphere, not a solid ball.
The conjecture essentially asks whether a 3-dimensional space that is curved in the 4th dimension, finite in volume, with no edges, and having no “holes”, can be stretched and shrunk to a hypersphere.
The Mathematician
For almost a century, the problem remained unsolved. The Generalized Poincaré Conjecture, the analogous conjecture for topological manifolds in other dimensions, was solved for all cases other than the 3rd dimension by 1986 (Milnor, 2004).
In 2000, the conjecture was included as part of the Millennium Prize Problems, a set of 7 challenging unsolved problems in modern mathematics. Providing a correct solution to any of the problems would yield the mathematician a prize of $1 million (Clay Mathematics Institute).
Enter Grigori Perelman. Perelman is a Russian mathematician who, unlike many other scientists and mathematicians, liked to work alone. Close friends described him as “ascetic” and “eclectic” (Nasar & Gruber, 2006). With a simple lifestyle, a firm dedication to problem-solving, and enough courage to publish a paper without seeking any review, Perelman posted his first of three instalments on the solution to the Poincaré Conjecture on the internet in 2002.
The Solution
In his first paper, Perelman summarized the use of Ricci flow in reducing closed manifolds. Ricci flow is most often presented as a differential equation
where g is a Riemannian metric, and R is the Ricci curvature (Perelman, 2002). For every point, the metric outputs a scalar number that describes the curvature and dictates how it should stretch or shrink; just as how a tangent line describes the slope and dictates how functions change. Ricci curvature expresses the curvature of the manifold in terms of the metric. The equation essentially states the change of the metric is proportional to the curvature. When a region is concave, Ricci curvature is negative, Ricci flow inflates it, and the metric increases; conversely, when a region is convex, Ricci curvature is positive, Ricci flow deflates it, and the metric decreases. A sphere has positive curvature everywhere, and will therefore always deform into a single point; if any manifold that undergoes Ricci flow deform into a point, then it must be homeomorphic to a sphere. However, in higher dimensions, Ricci flow sometimes fails and creates singularities (points that are not differentiable) before a single point is produced (Milnor, 2004).
The second paper addresses these singularities and introduces a method to bypass them. Called Ricci flow with surgery, Perelman proved that all singularities can be cut and replaced with spherical caps, and then restarting Ricci flow on the resulting two manifolds (Perelman, 2003a). If the resulting two manifolds tend to a single point, then they must be homeomorphic to two spheres, and two spheres connected must also be homeomorphic to a single sphere.
A natural question that arises next regards the number of surgeries needed. If one cut can be made to a manifold, it is logical that further cuts can be made to the resulting manifolds; and if an infinite amount of cuts are made, then Ricci flow with surgery might not necessarily reduce any manifold to a sphere. The third paper proves for any Riemannian metric, the solution to Ricci flow with surgery becomes extinct in finite time, meaning only a finite amount of cuts are needed (Perelman, 2003b).
As a result, any 3-manifold can be transformed into 3-spheres through Ricci flow with a finite amount of surgeries, and therefore is homeomorphic to a 3-sphere.
The Effect
Two years passed without a single flaw was found in Perelman’s papers. The proof was then deemed complete; Perelman was awarded a Fields medal and the $1 million from the Millienium Prize Fund – but he turned both down. Subsequently, Perelman retired from mathematics, partly due to his inability to handle fame, and partly due to controversy after fellow mathematicians claimed Perelman’s work as their own original work. He was “dismayed by the discipline’s lax ethics” (Nasar & Gruber, 2006).
As of 2020, the Poincaré Conjecture remains the only solved Millennium Prize problem (Clay Mathematics Institute).
Figure 2. The speeds of CFOP and Roux methods over 1 million solves, measured in slice turn metric.
Figure 3. The speeds of CFOP and Roux methods over 1 million solves, measured in half turn metric with adjustment factor.
Figure 4. The speeds of CFOP and Roux methods over 1 million solves, measured in slice turn metric with adjustment factor.
Figure 5. The speeds of CFOP and Roux methods over 1 million solves, measured in half turn metric, omitting regrips.
Figure 6. The speeds of CFOP and Roux methods over 1 million solves, measured in slice turn metric, omitting regrips.
