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].
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).
Well, certainly I have finally got the time to write something.
Grant Sanderson, otherwise known as 3Blue1Brown, is a Youtuber that posts videos about mathematics. If you haven’t seen any of his work, do check out his website and channel.
For those who have watched 3Blue1Brown videos, you will know how he presents math concepts: with the use of very clean animations. Well, today’s topic isn’t about his channel, it’s about the animations.
Visualizing Mathematics
More often than not, mathematics requires a lot of formulae and graphs, and sometimes those things change over time; there’s even an entire field of math dedicated to studying change. Static figures often don’t do justice to these concepts, and with the widespread usage of computer screens, it seems counterintuitive to try understanding mathematics only through textbooks.
This problem was exactly what I encountered when trying to summarize Perelman’s proof of the Poincaré in 1250 words. Diagrams are very important and useful, but do they translate well if I were to present to an audience? Is there a better medium?
Mathematics Animation Engine
There is apparently a way to present mathematical concepts in the way 3Blue1Brown does it. In fact, the engine that he developed for his videos is available on GitHub. There even is a community-maintained version, which I have been told, is updated more regularly.
The engine differs from traditional animation engines in the sense that it is mathematics first, animation second. With Manim, you can create accurate graphs and objects, and paths are easily tracked and traced. Everything is mathematically defined, and therefore it is more convenient to scientific animations than vector drawing software, where every endpoint, every curve has to be tweaked and adjusted.
However, there are downsides, the biggest of which is that Manim does not have a user interface. It is a very powerful tool, yet it is not user-friendly at all. The user has to manually write code in Python, run sections of it on a terminal, before obtaining a single piece of animated math. For those who are not familiar with computer science, you will most likely have to learn the syntax of Python, the lexicon of Manim, and how to translate math into code (i.e. the formatting of LaTeX).
On the other hand, as a user with about a year of experience in Python and Java each, Manim was relatively easy to pick up once you understood its strengths and limitations, despite being as convoluted and complicated as it can be. Next time if you ever think about doing a math/science project, don’t hesitate to try using Manim to make your presentation look sleek as hell.
If you want to take a look at my animations, I have made a PowerPoint containing all of them. To see the code that made these animations, head over to my GitHub here.
As I have written last week, my term 1 project involves writing about Perelman’s solution to the Poincaré conjecture. After some initial reading and a small discussion with my mentor, these are my first thoughts on the topic.
The Conjecture
The Poincaré conjecture, as conjectured by Henri Poincaré in 1904, states that:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
This might look very confusing, and that is why we are going to break this down into parts and explain each concept briefly.
Firstly, a manifold. A manifold is defined as “a topological space that resembles Euclidean space at each point”. A 2-manifold means that object looks like a plane locally, but does not necessarily need to be a 2-dimensional object, like a piece of paper, a Möbius strip, or even the surface of the Earth. To expand this concept one dimension higher, a 3-manifold will look like 3-dimensional space locally, but might be bent in another dimension (time?). The fabric of space itself can be considered a 3-manifold, since it is a 3-dimensional space (duh), but gravitational forces can bend it in respect to time.
Secondly, “simply connected”. Simply connected (or 1-connected, since we all like numbers), means that any closed path (like a circle) on that manifold can be contracted into a point. Applying to 2-manifolds, it means that the surface cannot have any holes. The surface of a sphere is simply connected; the surface of a doughnut is not.
Thirdly, “closed”. A closed manifold has no boundary, and is “compact” (imagine a closed interval). A 1-dimensional example is the circumference of a circle, where there is no “end” to the line, and the distance that line travels before it meets itself is finite.
Next, “homeomorphic”. In layman’s terms, an object is homeomorphic to another when they can be transformed between each other by stretching and bending it without creating holes. A classic example would be a coffee mug is homeomorphic to a doughnut, since they are both objects with one hole.
Lastly, 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 spacetime a fixed distance away from a single point.
So, the Poincaré conjecture can be understood as: space, if it is limited in volume, has no boundary, and has no “holes”, it is curved in another dimension, and is a hypersphere; an analogy for 2-manifolds would be how the surface of the Earth is limited in area, has no boundary, and isn’t a doughnut, therefore it must be spheric.
The Solution
Grigori Perelman, a Russian mathematician, provided the proof of the conjecture over 3 papers in 2002-2003. His proof utilized a method called Ricci flow with surgery, which successfully proved that those 3-manifolds can be reduced into 3-spheres. This problem regarding 3-manifolds is the last problem of its kind to be solved; all higher dimensions of the Generalized Poincaré conjecture have been proven true before the turn of the century.
Ricci flow, introduced by Richard Hamilton, is essentially a differential equation that changes a manifold based on its curvature over time. It tends to produce objects that are rounder in shape and reduced in volume, which is ideal in a problem that wants to reduce objects into a simpler one. However, as the differential equation proceeds through time, some objects might produce singularities, and since Ricci flow can only operate on smooth manifolds, not all 3-manifolds can be reduced to 3-spheres in this manner.
Perelman dealt with this scenario by introducing a “surgery” to cut, and then cap these singularities, and then perform Ricci flow again. He proved that all singularities can be dealt with this way without breaking homeomorphism.
A natural question to come up after this is the number of cuts needed. If singularities can be produced by a single Ricci flow process, it might happen again after cutting, potentially infinitely many times. Perelman spent most of his third paper proving that a finite number of cuts is enough for any object, and thus, Ricci flow can be used to reduce any simply connected, closed 3-manifold into a 3-sphere, and the Poincaré conjecture is proven true.
If you do have any insights on the Poincaré conjecture, please contact me on my discord server, I am in need of ideas to write this paper out.
The mood of the week so far has been, well, “academic”. First off, I had to choose my term 1 project this Tuesday, and I willingly chose to study Perelman’s proof (parts 1 | 2 | 3) of the Poincaré Conjecture; secondly, I willingly used part of my Friday to learn about the Euler-Lagrange equation (albeit a version for dummies, thank you, Morgan); so it would only be normal for me to finish this week by talking about APs.
The Rationale for Taking APs
AP, as you might already know, stands for Advanced Placement. It was originally conceived as an idea for high school students to jump ahead and learn material from first-year university courses without leaving the comfort zone of high school, but currently, so many people take it that it is barely considered “advanced”.
The first tip I can give you is: never ever be peer pressured into taking APs. This was never a competition about the number of APs you are taking, nor should it ever be a race to get the most 5s. You should always be taking subjects that you like, or are useful for you to have a preliminary grasp of the material before re-learning it in university.
But you might be asking, “you took 12 APs, why are you telling us not to take that many APs?” There are a few reasons: for one, I am used to being constantly stressed; secondly, I am able to take those courses comfortably; and thirdly, I have experienced the pains of having an overwhelming number of APs, and I do not wish for you to repeat it.
The Pitfalls of Too Much Work
An AP course usually requires a lot more work than a regular high school course. Most often, only paying attention in class and doing your homework will not be enough.
As the material is university-level, sometimes the class will also be taught univeristy-style. Preparing for the class by doing pre-readings is almost mandatory, and taking notes will be integral to scoring well on your tests. In extreme cases, sometimes entire sections of the syllabus will not be covered in class, due to the lack of time.
If you want to be prepared for university, AP courses are definitely a more than appropriate way to experience the culture and the material that you will soon encounter, without too many consequences. It is certainly a good way to challenge both your knowledge and your time management.
However, too many APs and you will end up falling behind in every course that you are taking. Prioritize your work, and realize that time is limited. Enjoy high school while you still can, because university is going to get a lot harder once you get in.
I would also reccommend a limit of 5 AP courses per year. If university students are reccommended to cap their course loads at 5 courses per term, and 6 is already stretching it, I see no reason for a high school student to take more than 5 APs, since it is, by then, a less than accurate representation of both university life and your academic capabilities.
Taking APs as University Credit
Some universities allow students to get certain first-year credit if they have attained a good enough score in AP; some even mandate it. I will reccommend you to only take the credit for the courses that are not in your major (i.e. electives, requirements), and leave the credit that is related to your major. I am a firm believer, like many others, that to solidify your understanding, material should be learnt more than once. Moreover, courses in university are often taught in a different manner than AP courses, that it would be useful to get a different perspective of the same material.
However, if you need to get certain requirements out of the way, certainly. But as I said before, I do not reccommend taking AP courses just to escape a certain university requirements.
Course Overview
In this section, I would like to go through the 12 APs that I have taken, and provide a little bit of background information and a small description of my experience in those courses. They will be sorted by subject.
Note: after writing this, I realized it was way too long. But there’s nothing else after this section, so if you aren’t inclined to read all this, it’s fine!
English Language and Composition
Lang is a course about the practical uses of English language. For most of the class, you will be studying speeches, public letters, and other forms of writing that are very commonly used. This is still a good course to take even if you do not plan to become a language arts major, since it also enhances your skills in constructing solid arguments and writing coherent essays.
The exam consists of a multiple choice section and 3 essays (40 minutes each). The three essays are, in order, a synthesis essay, which requires you to construct an argument and take a stance based on the multiple sources that they provided; a rhetorical analysis, which asks you how does the author/speaker convince their audience of their point; and an argumentative essay, which asks you to take a stance on a topic and construct an argument without any given material.
I would personally say that the rhetorical analysis is the most difficult of the three, because it does not only ask “what”, but also “how”. However, if you are a slow reader, the synthesis essay might also be a source of trouble.
English Literature and Composition
Lit is the other branch of English in AP. It is mostly a study of prose and poetry, old and new. Unless you are pretty into the beauty of the language, I do not reccommend taking this course over Lang because of the high amounts of reading and workload associated.
The exam is similar to Lang, with a multiple choice section and 3 essays, 40 minutes each. The three essays are, in order, a poetry analysis, where you are given a poem and asked to analyze its literary merit; a short prose analysis, where you are given a short story or an excerpt from a drama and asked to analyze its literary techniques; and an analysis of a longer work, where you are not provided with any material asked to examine a theme and use any supporting texts of your choice.
The exam is just difficult, and that I believe a lot of preparation is required of the student. I clearly did not prepare enough, and rightfully deserve my 3.
Microeconomics
Micro is the branch of economics that is more like math/statistics than social sciences, as compared to Macro. The class is mostly about charts and graphs, and the theories that drive demand and supply.
The exam involves a multiple choice section and a written section of 3 free response questions, where you are asked to apply the theories on the data that you are given, draw graphs and calculate certain amounts (without a calculator).
For me, this is the easier of the two economics, since I feel like the graphs make the theories more clear-cut, that there are way less ambiguity than Macro.
Macroeconomics
Macro is the exact opposite. It studies a lot more of the societal impacts of fluctuating demands and supply, and because it is applied to the actual world, there are a lot more factors to consider, and things are less straightforward.
The exam is functionally identical to that of Micro, with multiple choice and 3 free responses, but this time with more writing and explanation of the effects that a single action has on the economy.
Again, I am good with numbers, not descriptions, and I did not do as well on Macro as I did on Micro. It feels very subjective to me, but I guess that is also a very subjective, biased opinion.
Calculus BC
Calculus is split into two branches: AB & BC. AB covers the standard differential and integral calculus and their applications; BC covers that and more. Venturing into series and their relationship with functions, you can see more of the syllabus here.
The exam is split into four sections: two of which are multiple choice and two of which are written responses, and each type of questions has a calulator and a non-calculator section.
Most people will think that BC is a really hard course, but I think it is only because of the pace. While AB gets to cover differentiation and integration over an entire year, BC students have to fit that in half a year, as to leave time for the BC-exclusive material.
Physics 1
There are a total of four physics courses, and the starting point for all of them is Physics 1. This course covers the fundamentals of high school physics, and serves as a great introduction to almost all areas of physics used in modern day, from Newtonian mechanics to circuits and waves.
The exam is a multiple choice section followed by 5 free response questions. There are way less calculations than you expect, and it involves a lot of explanation of phenomenon using the known laws of physics.
I personally prefer calculations and derivations, but the breadth of the course material makes up for the undesirable exam. In the end, Physics 1 is still the course that truly transformed me into hoping to become a physics major.
Physics 2
Physics 2, together with the aforementioned Physics 1, are the algebra-based physics courses that AP provides. This course covers the physics beyond Newtonian mechanics, and delves into fluid mechanics, thermodynamics, modern physics (special relativity), etc.
The exam is similar to Physics 1, in the sense that it involves a multiple choice and a written section, but this time the written section has one less question.
Honestly, this course does not cover enough material, in my opinion, as compared to Physics 1.
Physics C: Mechanics
The ultimate challenge in AP Physics is split into two courses. Mechanics, as its name suggests, deals with Newtonian mechanics, and covers everything from dynamics to rotational to energy. This course is calculus based, so it is reccommended to be taken after you have gained an understanding of differential and integral calculus.
The exam is half-length, at 1.5 hours, involving a multiple choice section and 3 written questions. Time will be very tight, as questions often require you to submit answers as a mathematical expression, accompanied with the occasional writing.
I would say that Physics C is the hardest course out of all the APs, due to the sheer amount of material that you have to know for a half-length exam.
Physics C: Electricity & Magnetism
E&M is the bane of most students’ existence. If Mechanics was considered difficult, then E&M is even worse. Covering from electrostatics, to circuits, to electromagnetism, E&M is a series of hard-to-visualize concepts with really specific rules with calculus slapped onto it.
The exam is identical to Mechanics, and is traditionally taken immediately after it. Your brain will mostly be fried after the two exams.
Chem is the most popular AP science, usually with at least twice the amount of students as all the Physics C and Biology combined. It is not as intensive as the other two (and if you are going into UK universities, you will have to study a lot more), but it does provide a good enough basis, covering a wide variety of chemistry areas.
The exam is also multiple choice and then written, with the written part requiring less detail than that of Bio (see below). The questions include quite the number of data interpretation and analysis, and also explaining known phenomena.
Biology
Bio is another candidate for being the most difficult AP course, due to the sheer amount of material that one has to study and memorize. The notes that I took for this course is unfortunately not measures in pages, or thickness, but rather by the number of binders I filled and the total weight of the notes.
The exam is again, multiple choice and written. The written section goes beyond regurgitating knowledge, but also requires data analysis, which links back to theories and mechanisms that you learnt in class. A lot of dots need to be connected, and the time limit surely also doesn’t help.
Although I did say it is more than regurgitation, I did, however, study 16 hours over the weekend before the exam on Monday morning, and successfully produced a 5.
Computer Science A
Comp Sci A is a course that teaches the basics of Java. Other than that, it teaches nothing. I am not too big of a fan of this course due to the lack of depth of the syllabus, but I will take what I can take, since it is, in the end, a course about programming.
The exam is, as I repeat for the twelfth time, part multiple choice and part written. The written section requires you to write code based on a context that includes way too many words. And when I say “writing code”, I mean physically writing code with pen and paper.
The exam format is honestly the biggest deterrent for me, since in no way in real life will I ever write code without a computer.
Conclusion
I came out of my 12 AP exams with a total of 6 5s, 5 4s, and a single 3, spread out over my final 3 years in high school. Is it helpful? Certainly. Is it really needed? Not at all. We will all one day learn that knowledge if we are devoted enough to that subject, AP or not.
If you have actually read all that stuff, I congratulate you, because even I fall asleep when I am asked to do long readings such as this article.
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.
