TomZ Constrained Cube Ultimate

A Constrained Cube is a puzzle where the freedom to rotate the faces is restricted.This cube has five constrained faces. The white face cannot be turned at all, the red and orange faces can be turned by a quarter turn, the blue and green faces by a half a turn and the yellow face is unrestricted.

Turning was is a little stiff on my cube, especially the yellow face. Corner cutting is poor but it is not a puzzle for speed solving so I don't see this as a problem.

Solution

This puzzle is obviously more difficult to solve than a regular one but it is not as difficult as one might imagine. Complete freedom to turn the yellow face is a major boon, while the inability to turn the white face is little or no handicap after the white edges are solved. It is not a problem to flip the orientations of the green and blue centres allowing some flexibility.

Step 1: Solve the White Edges

Intuitive. 

Solve in such a way that it is possible to execute an M-move on the white/green/yellow/blue layer. This makes it easy to flip the green and blue centres later on. Also, solve the green and blue centres such that they can perform a full half turn.

The white face becomes the D-layer.

Step 2: First Two Layers

Block building or F2L techniques work here, but I had to work around the limitations of the constrained faces.

 

Step 3: Orientate the Yellow Edges

I try to do this at the same time as step 2 but it does not always work out that way.

Step 4: Solve the Yellow Corners

Regular algorithms work here but some inventiveness is necessary to work around the constraints. Obviously, the bottom layer cannot be used to rotate the corners.

Step 5: Permute the Yellow Edges

All that remains is the relatively simple task of putting the yellow edges into their solved positions.

It is a very satisfying puzzle to solve, not as difficult as it looks.

Katsuhiko Okamoto's Void Cube

Void Cube

When I saw this cube, I just had to have one. It is a very clever piece of design and engineering. It still works like a regular cube but the centre pieces are missing with a void that runs right through the puzzle. It is one Katsuhiko Okamoto's inventions.

Turning is not as smooth as a regular cube and corner cutting is out of the question, due to the way it is constructed.

The puzzle is slightly more difficult than a regular cube due to the lack of a centre piece to act as a reference but this is not much of a hurdle. The more interesting obstacle that this cube introduces is the possibility of a parity issue. If, after solving the top layer, the middle layer is solved next with the bottom layer done last, there is a 50% chance that the middle layer is rotated by 90°. This will prevent the final layer from being solvable.

Parity

To show more clearly the parity issue on the void cube, and the concept in general, please see the picture of the regular cube below. The centres on the middle layer are deliberately mis-solved by 90°, i.e. a qurter turn, out of alignment...

If we then try to solve the yellow face, we discover that it is impossible. To solve this layer, we need to swap two edge pieces...

Regardless of what algorithms are used to move the edges around, at least two will always be in the wrong place. Hence, we have an odd parity case. We can infer from this, that performing a quarter turn on the E-layer is an odd number of edge swaps. Counting them confirms this.

It is obvious on a regular cube but the lack of centres on a void cube makes this potential displacement of the centres invisible.

Solution

To avoid the possibility of an odd parity case rearing its ugly head, I solve this puzzle following the steps below. Obviously, the algorithms that work with a normal cube are applicable here, too.

Step 1 - Solve the White Face

This is an intuitive step.

 

Step 2 - Solve the Yellow Corners

This can be solved with one only algorithm or at the most two.

Step  3 - Solve the Yellow Edges

Again, nothing special about doing this.

You could solve three edges and then use the fourth one to rotate the middle layer's edges but you will need to ensure that you are rotating them to the correct orientation.

Step 4 - Solve the Middle Layer

Use the gap in the yellow layer to facilitate the process, permuting the middle pieces at the same time if possible.

This is a fun puzzle. It is possible solve using any method used for a standard cube but using an extra algorithm to solve an odd-case parity if it arises. I prefer to solve in such a way to avoid the parity issue.

Travels of a Rubik's Cube

During my walks that I have undertaken to maintain my physical and mental well-being I have taken a standard 3x3x3 cube and looked for places to photograph it. People who are local to the Blackburn-with-Darwen area might recognise some of the places.

Having a cube in my pocket means I have an excuse for a short breather. I can solve it in random places. Not being a speed cuber means I get a little longer.

Science not Pseudoscience: A Flat-Earth Challenge

A Flat-Earth Challenge

A common mantra used by subscribers to the flat-Earth concept is "I became a flat-Earther when I could not debunk it". I have yet to see an encompassing flat-Earth model that satisfactorily explains all of the astrophysical phenomena that we observe.

A Flat-Earth Model

There is one heliocentric model, just one. This model is used to describe our tiny piece of the cosmos. It is used to make predictions. It does this very well.
There is no singular and consistent flat-Earth model that can do the same. If the flat-Earth community is genuinely serious in its convictions then it should be able to present a model that explains the Universe we observe and can make accurate predictions. So where is it?

Here is a summary of all of the things that this hypothetical model must include and be able to explain. Just one failure and the model is incorrect and is disproved and discarded.

The Sun

The sun rises in the morning from below the horizon. It travels across the sky at a constant speed and maintains a constant angular size. In the evening, it sets below the horizon. Observers at the same latitude see the sunrise and set at the same time. All observers see the same face of the Sun.

The angular size of the sun does change by a minute amount during the course of the year. It is ever so slightly larger in January than it is in July.

The Moon

The Moon is slightly more complicated. While observers on earth see the same face of the Moon, this is only approximately true. The flat-Earth model needs to explain libration. We can, in fact, see just over half of the Moon's surface from the Earth, with a slightly different portion of the surface being visible depending on its position its orbit. Also, the angular size of the moon varies, giving rise to super moons, etc.

The Moon has phases. All observers on Earth witness the same phase.

Eclipses

There are two types of eclipse. A solar eclipse is when the disc of the sun completely or almost completely obscured. A lunar eclipse is when a shadow passes over the moon preventing the sun from illuminating the moon. The flat-Earth needs to not only explain these two phenomena but also be able to predict when they will occur.

The Celestial Sphere

North of the equator, the sky is seen to rotate in an anti-clockwise direction around the north celestial pole. South of the equator, it rotates clockwise around the south celestial pole. Specific stars are either visible or invisible depending on an observer's latitude. For example, no Southern stars can be seen from the Earth's North pole and Polaris cannot be seen south of the equator.

Forces and the Acceleration due to Gravity (or not-Gravity)

Gravity - throw something up in the air and it comes down again. Something at elevation possesses potential energy that converts to kinetic energy as it falls. Sit down on a chair and you can feel it pressing up against you. There is an acceleration of 9.8 m/s² at the Earth's surface. This drops off with altitude. There is also a slight change depending on latitude.

Tides - the waters of the world's oceans and, to a lesser extent, lakes rise and fall in a predictable pattern based related to the motions of the Sun and Moon in the sky.

Sun and Moon - What holds the Sun and Moon in place and causes their motion across the sky?

Atmosphere - the surface of the earth is covered in blanket that is a mixture of gases, mainly nitrogen and oxygen. The pressure of this mixture of gases decreases with altitude.

Earth's Angular Momentum

The principle of the conservation of angular momentum can be used to show that the Earth possesses angular momentum. In other words, the Earth is rotating. If a flat-Earth model is to pass muster it must explain the Coriolis effect. Equipment such as Foucault's pendulum and a laser gyroscope reveal this rotation, therefore, the flat-Earth model needs to be consistent with these pieces of equipment.

 

A Map of the World

Finally, we need to see a working map of the whole world. This map must be good enough for use as a navigational aid. All distances between any two locations must be accurate and precise. The flat-Earth maps we usually see are arimuthal equidistant projections. Distances from the pole (or centre) to other places on the mao are all correct, but distances between any two other locations are distorted.

Conclusion

Toppling the heliocentric model will require more than a collection of distorted "black swan" photographs. The flat-Earth community needs to present a singular model of the flat-Earth that is 100% consistent with the reality that we observe and has predictive power. Handwavery and evasion are not going to cut it. If you cannot explain all of the above with your model, your model is incomplete or wrong. Go away and come back when you have one that can.

Once you have succeeded, you can then work on advanced features of the model, such as vulcanism, plate tectonics, earthquake shadows (the ones explained by a hot iron core in the globe model), jet streams, the length of the day changing by a fraction of a second depending on the time of year, and any other phenomena you might wish to tackle.

Science Fiction not Sci-Fi: Saving the Doctor

Can Doctor Who be Saved?

I think it can. The scenario below is the result of a discussion and speculation with a fellow fan.

The Timeless Child, Retconned

The Rogue Doctor was not, in fact, the Doctor but a different Time Lord. This Time Lord has held a grudge, many grudges, against the Doctor for a very, very long time. Time Lords have long memories.

During his many lifetimes, many had tried to kill the Doctor. All had failed. Even a creature whose purpose was to assassinate him had failed. He even ended up marrying the woman!

The enemy Time Lord came up with an insidious plan. Instead of killing the Doctor, he would assassinate the Doctor's identity, his sense of self. An act of revenge so Machiavellian that Master and Rani would be forced to look on with envy. The first part of the plan was to affect the Twelfth Doctor's regeneration. For the first time in his life, the Doctor was female. He wished the Twelfth Doctor's visit with his first incarnation was part of the plan, to see his attitude towards women back then, but that was a happy accident.

Next, he created some clues for the Doctor to find. He knew he could count on the Doctor's curiosity. He created the concept of the Timeless Child. The enemy, too, took on a female form. She thought her plans would unravel after the Master got involved, but no, incredibly the Master bought into it. She could not believe her luck. He had hidden on Earth before, in human form. She did it again. She revealed herself as the Doctor. There were flaws in the plan, but Doctor missed them. The Rogue Doctor's TT capsule was a Mark VI, more advanced than the Doctor's Type 40. She had overlooked that the TARDIS assumed its police box shape after he had supposedly become the First Doctor and fled Gallifrey. Or was that part of the plan to make the Doctor doubt his memories?

After all of this time, he, or rather she, had had his revenge. Mortimus' meddling had prevailed. The Doctor was conflicted and no longer knew who she was. With Gallifrey lost, who was there to put her right?

Unmasking a Bankrobber - A Satire

"All righty, are we all clear on the plan?" Bill studied his gang's faces. He was a little nervous; it was his first time as the leader of a hold-up.
"Plan is simple enough," replied Lefty-Lou, "Piece of cake."
"I'm going to say this again," said Bill, "no screw-ups. No shooting, unless absolutely necessary. They look harder for murderers than bank robbers."
"We hear ya," said One-Eyed-Jake.
"Good!" The word shot out like a bullet from a gun. "Keep it clean."
"Let's rootin' tootin' do this!" Karen's excitement and keenness showed how green she was.
"Rootin' tootin'?" One-Eyed-Jack stared at Karen as best he could.
"What, you don't say 'Rootin' Tootin''?"
"No Karen, we don't," replied Lefty-Lou, embarrassed on behalf of his girlfriend.
"See you all at sunrise. Make sure you bring these masks..."
"Wait, what?" asked Karen.
"Masks, bring them," snapped Bill.
"You didn't say nothin' about masks, Lefty," said Karen.
The rest of the gang glared at Karen creating an awkward silence.
"Erm..." said Left-Lou, shifting on his chair and straightening his hair, "this is a hold-up. We are..." he mimed waving a gun, "...robbing a bank."
"I can't wear a mask!" Said Karen, horrified. "What are trying to do, kill me?"
"Look, little lady," growled Bill through gritted teeth, "We are bank robbers. Criminals. You have to wear a goddammed mask!"
Karen shook her head. "No. No. No. Let me clue you up on masks."
Lefty-Lou covered his face with his hand.
"Masks cover your face so you won't be recognised," said Bill, slamming his fist onto the table.
"No, couldn't possibly wear a mask," Karen persisted, "how do you expect me to rob a bank if I can't breathe properly?"
"If you're recognised, the sheriff will come after me," said Lefty-Lou, "and then Bill and One-Eyed-Jake."
"I wanna spend our money, not get banged up," said One-Eyed-Jake, his bad eye twitching.
"You either wear the mask, or you're out." Bill threw a mask at Karen.
"Oh, I see," said Karen, throwing her arms up in the air, "It's like that is it. I refuse to wear a mask and you're kicking me out? You're persecuting me because I want to breathe!"
The three men stared at her, but she refused to be intimidated.
"Look," she said, calming down, "if you want to cover your faces, go ahead. All I'm saying is, I need my face clear of obstructions. You wouldn't want me passing out, would you?"
"No, no, we wouldn't," said One-Eyed Jack.
"There, you see?" Said Karen, missing One-Eyed Jack's sarcasm. "No one knows I rob banks. How could I be recognised?"
"Change of plan," said Bill, "I'm moving the robbery back a day. Get plenty of rest, Karen. Sleep in as much as you want. Why don't you make us a pot of coffee?"
"I knew you would see sense," replied Karen. There was a skip in her step as she went into the kitchen.
Bill lent forward and whispered to his friends, "A three-way cut it is then."

Science not Pseudo-Science: Research Flat-Earth

"Research flat-Earth!" This is a mantra of the flat-Earth community. So, I thought, why not?

Flat-Earth

So what is this modern phenomenon of the "Flat-Earth"? The first thing that strikes the casual observer is that members of the flat-Earth community spend very little time trying to prove flatness and much of their time trying to disprove the globe. A basic pattern is repeated, a flerf makes a claim based on an observation and a misunderstanding of mathematics or misinterpretation of science. This claim is touted as proof that the Earth is flat. Then, a group of people debunk the claim. This claim then spread to various flat-Earth outlets. The debunkers are then subjected to ad hominem attacks. It is the same basic pattern as any pseudoscience-ideology.

Flat-Earthers (or Flerfs)

From what I can tell, there are three types of flerf in the community. 

There are people at the top, selling the concept and making money. They do not genuinely believe the flatness paradigm but are very good at dreaming up "proofs" that are good enough to fool enough people to build a following. 

Beneath them are the people who buy the concept. Their poor grasp of physics and mathematics means they are vulnerable to the spurious "poofs" of the flat-Earth sellers.

There is a third category, the Poes. These people create deliberately ridiculous content in the style of genuine material. I presume their motives are to discredit the flat-Earth movement from the inside or perhaps it is just mischief-making.

The Flat-Earth Model (or Lack thereof)

As I mentioned earlier, flerfs put most of the energy into attempting to disprove or debunk the globe or the heliocentric model rather than promote a flat-Earth model. The reason for this is simple. There is no flat-Earth model. Instead of one comprehensive model, a series of different models are required to explain astrophysical phenomena.

The Day/Night Model

To explain day and night, the sun tracks around the flat-Earth in a perfect circle lighting up a semi-circle (somehow). Another version has a more circular area lit by the sun surrounded by a crescent shape of darkness.

The Seasons Model

To explain the seasons the sun tracks around the flat-Earth spiralling between the tropic of Cancer and the larger tropic of Capricorn. Somehow, the sun can travel around a larger circle in the same time as a smaller one but still maintain the same speed across the sky. The Seasons model is inconsistent with the Day/Night model because we no longer have exactly half of the Earth illuminated.

Eclipses Model

The usual story is that the Sun and the Moon are the same size and equally far away. Unfortunately, we have eclipses; Lunar eclipses are particularly problematic as they require the Earth to be between the Sun and the Moon. To get around this, a Shadow Object is invoked (sometimes called the Anti-Moon) that passes in front of the Moon or the Sun, depending on the type of eclipse. Unless the object is causing an eclipse, it is totally invisible and undetectable, and has no discernable effect on any other astronomical body.

Gravity

A big problem for the flat-Earth concept is gravity. Gravity draws objects to the centre of mass and, therefore, cannot work on a flat-Earth. What is the solution? Easy, deny its existence.

The most common explanation is density and buoyancy. Objects fall or rise until they achieve a density-equilibrium with their surroundings. To shore up their concept, they cite the gravity-defying properties of helium balloons, etc. Problems that still need to be addressed are explanations for the directionality of the effect and what is causing the medium to remain on the Earth.

Some of the brighter sparks have figured out that density/buoyancy does not cut it to describe the acceleration due to "definitely-not-gravity". The Earth's crust has a very slight negative charge. Aha! Says an imaginative flerf, it is electromagnetism. The electromagnetic force pulls things to the ground. There are still a few things to work out here, for example, why do electrically charged objects fall at the same rate as uncharged ones? Why aren't negatively charged objects repelled? Why do magnetic and non-magnetic materials all fall at the same rate?

An interesting attempt at an explanation is Universal Acceleration. The basic idea is that an unknown force is constantly accelerating the whole (!) flat-Earth system. This draws on the equivalence principle (i.e. it is impossible to tell the difference between being in a gravitation field and being accelerated). The problem with this explanation is that the acceleration we experience diminishes the higher up we get.

Flat-Earth Mantras

Flerfs repeat a stock list of clichés so often that it almost seems like there is an official script.

Ask someone how they joined up and the answer will almost certainly be along the lines of: "When first I read about flat-Earth, like everyone else, I laughed. When I tried to debunk it, I found I couldn't. That is when I became a flat-Earther."

"Water always finds its level." In other words, water always forms a flat, planar surface rather than conforming to the forces applied to it.

"I trust my senses." Considering how easy it is to fool our senses, this seems like a perilous premise on which to build a world view.

"Spinning at 1,000 mph." There are two observations to make from this. Firstly, flerfs cannot get their heads around the concept of angular velocity. Secondly, they cannot cope with modern units. This is a particularly dirty trick that is played here. Moving at 1,000 mph sounds very fast. Rotating at a rate of 1 revolution per day, not so much.

"Water doesn't stick to a ball." To prove it, water is poured over a ball. We are probably deeper in Poe territory here rather than true-believer.

"Air can't resist a vacuum," but high-pressure air at low altitude can resist low-pressure air at high altitude.

"There are no genuine photographs of the globe" and "It's CGI". The technology behind computer-generated imagery is truly impressive. It was used to create pictures of the globe years before it was invented. The only pictures of the flat-Earth, I have ever seen, are drawings.

"We see farther than we should." This phrase is used when a flerf uses a picture to demonstrate a lack of understanding of refraction. They are particularly keen on the so-called black swan photographs. These pictures are always taken over water and while they show a picture of an object well-beyond the physical horizon it is within the apparent horizon explicable by our understanding of the behaviour of light through the atmosphere. While the subjects of the photographs are far away, they are not particularly distant compared with the scale of the Earth. Where is the photograph of the coast of the United States taken from Ireland?

"Research Flat-Earth," by which they mean, "consume our content".

Northern Hemisphere Chauvinism

Our modern civilisation originated on the Northern hemisphere. Our Western culture exists on that side of the globe. It is all too easy to forget that the seasons and the rotation of the celestial sphere are reversed on the other side of the equator. Flat-earth models require that the southern hemisphere behaves differently from the northern.

Crazy conspiracy theories pop up to get around this glaring problem. Who can forget the one that stated that Australia is a hoax? The Antarctic circle experiences the midnight sun in its summer, just as the Arctic does. To get around this, the Antarctic Treaty is invoked. No one is allowed to go there. Apparently, the governments of the world have banded together to militarise (contrary to the treaty) Antarctica. The hundreds of tourists who go there every year must be paid actors. The military sentinels of this part of the planet must have developed highly advanced stealth technology because no one has ever seen and photographed it.

What is in it for the Flat-Earthers?

At first glance, there seems little or no point in arguing about the shape of our planet. A group of people puts a lot of time and effort into it, so they must get something out of it.
It is said there are two types of flat-Earther: those that sell the T-shirts and those that buy them. This supplies one strong motive, money. Google is clamping down on charlatans, pseudoscience and woo practitioners, but there are also books, T-shirts and other merchandise.
The other motive is attention. It is not difficult to find narcissists who hold court over the faithful. They have limited knowledge of physics and mathematics. This is not a problem because, for them, flat-Earth is about their egos and has nothing to do with reality. A normal person, unfortunate enough to encounter one of these individuals, can expect gas-lighting, name-calling and not even a semblance of two-way discourse.

Why is this Dangerous?

There is no doubt that flat-Earth is a hoax. A group of conmen perpetuate it for profit. Does it really matter if my friend who struggled with mathematics at school believes the Earth is flat? Ridicule aside, it is not going to be detrimental to a person's everyday life. The danger lies in it being a step-stone to other, more harmful, hoaxes and conspiracy theories. Our hypothetical friend may find a sense of belonging in the flat-Earth community. It is not a stretch of the imagination to believe that this person may look elsewhere. Our mathematically-challenged friend may join the ranks of the anti-vaxxers, anti-GMO activists, climate change deniers, etc, etc, etc. This person we previously ridiculed has now become a threat.

Flat-Earth Industry

A flat-Earth industry has evolved on YouTube. We have videos promoting flat-Earth and other videos debunking them.
The pattern is consistent. A flerf posts a video that says "Here is an astrophysical phenomenon I don't understand, therefore, the globe/heliocentric model is wrong." A short time later a response video appears explaining the real science and mathematics in the flerf's video, often with a thick helping of ridicule.

 

Debunking Flat-Earth without Equations

1) Stars

There are two hemispheres of stars, North and South. The behaviour of one hemisphere exactly mirrors the other. In the North, they move around the sky in an anti-clockwise direction and in the South, clockwise.

1) Polaris

The angle of the position of Polaris, to a good approximation, gives the latitude of the observer in the Northern hemisphere. It cannot be seen from the Southern hemisphere.

3) Eclipses

Eclipses do not work with a flat-Earth. We are given advance warning of eclipses from astronomers who subscribe to a global Earth and the heliocentric model. We do not get predictions of eclipses from the Flat-Earth Society.

4) Maps

There is no working map of a flat-Earth. All two-dimensional maps are projections that distort the true shape of landmasses and oceans.

5) Sunset and Sunrise

We observe the sun appear from below the horizon in the East, travel across the sky and drop down below the horizon again, in the West. What it does not do is trace a circle above us in the sky, getting larger and smaller as it approaches and recedes.