Shepherd Cube

The Shepherd Cube was originally created by Alistair Shepherd with hearts. The current design uses is arrows. It is the ultimate orientation cube, the solved state having arrows all pointed in the same direction on each face.

Shepherd Cubes available for sale are rare, but the stickers are readily available and will be sold by your preferred supplier of custom stickers.

The secret to the puzzle is that one corner has the arrows circling in a clockwise direction...

...the diametrically opposed corner has the arrows circling anti-clockwise.

The consequence of this is that opposite faces have arrows pointing in opposite directions. Knowing this lays the foundation to a solution. These two corners are referred to as "starting corners" in the solution, below.

Solution

This is a challenging puzzle to solve but not as difficult as I anticipated. You do need good spacial awareness to work out where the arrows need to point, and a good understanding of how parity odd-cases can arise and how to deal with them.

Step 1: Find a Starting Corner and Orientate the Centres

Find one of the corners where the arrows circle either clockwise or anti-clockwise. This is the starting point and acts as the initial reference point for the next part of the solve.

Step 2: Solve the Edges Adjacent to the Starter Corner

The most difficult aspect of this is finding the correct pieces.

There is now a 2x2x2 solved section.

Step 3: Extend the Solved 2x2x2 Block to a 2x2x3 Block

I prefer to solve the corner first and then add the edges. There are three directions in which to extend. There is a chance that might deliver the right corner in one of the cases.

Step 4: Complete the First Two Layers

There are two directions left into which to expand the solved section. Serendipity is less likely to deliver an easy start to this step.

Step 5: Solve Final Layer Corners

The first corner is the starting corner. The others follow from that one. There is a 50% chance of an parity odd case that needs to be resolved.

Step 6: Solve Final Edges

Finally, solve the edges. It is possible to encounter a parity odd case due to edges being the same.

Parity Odd Cases

Odd parity cases are possible. The centres may be misaligned by a quarter turn, similar to the Void Cube. There are also some edges that are interchangeable. The two starting corners have no orientation, therefore, it is possible to have one cornered rotated out of position.

Science not Pseudoscience: Relative Density Disequilibium and Gravity

Falling

The concept of flat-Earth has a major problem that its proponents must overcome: the fact that things fall. We have given this phenomenon the name of gravity. At its most basic level of understanding a smaller object is drawn towards the centre of mass of a larger object. This clearly cannot be the case for a flat Earth.

Undaunted, the flat-Earth community came up with an alternative explanation for falling: relative density disequilibrium. An object, when in a medium of a different density, experiences a force that drives it to find its density equilibrium. I have yet to find someone who can explain the source of the energy differential responsible for this force. I have found no equations that describe this phenomenon.

I have seen flerfs cite buoyancy to explain the behaviour of objects in different media. Unfortunately for the density disequilibrium proponents, buoyancy is a gravitational effect. If you deny a constant downward acceleration, you automagically deny buoyancy, too. More on this later.

Even without mathematics, we can use these two explanations to make predictions. This allows us to perform experiments to find out which one is the better description of our reality.

Relative Density versus Gravity

There are many ways that we can test gravity versus density disequilibrium. First a quick summary of the histories of these two concepts.

Gravity

Johannes Kepler figured out the laws of planetary motion. These were early steps towards our understanding of gravity and how it affects the orbits of astronomical bodies.

Isaac Newton came up with an equation that described, in most cases, how bodies behaved under the influence of gravity. However, he had no idea what caused gravity. He was particularly troubled by the action at a distance.

Einstein realised that gravity is space-time curvature. His equations are much more complicated than Newton's but Newton's equation is a good enough approximation in most circumstances. 

The picture is still not complete because quantum mechanics is absent from our description of gravity.

Relative Density Disequilibrium

The history of density disequilibrium is not documented. I was unable to discover whom I should credit with its origination.

Falsifying the Hypotheses

Buoyancy versus Finding Equilibrium

Buoyancy is a consequence of the acceleration due to gravity. Put an object into a fluid and it displaces that fluid. If the weight of the fluid displaced is less than the object then the buoyant force is greater than the weight and the object floats (e.g. ships on water) or is displaced upwards (e.g. helium balloons).

Strictly speaking, there is no buoyancy with relative density disequilibrium. There is only equilibrium and disequilibrium. Once an object has found its density in an environment, there are no forces acting upon it. There is no downward acceleration and no buoyant force.

The Test: place an object, less dense than water, in water. 

• If gravity is correct, the object will sink into water until a weight of water is displaced equal to the weight of the object
• If relative density disequilibrium is correct, the object sits will sit on top of the water.

Pressure

Imagine a quantity of a fluid of constant temperature, such that there are no density variations and it is in thermal equilibrium. A consequence of the acceleration due to gravity is that the weight of fluid higher up presses down on the fluid lower down. This results in a pressure gradient. Higher pressure at the bottom, lower pressure at the top.

With relative density disequilibrium, the fluid at the top is in density equilibrium with the fluid at the bottom, therefore, there are no forces acting on it. There is no pressure gradient.

The Test: create a long tube with holes placed at regular intervals along its length. Stand the tube vertically and fill with water.

• If gravity is correct, the water shooting out of the lower holes will be under higher pressure than the higher ones and, consequently, will shoot out at a higher speed. The water from the lower holes will project farther than the higher ones.
  
• If relative density disequilibrium is correct, there is no pressure gradient the water will come out at the same speed from each hole. The trajectories of the ejected water will all look the same until the level of the water in the tube reaches the holes.

Another Test: the atmosphere is a fluid mixture of different gases.

• If gravity is correct, we will observe a pressure gradient. Higher altitudes will have lower pressures than lower ones. There will be no definite boundary with the vacuum of space but a gradual reduction to the pressure of gas in the Solar System.
  
• If relative density disequilibrium is correct, then the only pressure gradient we shall observe is due to hot air rising. There will be a definite boundary with space (assuming no container). Air molecules will feel a force that will try to arrange them by density. Denser gases will tend accumulate at lower altitudes when weather conditions are not mixing them.
  

I find it both interesting and ironic that many flerfs claim it impossible to have a boundary between gas pressure and a vacuum without a physical barrier, yet their model predicts one.

I have not done the mathematics but molecules intrinsically seeking to arrange themselves by density does seem to violate the Second Law of Thermodynamics. If this is the case, then that is truly ironic.

Density and Weight

The weight of an object, according to gravity, is its mass multiplied by acceleration. It is as simple as that. For an objects apparent weight, it is necessary to take into consideration the buoyant force acting on it as well.

I have not seen any mathematics relating to weight and relative density disequilibrium. I don't know how weight is calculated in this model. If anyone knows, please let me know. What I do know is that the disequilibrium force is proportional to density as well as mass.

The Test: find an easily compressible material. Put equal amounts of this material on a balance. We know we have the same weight on either side. Compress the material on one side, thereby, increasing its density.

• If acceleration due to gravity is correct, the only property that affects the weight is mass. After compressing one, the two samples of material retain the same weight and remain balanced.
• If relative density disequilibrium is correct, not only mass but also density affects weight. The compressed sample will weigh more and the balance tips on this side. 

Free-Fall

An object that is falling, according to the Theory of General Relativity, does not feel gravity and has no weight. This same object is still in a state of relative density disequilibrium.

The Test: take two immiscible liquids, say oil and water. Water added to oil sinks to the bottom. Oil added to water sits on top. Shake oil and water and it separates back into the two layers. But what about in free-fall?

• If acceleration due to gravity is correct, the oil and water have no weight and will not reform into two layers.
• If relative density disequilibrium is correct, the oil and water will seek their density equilibrium and separate into two layers.
  

Axis Cube

The Fisher and Windmill cubes don't offer a significantly higher challenge to a person proficient in solving a standard 3x3x3. The only additional aspect is controlling the orientations of the pieces that behave like centres.

The Axis Cube, also known as the Axel Cube, is a significant step up in difficulty. It is still based in the 3x3x3 mechanism but the shape modification makes the puzzle confusing, especially when picking the puzzle up for the first time. Familiarity should be gained with the workings of the puzzle; that is knowing which pieces behave like edges, corners and centres, and understanding what corresponds to a face on a normal cube. Once this hurdle is crossed, it should be easier to solve.

The way the pieces are cut means that a equivalent to a face, as we understand it from a regular cube, is not one colour and not a side of the cube. The picture below shows a "face" rotated by 180°.

 

Solution

The puzzle shape shifts into a chaotic jumble when scrambled. It can be confusing when figuring out which pieces behave like corners, edges and centres, and how they go together during a solve.

Step 1: Cross

Intuitive step but care needs to be taken to ensure that the centres on the E-layer are orientated correctly as the edges of the cross are put into place.

Step 2: First Two Layers (F2L)

Normal F2L and block building techniques work here as one might expect. Recognition will be an issue at first until familiarity is built up with the shaped pieces.

Step 3: Edges Last Layer

At this point, I see which is easier to solve first, edges or corners. If there is no advantage either way I go for corners. During the solve when I was photographing for it this post, I decided to go with edges, because that was easier.

Step 4: Solve the Corners

Next solve the corners.

Step 5: Orientate Last Layer Centre

The way I solve this puzzle, sometimes leaves the final centre rotated by 180°. So there might be one final step to finish off the solve.

Windmill Cube

After the Fisher cube, the next shape modification I got was the Windmill Cube, also known as the Fenghuolin cube. I call it the Windmill Cube as I am don't know the correct pronunciation of "Fenghuolin".

It is called a Windmill cube because it is possible to create this windmill pattern...

I think this is a better puzzle than the Fisher Cube because the edges on the equatorial layer must be orientated correctly during the solve. This does eliminate the possibility of an odd-parity case when orientating the edges on the final layer.

 

Solution

Solving this puzzle is not difficult for anyone who can solve a standard 3x3x3. The only additional complication, except for solving some edges by shape rather than colour, is the orientation of the centres on the equatorial layer.

The scrambled puzzle shape shifts.

Step 1: Cross

Solve the white edges. Intuitive

Step 2: First Two Layers (F2L)

Normal F2L and block building techniques work here. The only difference is that the edges are solved by shape rather than by colour. It is not difficult once you are used to it.

Step 3: Orientate Yellow Edges

I orientate the final layer edges at this point, to make the puzzle easier to handle for the rest of the solve.

Step 4: Solve Final Layer Corners

Care needs to be taken to avoid rotating the centres on the equatorial layer.

Step 5: Permute Yellow Edges

Again, care needs to be taken to avoid rotating the centres on the equatorial layer.

And there it is solved.

 

Fisher Cube

The Fisher cube was created by Tony Fisher in the 1980s. If this was not the first shape modification of the cube, it was one of the first. It is not a bad first puzzle to try, if you want to play with something a little different from a standard 3x3x3.

On the U- and D-faces the corners behave like edges and vice versa. On the E-layer, the edges behave like centres and vice versa. 

 

Solving the Fisher Cube

Solving a Fisher Cube is not a whole lot different from solving a normal cube. After getting used to the configuration the only additional complication is ensuring the pieces that behave like centres are orientated correctly.

When scrambled, the puzzle shape shifts into a chaotic shape.

Step 1: White Cross

Intuitive. The trick here is to align the pieces that behave like centres on the E-layer.

Step 2: First Two Layers (F2L)

Normal F2L techniques work here. If anything it is slightly easier than on a standard 3x3x3 because the pieces that behave like edges on the E-layer have no orientation.

Step 3: Orientate Yellow Edges

I this stage I orientate the final layer-edges. I leave permuting them until later. I find the puzzle easier to handle with the edges facing the right way.

 There is the additional complication of a possible odd case parity where there could be one or three of the edges rotated. This is very easy to fix: just orientate one of them with an E-layer edge.

Step 4: Solve Corners 

I then do things the old fashioned way of permuting the corners first, before rotating them. I don't know which advanced algorithms affect the centres on the E-layer and which don't. I should much rather keep things simple and use basic algorithms that I know are super cube safe.

Step 5: Permute Edges

The trick here is do this without rotating the pieces that behave like edges on the E-layer. I do it using Sune algorithms.

TomZ Constrained Cube 90

As the name suggests, each face is constrained to turn by a quarter turn. Constrained Cubes Ultimate and 180 had enough freedom of movement to allow the use of normal algorithms, albeit adapted to contend with the puzzles' turning limitations. This is no longer the case with Constrained Cube 90. Losing the option to execute a Sune algorithm removes many solving options from rotating corners to permuting edges.

 

Solution

Step 1: Solve the White Edges

Intuitive

I have not figured out if this is the optimum way of solving the puzzle but I solve the white cross in such a way that one edge can be turned in the opposite direction from the other three. For example, one turns clockwise, the others anti-clockwise. This gives me one corner where I can execute a sledgehammer algorithm.

White becomes the bottom face.

Step 2: Solve the E-Layer Edges

Intuitive.

This is extremely fiddly. Moving edges around the puzzle is not easy with turning options so constrained.

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.

Step 3: Permute the Yellow Edges

This is a challenging step that requires some thought.

Step 4: Permute the Corners

This is fiddly but not as difficult as as it looks.

Step 5: Orientate the Corners

This step is not that difficult, especially compared with the steps.

This is a very challenging puzzle and, hence, all the more satisfying to solve. It is least an order of magnitude more difficult to solve than its brothers, Constrained Cubes Ultimate and 180. I would advise becoming familiar with at least one of those before tackling this puzzle.

TomZ Constrained Cube 180

As the name suggests, all of the sides on this puzzle are constrained to a half turn. It is slightly more difficult than the Constrained Cube Ultimate.

 

Solution

I did not find it necessary to do anything outlandish to solve the puzzle, following steps that would be familiar to veteran twisty puzzle-solvers.

 

Step 1: Solve the White Edges

Intuitive. The easiest way is to get the edges into position and then double turn the requisite face to put the edge into position.

 

I prefer to have the green, orange, blue and red centres fully turned in one direction or the other. The centres can be flipped later on to allow flexibility in turning. I don't worry about the white centre but it will affect the yellow centre later on. If white is fully turned, then the yellow centre will be in the middle of its turning arc when solved and vice versa.

 

Step 2: First Two Layers

Normal block building or F2L techniques work here but with the need to work around the constrained faces. It is fiddly but I did not find it too difficult.

Step 3: Orientate Yellow Edges

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

 

Step 4: Solve the Yellow Corners

Normal algorithms can be used here but attention must be paid to how faces turn. Some flipping of centres may be necessary.

 

Step 5: Permute the Yellow Edges

The final step is a little fiddly but not too difficult.

This is a satisfying puzzle to solve. Compared with the Constrained Cube Ultimate, the loss of freedom on the yellow face has a greater impact than gaining the ability to turn the white face. This makes the puzzle slightly more difficult to solve.