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.