# Rubik’s Cube | The Carbon Copy Model

### New Rubik's Cube Views

Seeing the Rubik's Cube differently opens up new avenues of knowledge.

1. The Cell Model – From 27 to ?
2. The 3 Cell Roles – It’s How We Roll
3. The Roll Model – Going Places

### The Diagonal Cells Network

If you want to master the Rubik's Cube, discover its secrets, ( and learn some simple maths ) – you’ve come to the right place.

### Same Recipe – Different Page

It’s the same recipe – page after page.

Turn something boring and obvious – into something profound.

Here is an example.

## Boring But True

Boring But True ➜ The ( middle left high ) edge is next door to the ( near left high ) corner.

The drab and boring truth turns into a network linking all 27 cells to their next door neighbour! Read More »

### The 0 1 and 2 Cell Separations

Boring But True ➜ The midpoints of the ( near, left, low ) corner and the ( near, middle, high ) edge, are separated by

1. 0 cell widths on the axis ➜ near and near
2. 1 cell width  on the axis ➜ left and middle
3. 2 cell widths on the axis ➜ low and high

The network connects every cell pair that is 0, 1 and 2 cell widths apart.

### Strategy – Do Loads – Without Doing Much

Now children can solve the Rubik's Cube – even when you can count their years of life on the fingers of one hand. How! It’s thanks to a surprising strategy

1. convey a lotwithout saying much
2. know a lotwithout knowing much
3. do a lotwithout doing much

QuestionHow can we say a lot without saying much?
Let’s use the example in the above section to answer this question.

### Which cell pairs are 0, 1 and 2 Cell Widths Apart?

Tell me the cell pairs separated by 0 cell widths on one axis, 1 cell width on another and 2 cell widths on the third.

Let’s jot down 5 – then stop and think.

1. ( near left low ) and ( near middle high ) ➜ Offsets – x=0, y=1, z=2
2. ( near left low ) and ( far left middle ) ➜ Offsets – x=2, y=0, z=1
3. ( far middle high ) and ( middle middle low ) ➜ Offsets – x=1, y=0, z=2
4. ( near left high ) and ( near right middle ) ➜ Offsets – x=0, y=2, z=1
5. ( far left middle ) and ( near middle middle ) ➜ Offsets – x=2, y=1, z=0

Stop! How many pairs? Each corner pairs with 6 edges. That’s 48 pairs.

Each edge pairs with 4 corners and 2 centres. We’ve counted the corner pairings so the 12 edges contribute 12 × 2 = 24 pairs.

We must list 72 cell pairs ( 48 + 24 ). Our strategy is to convey a lot, without saying much – so we can’t list 72 cell pairs. Help!

### The Unambiguous Global Truths

Let’s fast fwd to the answer. We can convey a lot without saying much in two ways. Either 12 sentences, or 7 joined up dots!

1. corner ➜ x diagonal y edge and corner ➜ x diagonal z edge
2. corner ➜ y diagonal x edge and corner ➜ y diagonal z edge
3. corner ➜ z diagonal x edge and corner ➜ z diagonal y edge
4. x edge ➜ x diagonal y centre and x edge ➜ x diagonal z centre
5. y edge ➜ y diagonal x centre and y edge ➜ y diagonal z centre
6. z edge ➜ z diagonal x centre and z edge ➜ z diagonal y centre

These statements are unambiguous. A corner has one and only one x diagonal y edge – This global truth replaces 8 local truths. But we can do better.

### Just 7 Connected Dots

It conveys a lot without saying much. All 72 cell pairs are here – the 12 sentences are here – with 7 dots and 9 lines – it’s all here!

We’ve stuck to our strategy like bees to honey. We’ve said loads – without saying much.

### From Pascal’s Triangle to Rubik's Cube

If your multiply Pascal’s Triangle by the 3rd row of Pascal’s Triangle you get the 27 Rubik's Cube cells.

But which cells go where?

### The Cell Triangle Dilemma

I have found only one solution to this dilemma. Could there be 2, 3, 4 … or more?

The 6 Rubik's Cube cells are the highlight of this fascinating view. What do they mean? »

### The Trinomial Coefficients

It is rare, to find both

1. the binomial coefficients { 1, 3, 3, 1 } and
2. the trinomial coefficients { 1, 3, 6, 7, 6, 3, 1 }

in the same place. The trinomial coefficients allow us to cluster the 27 Rubik's Cube cells in a 1 + 3 + 6 + 7 + 6 + 3 + 1 arrangement.