# Rubik’s Cube | The Carbon Copy Model

### New Rubik's Cube Views

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

**The Cell Model – From 27 to ?****The 3 Cell Roles – It’s How We Roll****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.

*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

**0 cell widths**on the axis ➜ near and near**1 cell width**on the axis ➜ left and middle**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

**convey a lot**–*without saying much**know a lot*–**without knowing much****do a lot**–*without doing much*

**Question** – *How 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.

**( near left low )**and**( near middle high )**➜ Offsets – x=0, y=1, z=2**( near left low )**and**( far left middle )**➜ Offsets – x=2, y=0, z=1**( far middle high )**and**( middle middle low )**➜ Offsets – x=1, y=0, z=2**( near left high )**and**( near right middle )**➜ Offsets – x=0, y=2, z=1**( 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!

**corner ➜ x diagonal y edge**and**corner ➜ x diagonal z edge****corner ➜ y diagonal x edge**and**corner ➜ y diagonal z edge****corner ➜ z diagonal x edge**and**corner ➜ z diagonal y edge****x edge ➜ x diagonal y centre**and**x edge ➜ x diagonal z centre****y edge ➜ y diagonal x centre**and**y edge ➜ y diagonal z centre****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 3^{rd} 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

- the
**binomial coefficients**{ 1, 3, 3, 1 }*and* - 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.