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.