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Christopher Mowla Math 3900 November 18, 2011. The Rubik’s Cube. Rubik’s Cubes of Order n. A Rubik’s Cube of order n is referred to as the n x n x n cube, where A cube with n = an even integer value is referred to as an even cube .

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Christopher Mowla

Math 3900

November 18, 2011



Rubik s cubes of order n l.jpg
Rubik’s Cubes of Order n

  • A Rubik’s Cube of order n is referred to as the nxnxn cube, where

  • A cube with n = an even integer value is referred to as an even cube.

  • A cube with n = an odd integer value is referred to as an odd cube.

  • A cube of size n = 4 and larger are referred to as big cubes.



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n = 2


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n = 4


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n = 5


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n = 6


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n = 7


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n = 9


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n = 11



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Corners: They each have 3 stickers.


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Edges: They each have 2 stickers.


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(Fixed) centers: They each have 1 sticker.



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Piece Types of the nxnxn

  • Unlike the common 3x3x3 Rubik’s Cube size, there are 4 additional types of pieces on the nxnxn cube, in general.

  • Therefore, there is a total of 7 different pieces types which can be seen on larger cube sizes.

  • Side Note: We will exclude the 1x1x1 cube size for consistency (it’s an exception in more than one respect).


Types of pieces 7 l.jpg
Types of Pieces (7)

  • Corners

  • Edges

    • Middle Edges

    • Wing Edges

  • Centers

    • Fixed Centers

    • Non-Fixed Centers

      • X-center pieces, oblique center pieces, and + center pieces.


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Constructing the n x n x n l.jpg
Constructing the all cube sizes fromnxnxn


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Odd Cube Size Construction all cube sizes from


2x2x2 3x3x3 l.jpg
2x2x2 all cube sizes from 3x3x3


3x3x3 5x5x5 l.jpg
3x3x3 all cube sizes from 5x5x5


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New piece types on the 5x5x5 all cube sizes from

  • There are three new piece types on the 5x5x5: wing edges, X-center pieces, and + center pieces.


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Wing Edges all cube sizes from



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X-Center Pieces due to their symmetry.)


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+ Center Pieces because they form an X about the composite center.


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5x5x5 T-center pieces as well) because they form a plus sign about the big cube center. 7x7x7


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Wing Edges T-center pieces as well) because they form a plus sign about the big cube center.


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  • For the 7x7x7, we have a second set of 24 wing edges. T-center pieces as well) because they form a plus sign about the big cube center.

  • The term orbit is used to differentiate different sets of wing edges. It means where the pieces are able to move.

    • On the 7x7x7 cube, we have 2 orbits of wing edges.


X center pieces38 l.jpg
X Center Pieces T-center pieces as well) because they form a plus sign about the big cube center.


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+ Center Pieces T-center pieces as well) because they form a plus sign about the big cube center.


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New piece type on the 7x7x7 T-center pieces as well) because they form a plus sign about the big cube center.


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Oblique Center Pieces T-center pieces as well) because they form a plus sign about the big cube center.


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  • The term T-center pieces as well) because they form a plus sign about the big cube center.oblique is used to describe these center pieces because they are neither X-center pieces nor + center pieces.


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Permutations T-center pieces as well) because they form a plus sign about the big cube center.

  • A permutation is an arrangement of objects of the same type in some order.

  • Permutations can be decomposed into cycles.


Definition of a cycle l.jpg
Definition of a Cycle T-center pieces as well) because they form a plus sign about the big cube center.

  • An n-cycle is moving n pieces (2 or more) of the same type (i.e. edges, corners, or centers) at the same time so that, when the algorithm generating the cycle is repeated exactlyn times (and not until then), the cube will be restored to the original state it was in.


Examples of n cycles l.jpg
Examples of T-center pieces as well) because they form a plus sign about the big cube center.N-Cycles


3 cycles permutations of middle edges and corners on the 3x3x3 l.jpg
3-Cycles T-center pieces as well) because they form a plus sign about the big cube center.(Permutations of Middle Edges and Corners on the 3x3x3)


2 cycles permutations of wing edges on the 5x5x5 l.jpg
2-Cycles T-center pieces as well) because they form a plus sign about the big cube center.(Permutations of Wing Edges on the 5x5x5)


4 cycles permutations of wing edges on the 5x5x5 l.jpg
4-Cycles T-center pieces as well) because they form a plus sign about the big cube center.(Permutations of Wing Edges on the 5x5x5)


Combinations of disjoint cycles l.jpg
Combinations of Disjoint Cycles T-center pieces as well) because they form a plus sign about the big cube center.

  • Definition:If an algorithm affects 4 or more pieces of the same type on a cube (corners, edges, or centers), all of the pieces affected need not be part of an n-cycle.

  • In other words, n pieces of the same type affected by an algorithm  an n-cycle of pieces.


2 2 cycles permutations of middle edges on the 3x3x3 l.jpg
2 2-Cycles T-center pieces as well) because they form a plus sign about the big cube center.(Permutations of Middle Edges on the 3x3x3)


2 2 cycles permutations of wing edges on the 5x5x5 l.jpg
2 2-Cycles T-center pieces as well) because they form a plus sign about the big cube center.(Permutations of Wing Edges on the 5x5x5)


2 2 cycles permutations of wing edges on the 5x5x553 l.jpg
2 2-Cycles T-center pieces as well) because they form a plus sign about the big cube center.(Permutations of Wing Edges on the 5x5x5)


2 3 cycles permutations of wing edges on the 5x5x5 l.jpg
2 3-Cycles T-center pieces as well) because they form a plus sign about the big cube center.(Permutations of Wing Edges on the 5x5x5)


Calculating the number of permutations positions on the n x n x n cube for n 1 l.jpg
Calculating the Number of Permutations (Positions) on the T-center pieces as well) because they form a plus sign about the big cube center.nxnxn Cube (for n>1)

  • By the multiplication rule of probability, the main idea is to determine the number of possible permutations for each piece type and multiply them all together.


More background information l.jpg
More Background Information T-center pieces as well) because they form a plus sign about the big cube center.


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Permutation Notation T-center pieces as well) because they form a plus sign about the big cube center.

  • Permutations are bijective maps (one to one and onto) and therefore can be represented by:


Classifying permutations l.jpg
Classifying Permutations T-center pieces as well) because they form a plus sign about the big cube center.

  • A permutation can be either even or odd.

    • An even permutation is a permutation in which can be solved (achieve the identity) in an even number of 2-cycle swaps.

      • The identity is an even permutation.

    • An odd permutation is a permutation in which can be solved (achieve the identity) in an odd number of 2-cycle swaps.


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Rubik’s Cube Interpretation of Permutations to the number of odd permutations for sets of 2 or more objects.

  • Permutations on the nxnxn Rubik’s Cube mean how many possible positions there are for each piece type combined with the rest of the other types of pieces.


A second meaning of the term permutation on rubik s cubes l.jpg

As with the mathematical definition of permutations, the term permutation is also used to describe the location where each piece is located.

The first row of the permutation notation can be assigned to fixed slots on a cube.For example, the slots containing the wing edges in the last layer of a 5x5x5 can correspond to the permutation notation as follows:

A second meaning of the term “permutation” on Rubik’s Cubes


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Each number on the cube designates the slot number (the first row of the 2-line permutation notation) and f(1) is the wing edge piece in slot 1, f(2) is the wing edge piece in slot 2, etc.

6

5

7

4

8

3

1

2


Orientation l.jpg
Orientation first row of the 2-line permutation notation) and f(1) is the wing edge piece in slot 1, f(2) is the wing edge piece in slot 2, etc.

  • The term orientation refers to how a particular piece type on a cube is twisted in its location.


Orientation of corners l.jpg
Orientation of Corners first row of the 2-line permutation notation) and f(1) is the wing edge piece in slot 1, f(2) is the wing edge piece in slot 2, etc.

  • Corners can be twisted 3 different ways, and thus have 3 orientations each independently.


Orientation of middle edges l.jpg
Orientation of Middle Edges first row of the 2-line permutation notation) and f(1) is the wing edge piece in slot 1, f(2) is the wing edge piece in slot 2, etc.

  • Edges can be oriented in two ways: flipped or not flipped.


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The corners l.jpg
The Corners edge that is in the same composite edge.

  • There are 8 Corners.8!

  • Each corner can be oriented (twisted) in 3 ways independently.


Laws of the cube 1 l.jpg
Laws of the Cube (1) edge that is in the same composite edge.

  • Actually, by what is commonly referred to as “cube laws,” only 1/3 of the 3^8 orientations of corners are possible.

  • If we go back to the illustration of the three ways a corner can be twisted,


Slide74 l.jpg

  • Suppose the left most image represents the identity. Then: edge that is in the same composite edge.

    • The corner in the left image is turned clockwise 0 times from the identity.

    • The corner in the middle image is turned clockwise 1 time from the identity.

    • The corner in the right image is turned clockwise 2 times from the identity.

0

1

2


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  • Once one chooses a point of reference, that is, one fixes one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.For example, an impossible state is to have the entire cube solved except for one corner twisted, since 1 is not evenly divisible by 3.


Slide76 l.jpg

Orange one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • Neither can there be two corners twisted clockwise once since 1+1 = 2, which is not evenly divisible by 3.

  • On the other hand, you could have one corner twisted clockwise and the other twisted clockwise twice (or anti clockwise) since 1+2 = 3.

Orange


The corners77 l.jpg
The Corners one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • There are 8 Corners.8!

  • Each corner can be oriented (twisted) in 3 ways independently.

  • Thus we divide this result by 3 to get


Even cube symmetry l.jpg
Even Cube Symmetry one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • On even cubes, we can always fix a corner to be solved (in the identity) and eliminate the number of permutations it contributes to the whole.

    • This is because on even cubes, there are no fixed centers and therefore there is no unique way to set the cube on a table, when considering all possible ways it can be scrambled.

    • Since each corner can be placed in 8 different locations and twisted in 3 different ways, we divide the total number of corner permutations by to have a value of zero for odd values of n.


Corners conclusion l.jpg
Corners Conclusion one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • Therefore, the total number of possible permutations of corners on the nxnxn cube, taking into account the cube law and even cube symmetry is:


Middle edges odd cubes only l.jpg
Middle Edges (Odd Cubes Only) one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • There are 12 middle edges.

  • Each edge can be flipped in two ways independently.


Laws of the cube 2 l.jpg
Laws of the Cube (2) one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • Similar to the reduction of the possible orientations of the corners, by the “cube laws,” only 1/2 of the 2^12 orientations of middle edges are possible.

  • Only an even number of middle edges can be flipped.


Middle edges odd cubes only82 l.jpg
Middle Edges (Odd Cubes Only) one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • There are 12 middle edges.

  • Each edge can be flipped in two ways independently.

  • By the “cube law,” we divide this number by 2 to have


Laws of the cube 3 l.jpg
Laws of the Cube (3) one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • It also turns out that the middle edges are in an odd permutation if and only if the corners are in an odd permutation.

  • Recall that odd permutations make up half of all permutations of 3 or more objects.

  • Since half of the permutations of two different piece types are dependent on each other, only half of the permutations of one of them is allowed on the cube.


Middle edges final l.jpg
Middle Edges Final one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • Since middle edges only are in odd cubes (and thus the presence of both the corners and middle edges only occurs on odd cubes to cause the ½ reduction), we divide what we had previously by 2 and raise it n mod 2 to have a value of zero for even cubes.


Wing edges85 l.jpg
Wing Edges one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • Whether even or odd cube sizes, big cube sizes (larger than the 3x3x3) have sets (orbits) of 24 wing edges.


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  • For example, the 6x6x6 has one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3. orbits of wing edges.


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  • The 7x7x7 has one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.orbits of wing edges.


Slide88 l.jpg

  • Since there are orbits of wing edges on the one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.nxnxn cube, containing 24 wing edges each, the total number of permutations of the wing edges is:


X center pieces center pieces and oblique center pieces l.jpg
X-Center Pieces, + Center pieces, and Oblique Center Pieces. one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.

  • Similar to wing edges, big cubes have a set number of orbits of all non-fixed center pieces.

  • To determine the number of orbits for the nxnxn, let’s observe a large big cube center.


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  • Recall that the term “orbit” refers to where pieces can move to.

  • Therefore, since all 4 quadrants can move to each other, we need only consider one quadrant to count the number of orbits of non-fixed center pieces.

  • Clearly for even cube sizes, the number of non-fixed center orbits is the amount of squares in each quadrant:


Slide94 l.jpg

For odd cubes, notice that the white cross also consists of 4 symmetrically equivalent blocks. If we rotate the face 3 times, all of these

blocks will go to each other.

(These blocks make up the orbits of the + center pieces.)


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  • Note that there are 4 of each non-fixed center piece type (since we divided the composite center into 4 equal parts) in every one of the 6 faces of the cube, or 4(6) = 24 non-fixed center pieces in each orbit.

  • This gives the total number of possible permutations of non-fixed center pieces on odd and even cubes to be:


Non fixed centers final l.jpg

Lastly, just as there were 4 pieces of each type in each face.

Each face is a different color.

Non-fixed center pieces are indistinguishable.

Therefore, we must divide by:

This gives us a final number of:

Non-Fixed Centers Final


The formula l.jpg
The Formula face.

  • Multiplying the total number of permutations for the corners, middle edges, wing edges, and non-fixed center pieces all together, we achieve the formula for the number of positions of the nxnxn Rubik’s Cube.


Examples l.jpg
Examples face.

  • n =2: 3,674,160

  • n = 3: 43, 252, 003, 274, 489, 856, 000.

  • n = 4: Approximately 7.40 x 10 ^45.

  • n = 7: Approximately 1.95 x 10^160.

  • n = 11: Approximately 1.09 x 10^425.

  • n = 100: Approximately 2.35 x 10^ 38, 415


Supercubes l.jpg
Supercubes face.

  • Probably everyone has either seen or heard of the Sudoku 3x3x3 cube:


Supercubes101 l.jpg

Notice that the numbers of the center pieces are all turned to match the direction of the numbers on the corners and middle edges perfectly.

Supercubes


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Definition to match the direction of the numbers on the corners and middle edges perfectly.

  • An nxnxnsupercube is similar to the nxnxn cube, except that all center piece types, both the fixed center pieces and the 3 non-fixed center pieces types (X-center pieces, + center pieces, and oblique center pieces) are distinguishable.


Slide107 l.jpg

  • In other words, the regular 6-colored to match the direction of the numbers on the corners and middle edges perfectly.nxnxn cube and the nxnxn supercube have the same number of permutations for the “Cage” portion of the cube.


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Calculating the number of positions of the n x n x n supercube l.jpg
Calculating the Number of Positions of the sizes begin with nxnxn supercube.

  • We merely need to multiply the formula for the regular nxnxn by a factor determined by the additional number of ways each color of center pieces can swap with each other (in any permutation from the formula for the regular 6-colored nxnxn cube).


Fixed centers odd cubes only l.jpg
Fixed Centers (Odd Cubes Only) sizes begin with

  • There are 6 fixed centers on every odd cube.

  • Each center can be rotated in 4 directions in its location.

  • This gives total permutations.


Laws of the cube 4 l.jpg
Laws of the Cube (4) sizes begin with

  • Just as the middle edges are in an odd permutation if and only if the corners are in an odd permutation, it carries on to fixed centers as well.

  • That is, middle edges, corners, and fixed center pieces are all dependent on each other. If one of them has an odd permutation (or even permutation), so do the other two.


Fixed centers odd cubes only112 l.jpg

There are 6 fixed centers on every odd cube. sizes begin with

Each center can be rotated in 4 directions in its location.

This gives total permutations.

By the “cube law,” we divide this number by 2, because only half of the fixed center permutations are allowed:

Fixed Centers (Odd Cubes Only)


Non fixed center pieces l.jpg

From calculating the formula for the regular sizes begin with nxnxn, we found that there are:orbits of non-fixed center pieces.

In each orbit, there were 24/6 = 4 center pieces of each color.

Now that center pieces ARE distinguishable, we have an additional 4! permutations of each color. There are 6 colors, which gives us

Non-Fixed Center Pieces


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Laws of the cube 5 l.jpg
Laws of the Cube (5) since that is the number of different sets we are multiplying together:

  • Now that non-fixed center pieces are unique to their positions,

    • An odd permutation of wing edges exists if and only if an odd permutation exists in the + center pieces and oblique center pieces.

    • An odd permutation of corners exists if and only if there is an odd permutation all of the X-center pieces and at least+ center and oblique center pieces.


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  • Despite the complexity of these relationships, the total number of permutations shared between the non-fixed center pieces and the corners and wing edges is just:

  • Therefore, we divide by this amount:, which means we only consider half of the permutations of each orbit of non-fixed centers.


The formula for the supercube l.jpg
The Formula for the Supercube number of permutations shared between the non-fixed center pieces and the corners and wing edges is just:

  • We multiply the formula for the regular 6-colored nxnxn by all of the factors just discussed to achieve the formula for the supercube:


The factor increase examples l.jpg
The Factor Increase (Examples) number of permutations shared between the non-fixed center pieces and the corners and wing edges is just:

  • The number of permutations of the regular nxnxn are increased by:

  • n = 2: 1

  • n = 3: 2,048.

  • n = 4: 95, 551, 488.

  • n = 7: Approx. 1.56 x 10 ^ 51.

  • n = 11: Approx. 8.24 x 10 ^ 162.

  • n = 100: Approx. 3.55 x 10 ^ 19, 160.


Relationship between wing edges and non fixed center pieces l.jpg
Relationship Between Wing Edges and Non-Fixed Center Pieces. number of permutations shared between the non-fixed center pieces and the corners and wing edges is just:

  • We need to find a formula to represent the total number of non-fixed center orbits in an odd permutation for a given number of orbits of wings in an odd permutation.


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  • We will use a 13x13x13. number of permutations shared between the non-fixed center pieces and the corners and wing edges is just:


Slide121 l.jpg


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  • Reflecting this to the top, clearly we have orbits in an odd permutation now in the yellow column (just focus on the yellow column, not its corresponding yellow row).(n-2)-2-2 center orbits left with an odd permutation in both slices. That is, 2(n-2-2(2)).


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Example l.jpg
Example arbitrary wing edge orbits, the number of non-fixed center orbits in an odd permutation with the wings would be the same for any combination of the same number of wing edge orbits.

  • On the 1,000 x 1,000 x 1,000 cube, if w = 65 orbits of wings have an odd permutation, thenorbits of non-fixed center orbits (consisting of + and oblique centers) has an odd permutation.


The maximum formula l.jpg
The Maximum Formula arbitrary wing edge orbits, the number of non-fixed center orbits in an odd permutation with the wings would be the same for any combination of the same number of wing edge orbits.

  • It’s useful for us to know the maximum number of non-fixed center orbits that can be in an odd permutation for a given size n.


Finding a maximum formula by the use of calculus l.jpg
Finding a Maximum Formula By the Use of Calculus arbitrary wing edge orbits, the number of non-fixed center orbits in an odd permutation with the wings would be the same for any combination of the same number of wing edge orbits.

  • The formula is NOT a 2 variable function becauseThus, the maximum value for a given n is the maximum value achieved by at least one w from that domain.


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  • We can differentiate with respect to arbitrary wing edge orbits, the number of non-fixed center orbits in an odd permutation with the wings would be the same for any combination of the same number of wing edge orbits.w, since that is the changing domain:

  • To verify that we will obtain the maximum value, we differentiate once more.


Slide145 l.jpg

Setting the first derivative equal to zero to find the critical point, we have:

Substituting w into , we get

By comparing the formula to the true maximum values, we just need to take the floor of this.


Slide146 l.jpg

We can see a 3-D graph representation of the formula critical point, we have:

very well by rewriting as .

n

W


Slide147 l.jpg

w critical point, we have:

n

  • The graph of the maximum formula

w

w

n

n


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Solutions for even cubes divisible by 4 l.jpg
Solutions for Even Cubes Divisible by 4 multiplied 1/8 by cosine.

  • Using the quadratic formula, we get:

  • Both solutions work! Therefore w is not unique for this cube size class.


Solutions for even cubes not divisible by 4 l.jpg
Solutions for Even Cubes Not Divisible by 4 multiplied 1/8 by cosine.

  • Using the quadratic formula, we get:

  • This is equal to the critical point we found, which is not surprising since the original formula worked for this cube class from the start.


Solutions for odd cubes l.jpg
Solutions for Odd Cubes multiplied 1/8 by cosine.

  • Using the quadratic formula,

    • w1 is true for every other odd integer starting with n = 5.

    • w2 is true for every other odd integer starting with n = 7.


Issues l.jpg
Issues multiplied 1/8 by cosine.

  • We do not have a single representation for the value of w that yields the maximum value.

  • If we approach this problem using the following 4th order non-homogeneous recursion relationwith the initial conditions:


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  • With ingenuity, we can find the solution: multiplied 1/8 by cosine.

  • Comparing with the original formula

  • Clearly , and we canrepresent w equal to that for all size cubes. However, we note that n evenly divisible by 4 has two solutions:



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Laws of the cube 5158 l.jpg
Laws of the Cube (5) multiplied 1/8 by cosine.

  • Now that non-fixed center pieces are unique to their positions,

    • An odd permutation of wing edges exists if and only if an odd permutation exists in the + center pieces and oblique center pieces.

    • An odd permutation of corners exists if and only if there is an odd permutation all of the X-center pieces and at least+ center and oblique center pieces.


Slide159 l.jpg

  • As we first were deriving , multiplied 1/8 by cosine.we learned that the X-center pieces were in a 2 4-cycle when inner layer slices were turned. Hence we should be able to understand the first bullet completely.

  • The second bullet is merely stating that X-center pieces are in an odd permutation if the corners are in an odd permutation.

  • We can see this using a similar image as before, just extrapolating the permutation to the outer layer.


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  • The only item left to interpret in the second bullet is: permutation) too.

  • The total non-fixed center pieces (besides X-center orbits which is equal to can be no less than the total number of non-fixed center orbits minus the maximum number of center orbits which can be in an odd permutation with wings, M(n).


References l.jpg
References permutation) too.

  • Cube Lawshttp://www.ryanheise.com/cube/cube_laws.html

  • Number of Permutationshttp://en.wikipedia.org/wiki/Rubik%27s_cube


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