**A Very Challenging Game**

Sebelum:

Selepas:

**The mathematics of the 3x3 Cube :**

As the centre pieces of each face of the Cube do not move, the total number of possible configurations is calculated by multiplying the number of possible arrangements of the corner pieces by the number of possible arrangements of edge pieces.

There are 8 corner pieces, so the number of possible arrangements equals 8! (or 8x7x6x5x4x3x2x1), which is 40320. Each corner piece has 3 different orientations, so this figure must be multiplied by 38 (3x3x3x3x3x3x3x3), which equals 6561. But when the Cube is almost complete, the number of possible moves diminishes, so the equation must be adjusted. In this case, once the second from last corner piece is placed, the last piece can have only one automatic orientation, so 6561 must be divided by 37, which is 2187. Finally, the total possible arrangements of corner pieces

40320 x 2187 = 88,179,840.

With the 12 Edge Pieces, the number of possible arrangements equals 12! (12x11x10…), which is 479,001,600. However, unlike corner pieces, it is impossible to exchange just two edge pieces, so once the third from last is placed, the remaining two can have only one possible arrangement, which means this figure must be divided by 2, leaving 239,500,800. Each edge piece has two different orientations, so this must now be multiplied by 212, which gives 6561. This figure must also be adjusted because once the third from last edge piece is placed, one of the remaining two can be reoriented but the last will always have a fixed orientation in relation to it. So 6561 must be divided by 211, which is 2048. Finally, the total possible arrangements of edge pieces

239,500,800 x 2048 = 490,497,638,400.

So the total possible arrangements of Rubik's Cube

88,179,840 x 490,497,638,400 = 43,252,003,274,489,856,000.

Or, to put it another way,

4.3 times 10 to the power of 19

Many 3×3×3 Rubik's Cube enthusiasts use a notation developed by David Singmaster to denote a sequence of moves, referred to as "Singmaster notation".[21] Its relative nature allows algorithms to be written in such a way that they can be applied regardless of which side is designated the top or how the colours are organised on a particular cube.

F (Front): the side currently facing you

B (Back): the side opposite the front

U (Up): the side above or on top of the front side

D (Down): the side opposite the top, underneath the Cube

L (Left): the side directly to the left of the front

R (Right): the side directly to the right of the front

ƒ (Front two layers): the side facing you and the corresponding middle layer

b (Back two layers): the side opposite the front and the corresponding middle layer

u (Up two layers) : the top side and the corresponding middle layer

d (Down two layers) : the bottom layer and the corresponding middle layer

l (Left two layers) : the side to the left of the front and the corresponding middle layer

r (Right two layers) : the side to the right of the front and the corresponding middle layer

x (rotate): rotate the Cube up (as in R)

y (rotate): rotate the Cube to the counter-clockwise (as in U)

z (rotate): rotate the Cube clockwise (as in F)

When a prime symbol ( ′ ) follows a letter, it denotes face counter-clockwise, while a letter without a prime symbol denotes a clockwise turn. A letter followed by a 2 (occasionally a superscript 2) denotes two turns, or a 180-degree turn. R is right side clockwise, but R' is right side counter-clockwise. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes. When x, y or z are primed, it is an indication that the cube must be rotated in the opposite direction. When they are squared, the cube must be rotated twice.

For methods using middle-layer turns (particularly corners-first methods) there is a generally accepted "MES" extension to the notation where letters M, E, and S denote middle layer turns. It was used e.g. in Marc Waterman's Algorithm.[22]

M (Middle): the layer between L and R, turn direction as L (top-down)

E (Equator): the layer between U and D, turn direction as D (left-right)

S (Standing): the layer between F and B, turn direction as F

The 4×4×4 and larger cubes use an extended notation to refer to the additional middle layers. Generally speaking, uppercase letters (F B U D L R) refer to the outermost portions of the cube (called faces). Lowercase letters (ƒ b u d ℓ r) refer to the inner portions of the cube (called slices). An asterisk (L*), a number in front of it (2L), or two layers in parenthesis (Lℓ), means to turn the two layers at the same time (both the inner and the outer left faces) For example: (Rr)' ℓ2 ƒ' means to turn the two rightmost layers counterclockwise, then the left inner layer twice, and then the inner front layer counterclockwise.

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