Algebra. and the Rubik’s Cube. Scott Vaughen, Professor of Mathematics Miami Dade College. Group Theory. Group theory is a branch of modern algebra. It could be called the mathematical language of symmetry. The first principles of group theory were
and the Rubik’s Cube
Scott Vaughen, Professor of Mathematics
Miami Dade College
language of symmetry.
discovered by Evariste Galois in the 1830s.
The exact circumstances of the duel may never be known.
are equations of degree 2 (because 2 is the highest exponent).
involving only the operations of arithmetic (adding, subtracting, multiplying and dividing) and use of square roots.
e*a = a*e = a.
a*a-1 = a-1*a = e.
(a*b)*c = a*(b*c)
The set of integers with the operation of addition is an example of a
x + 0 = 0 + x = x
The set of rational numbers without 0 and the operation of multiplication
is an example of a group:
x*1 = 1*x = x
Note: 0 is not included in this group because it would have no inverse by multiplication (that is, there is no number times 0 that would produce the identity 1).
The set of all possible rotations and reflections of the stop sign, with
composition as the operation, is an example of a group:
This is called the dihedral group (of the octagon) with 16 elements.
To verify that this is really a group, we can check each condition…
This is analogous to the algebraic equation 1*x = x.
The Rubik’s cube is another example of a group:
The number of elements in the Rubik’s cube group is 43,252,003,274,489,856,000 because there are this many possible distinct permutations of the cube that can be reached by legal moves.
A legal move is any combination of turns of any of the faces. Illegal moves can result from breaking the cube apart or peeling off the stickers.
Any two legal moves can be combined and the result would be one of those distinct permutations.
We can check that the properties of a group are satisfied…
To verify that the Rubik’s cube group satisfies the three properties of a
Consider: (5 – 3) – 2 = 5 – (3 – 2). This is not true, because
0 (on the left) is not equal to 4 (on the right).
This means subtraction with the integers is not a group.
therefore, the group of integers with addition is Abelian.
therefore, the group of rationals (without 0) is Abelian.
Apply move R followed by D Applying move D followed by R.
The fact that the operation “followed by” in the Rubik’s cube group is not commutative is what makes the puzzle so difficult (aside from the large number of permutations). Imagine if it were commutative – this would mean that to solve the puzzle it would not matter what order the moves were applied: simply rotating each face an appropriate number of times would suffice!
For example, we solve the equation 3x + 10 = 22 as follows
Apply the additive inverse of 10 to both sides to isolate the 3x term
Apply the multiplicative inverse of 3 to both sides to isolate the factor x.
In this example, we are using the group of integers with addition …
because we used 10 and its inverse -10
And we are using the group of rationals and multiplication …
because we used 3 and its inverse 1/3.