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First off, a metric is a function (call it D) that assigns a

non-negitive number to every pair of points in some set. This number

is to be thought off as the distance between those two points. The

function must satisfy the following:

For all a, b and c 1) D(a,b) >= 0 2) D(a,b) = D(b,a) 3) D(a,b) = 0 if and only if a = b 4) D(a,b) + D(b,c) >= D(a,c)

(Number 4 is usually called the "triangle inequality". It is the

constraint that most makes D act like a distance, and not something

random.)

We wish to construct a metric on the set of all attainable cube

configurations. So from now on, those lower case letters will

represent cube configurations.

Now we have recently been talking a lot about methods of counting the

number of "twists" that it takes to perform certain manipulations of

the cube. We have been looking for a function (call it T) that

assigns a non-negitive integer to each manipulation. I claim that it

is obvious that any such function should satisfy the following:

For all M and N 1) T(M) >= 0 3) T(M) = 0 if and only if M = I (I is the identity manipulation) 4) T(M) + T(N) >= T(MN)

(We adopt the convention of using upper case letters to represent

manipulations. Also we shall use M' to denote the inverse manipulation

from M.)

Now manipulations can be applied to configurations to yeild other

configurations. We use aM to denote the configuration resulting from

applyint the manipulation M to the configuration a. (Note that

(aM)N = a(MN), so we may omit the parens and simply write aMN.)

Now how may we use our twist measuring function T to obtain a metric

on the configurations? Again I think it is obvious that we wish the

relationship D(a,aN) = T(N) to be true for all configurations a, and

all manipulations N.

It is easy to show that given that D(a,aN) = T(N), metric property

number 1 is equivalent to twist measure property number 1. Similarly

for numbers 3 and 4. But what about metric property number 2?

Well, if T(N) = D(a,aN), and D(a,aN) = D(aN,a) (property 2!), and a = aNN', then we have that T(N) = D(aN,aNN') = T(N'). So the missing property of twist measures must be that T(N) = T(N').

So this means that if we agree that T(L) = 1, and we like metrics

(how can we use words like "distance" unless we have a metric?), then

T(LLL) = T(L') = T(L) = 1. We can argue about T(LL) some other time!