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PROFESSOR: The easiest way--
there are lots of ways to solve

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this problem, but I think the
easiest and most elegant way

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to solve it is--

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the thing that makes
it hard is the fact

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that these cubes can be
rotated and moved around,

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so the easiest thing is,
imagine-- let's just sit

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a cube, pick one,
put it on the table,

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and let's paint it
with six colors.

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And let's pick the first face.

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Maybe we paint the top face.

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Pick a color for that.

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Well, we have six choices.

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OK, now we got to
paint another face.

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Maybe we paint
the one facing us.

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We have five choices left for
the color we choose there,

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because we're not allowed
to paint two faces the same.

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And then we proceed on down.

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We have four choices
for the next--

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3-2-1.

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And so, we get 6 times, 5
times, 4 times, 3 times, 2--

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or 6 factorial.

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That part's easy.

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The hard part of
the problem is--

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but what if we rotate that cube?

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We could have made
different painting choices

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and wound up with the same
cube if we rotated it,

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and so the solution
to that problem

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is what's sometimes known
as the shepherd's rule.

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If you want to count
a flock of sheep--

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one easy way or maybe not
so easy way to count them

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is count their legs
and divide by 4.

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In order to count the different
cubes that we can count,

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I think the easiest
way to do it is,

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first, count all the ways you
could paint it if you fixed it.

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We figured that out.

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That was 6 factorial--

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and then figure out how
many different orientations

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are there of the cube?

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And that takes a little bit
of thought to figure out,

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but if you imagine sort of
picking up the cube by one

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edge with your fingers--

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maybe grab an edge.

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Sort of as long as you've
got a hold of the cube,

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it can't move around, so you've
sort of fixed in orientation.

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Well, how many edges
could you have grabbed?

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Well, if you think
about it for a minute,

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there's like four along the
top, four along the bottom,

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and then four along the side.

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So there's 12.

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And for each of those edges,
you could have grabbed it

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with your thumb on either side.

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You could grab it like
that or like that.

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So if I grab it like
that, I could twist it.

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So there's 12 times 2--

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24 different orientations.

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There's a lot of other
ways to prove that,

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but that's maybe the
simplest one I can think of.

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And so, then to count the
number of distinct cubes,

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you simply divide 6 factorial
by 24, and you get 30.

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There you have it.