Seeing as the going
claim is that there are only 36 ways to paint a
tetrahedron using four
unique colors, I'll
endeavor to
count them. Below I have '
drawn' a
tetrahedron for visualization purposes:
,|,
,7``\'VA,
,7` |, `'VA,
,7` `\ `'VA,
,7` |, `'VA,
,7` `\ `'VA,
,7` |, `'VA,
,7` `\ ,..ooOOTK`
,7` |,.ooOOT''` AV
,7` ,..ooOOT`\` /7
,7` ,..ooOOT''` |, AV
,T,..ooOOT''` `\ /7
`'TTs., |, AV
`'TTs., `\ /7
`'TTs., |, AV
`'TTs., `\ /7
`'TTs., |, AV
`'TTs.,\/7
`'T`
There are two ways to paint a T (tetrahedron) with a unique color for each
side; one is the
mirror of the other.
There are 24 ways to paint a T with three unique colors. Four
possibilities for the two that are the same, three for the first unique, and two for the second unique = 4*3*2 or 24. It is not
possible to create
duplicates, since duplicates are taken care of by
successively reducing the number of possible colors for the next
consecutive side.
There are 12 ways to paint a T where three sides are the same: 4 possibilities for the three same sides, three remaining possibilities for the remaining side for a total of 4*3=12. Again, duplicates do not exist.
There are four ways to paint a T where all the sides are the same (if it is
imagined that four is the total amount of colors available).
Now, to total the
results: 2+24+12+4=42. Hmm. I don't count 36.
Theory and
reality not quite in
sync, or are my
figures
wrong? Just a
side note, if
mirror images aren't
allowed, this doesn't quite cut the numbers in
half, but close. They would become 1+12+12+4=29.
Update: 15:51 Tue Aug 22 2000
In reply to mblase, umm, I still have to disagree, and even add some more to my list. Remember, a mirror image is one such that no
rotation can
achieve it while still within the same
dimension (
i.e. a left hand cannot be rotated into a right hand while still in the
third dimension). So my original count for three unique colors is still
accurate. For two unique colors, I
concede the
point. So that's another six onto the original, or 48, or another six onto the secondary, or 35. I have done the work on paper and I believe it stands as I have stated it.
Update again.
Now I see the
light. It took quite a bit of imagination (something I have trouble with . . .). The count is 36. So much for counting accuracy . . . The operative case here is a tetrahedron viewed on an
edge. Two unique colors are on the
faces visible, a third unique color is on the faces that aren't. The two visible
faces can be rotated either way 180 degrees to produce a mirror image (so much for my mirror "
logic"
above), so there are only 12 instead of 24, and the count is 36 (or 35 if mirrors aren't allowed . . .;).
Final note: I don't
intend to "
fix" the
mistakes above . .
rather I wish this to represent a
process of
thought that may (or equally well may not) be useful to someone wishing to
understand. My mistakes above are
legitimate; I really did think I was right for quite some time (an
hour at least . . .).