another call for reminiscences ...
) From: email@example.com (Gerry Wildenberg)
) Please remove me from the mailing list.
yeah, remember back in those days when people were actually asking to
be ADDED to the mailing list?
btw, administrivia should be sent to "firstname.lastname@example.org".
thank you for your cooperation. ^^^^^^^
From: email@example.com (Bill Taylor)
This eightfold way is just a commutator of a face move and (a commutator of
two face moves)); so it turns out to be group-theoretically natural, as
commutators do "as little as possible".
here's the way i describe this. if sigma is a permutation on n symbols,
(say 1, 2, 3, ... , n), define the "support" of sigma to be those
integers which are NOT fixed by sigma. if tau is another permutation
on the same set, such that supp(sigma) and supp(tau) are disjoint,
then sigma and tau commute (i.e. the commutator is the identity).
if supp(sigma) intersect supp(tau) has just one element, then the
commutator is a three-cycle. as a rule of thumb, the smaller the
intersection of the supports, the smaller the support of the commutator.
in bill's example, ( R~ U L U~ R U L~ U~ ) the two permutations are
"R" and "U L U~", which only affect one corner in common. (actually,
to consider the cube as a permutation group, each corner is really
3 objects, one for each orientation.) but the analogy works well.
this idea is also helpful for creating three-cycles of corner-edge pairs
as well. on the 5x5x5 cube, you can make three-cycles of large blocks.
in fact, a larger cube is probably a better visual aid for understanding/
explaining this concept. another good commutator to try is with the
two sequences "B1 D2 B3" and "R1 U2 R3", which affect two corners
in common. (this is a fairly well-known maneuver.)
"I couldn't remember how to do it, but my fingers could !!", he said. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ This was my experience too, a few years ago. It's quite uncanny, like starting to ride a bike again after decades of not doing; only more so.
i've also experienced the exact same thing. it seems as though my hands
remember a short sequence by which i've just conjugated some maneuver,
which helps at the end of the conjugation. also in these commutators,
after i've done A and working on B, it feels as though my hands are
anxious to undo A, almost as if the cube is spring loaded and will just
[lame story about a friend pulling some hoax deleted]
actually, i think i FINALLY understand the story. the idea is that not
only is "brent" going to solve the cube behind his back, but he's also
going to do it WITHOUT first looking at it. actually, the story isn't
nearly as lame as i first thought, after realizing this.
when i visited mit in 1984(?), i saw joe killian do the real trick.
i certainly would have complained if i hadn't been allowed to scramble
the cube exactly as i wished. in fact, i may have even insisted that
he use MY cube (not too sure, though), just to be certain that the
surface hasn't been textured in any way. it was quite impressive.
about 5 minutes of studying, then behind the back without peeking.
he said that all it took was a good system of remembering where all
the pieces are. but i don't know what his system was.
by the way, bill, this "brent" wouldn't happen to be your friend who can
do the cube in 0.87 seconds, would he? :-)
and speaking of tall tales, let's see if anyone can top this one:
back in the days when i was into speed, er, speed CUBING, i'd solve
the cube maybe 200 times every day. for some reason, i got into the
habit of scrambling it behind my back (probably from listening to too
many complaints like "you're just watching all the moves you're doing!"
yeah, such a good complaint deserves such a fine solution.) well,
anyway, one time i stopped scrambling it, and as usual, i get 15
second to study it (standard racing rules). however, much to my
surprise, the cube was quite UNscrambled! how could this possibly be?
well, the only explanation is certainly that after scrambling the cube
thousands of times, my hands began to get into a rhythm (maybe even a rut).
they'd just do the same sequence over and over again. depending upon
my concentraion level, i'd find that sometimes i needed to make a
conscious effort to vary the sequence. in fact, at least once i got
a pattern that i'd previously seen: it was 4 dots with 6 corners twisted
(hence has order 6). so it's not too implausible.
like bill, and unlike dik, i spent quite some time struggling with
the cube before i finally solved it; probably about 6 or 7 weeks.
in fact for some time, probably about 2 weeks, i was convinced that
it couldn't be done, except by very dumb luck (as in story above).
of course, in those days, i hadn't heard of cube-lovers, hadn't even
seen a computer, didn't know the furst-hopcroft-luks algorithm, hadn't
even heard of anyone who could solve it ... but i was just a high school
freshman (age 14) at the time. i didn't even know what a group was!
this was shortly before the big craze started here in the u.s. (late 1980).
at school, some friends and i talked about it, but the main questions were:
how was it made, and how many combinations did it REALLY have? i was truly
convinced that trying to solve it would be futile. there was also a
shortage of them at the time, so i didn't get one until xmas. in fact,
i remember the tv commercial that ideal put out. they didn't even make
it clear that it actually turned in all possible directions! we had all
sorts of ridiculous diagrams and ideas of cables and magnets, but none of
them quite worked. and how could it turn in all directions? i heard of a
bookstore somewhere that had one on display (but were otherwise sold out),
so i went to see it. i remember spending a few minutes twisting it to
find an axis that wouldn't turn! in fact, i could keep turning the same
face in the same direction, around and around and around ... and the
cables inside never got caught!
sometimes i'm amazed at just how stupid i can be when i try ...
anyway, the story about how i finally figured out how to solve it isn't
nearly as interesting. after i first heard about people that could do it,
i started to work on it more seriously. the key ideas were: get all
the corners, (here was something that you could do, and then still do
more without destroying what's already done. but this was hard and
usually took more dumb luck and/or persistence.) then two opposite layers.
(again, the middle slice still can turn, even with half turns on the
sides F, B, R, L.) it took several days to flip the last two edges on
the middle layer. (i just kept picking a different pair of opposite
layers to solve and stumbled across a U layer monoflip in the process.
of course, it took months before i realized what was actually happening.)
also figuring out how to take it apart (and finally seeing how it was made)
was helpful, 'cause then i could experiment easily.
well, i've droned on long enough. anyone else got any interesting stories?