Date: 3 August 1981 22:34-EDT
From: Alan Bawden
Subject: The Archive
To: CUBE-LOVERS at MIT-MC
Those of you who look through the archives of old Cube-Lovers mail
will notice that I have split off a new section of the archive. The
mail now lives in:
MC:ALAN;CUBE MAIL0 ;oldest mail in foward order
MC:ALAN;CUBE MAIL1 ;next oldest mail in foward order
MC:ALAN;CUBE MAIL2 ;more of same
MC:ALAN;CUBE MAIL ;recent mail in reverse order
Date: 4 August 1981 01:21-EDT
From: Chris C. Worrell
Subject: Poll
To: CUBE-LOVERS at MIT-MC
In line with the recent cubeing contest in Boston, I suggest a poll.
Such questions as, will be included:
1. age
2. average solving time
3 occupation (if student, undergrad or grad and major)
4. solving method
5. how long it took you physically working (playing) with the cube (or some
facimile, like a program) to solve the cube the first time, and reliably
6. how long have you been working with the cube (say since Jan. '81)
7. max #of qtw. in method, avg. # of qtw. in method
I would not suggest anybody send their answers to the list, because
1. it would generate a lot of semi-useless garbage in the list,and
2. you would lose anonymity (which is one of the aspects of a poll)
If anyone can suggest how this could be implemented please pass it on.
Chris Worrell
Date: 4 August 1981 01:58-EDT
From: Chris C. Worrell
Subject: Identities...
To: CUBE-LOVERS at MIT-MC
A few corrections to my list:
I16-13 has 18 qtw not 16
I18-4 has 20 qtw not 18 and has supergroup effect
I18-5 is not distinct from I18-6 so may be taken off the list
Additions:
I16-15 U'F'UBU'FUB'URU'L'UR'U'L courtesy of Jerry Agin
I16-16 FLF'RFL'F'R'B'LBR'B'L'BR
I16-17 LD'R'DL'D'RUR'DLD'RDL'U'
I18-9 SAME AS I16-13
I18-10 RDLD'R'DU'L'D'LUD'R'DL'D'RD
Concerning Sources:
I found none of these by exhaustive search so I believe that this is far
from a complete list esp. in 18 region.
I used several methods to derive these identities
1. from two processes which have same effect , I concatanate the inverse of
the second to the first, and remove noops.
2. from 1 process which affects only the d face or not the U face,
I twist both the D and U faces, run the reverse process (maybe with an
orientation change) then untwist the D and U faces.
3. examine old identities for similarities shift/rotate/invert/reflect
appropriatly concatanate and remove noops.
Chris Worrell
Date: 4 Aug 1981 1123-PDT
From: Alan R. Katz
Subject: 10 sided "cube"
To: cube-lovers at MIT-MC
cc: katz at USC-ISIF
I have a 10 sided "cube", which is made by "Wonderful Puzzler" (they also
make crappy cheapo regular cubes). The guts are essentially the same as
the regular cube, but the corners are cut off.
If you look down from the top, you see an octogon. The edges are all the
same as a regular cube, but since the corners are cut, they are one color
(thus the 10 colors). For example, there are red, blue, and orange
faces. On an ordinary cube you would have a red-blue-orange corner cubie,
but on this in its place is a pink face. To make this clearer, here is
the coloring of the thing:
red
*
light-blue**gold**yellow**blue**pink**orange**violet**green
*
white
(red on top, white on bottom, looking at the blue face, back face is green,
right face is orange, left face is gold).
The interesting thing about this is that unlike the ordinary cube, every cube
does not have a place, you dont know that the pink corner goes between the
blue and orange faces (in an ordinary cube it is the red-blue-orange corner
so you know where it goes). To solve it, you just put the corners in some
order, solve it using the usual transformations, and then if you get a
"parity error" you must go back to the top layer but switch two of the corners
and solve it again. Thus in general you have to solve it almost 2 times!
(almost because you dont have to redo the top layer or half of the second
layer (this assumes you solve top down, which I do.)).
What I mean by parity error is that if the corners are switched you can get
a configuration that in an ordinary cube would tell you the cube is put
together wrong. For example, you can be solving it and get to a point where
an odd number of edges must be fliped.
There may be a transformation to flip an odd number of edges with this cube,
but I have not found it. Anyway its more interesting to solve and it
changes it shape in general with each transformation. (unlike the cube
which stays a cube; this is a octagonal prism).
Alan
-------
Date: 5 Aug 1981 0948-PDT
From: ISAACS at SRI-KL
Subject: 10-sided cube
To: cube-lovers at MIT-MC
The 10-sided cube was discussed a couple of months ago. The main result
was that the easiest way to solve it was "side across" - that is, don't
start from the octagonal face, but from one of the "sides". Then the
last layer should be solvable except for a possible edge-flip. Note that
there are two new difficulties with this cube: the one mentioned, and the
single edge flip. SPOILER! This is, of course, an optical illusion, brought
about by the fact that the second edge, which of course is also flipped, is
a uni-colored edge, and you can't see it.
--- Stan Isaacs
-------
Date: 7 Aug 1981 1005-PDT
From: ISAACS at SRI-KL
Subject: BARREL AVAILABLE
To: CUBE-LOVERS at MIT-MC
The "Magic Barrel" or "Ten Billion" or whatever its name is, that was
talked about in this digest several months ago, seems to have finally gotten
on the American market. In the Bay Area, at the Stanford shopping center,
they have them in stock at Macy's, and will soon have them at Games and Things.
Games and Things also has the smallest cubes I've seen - about 3/4 inch on
a side. That makes at least 4 different small sizes, from 1 1/2 inch down.
--- Stan
-------
Date: 10 Aug 1981 0841-PDT
From: Tom Davis
Subject: Barrel Puzzle instructions
To: cube-lovers at MIT-MC
I just purchased one of the "Wonderful Barrel" puzzles mentioned in
Stan Isaac's note, and I found the enclosed set of instructions to be
a classic. I figured I'd pass them on, in case some of the members of
this list don't buy the puzzle itself. The first couple of paragraphs
are not so great, but it gets better toward the end.
@begin(verbatim)
"Wonderful Barrel" is a kind of mental game plate also can be regarded
as an indoor ornament. The red, orange, yellow, green, blue, black
color balls which is inner in barrel can create more than 10 billion
combination to reach a solution and finally be aligned on 5 straight
lines, yet each straight line are with 4 same color balls.
* PLAY RULE *
Original alignment circumstances is as follow: there are three black
balls are set on SHELTER but each black ball is above a shore of
PLUNGER; the 5 straight lines are up the SHELTER with different color
from one another, each line contains 4 same color balls. The 5
straight lines are aligned in clock-wise with blue, green, yellow,
orange, red color.
As you know, the original alignment is orderly, now you can
circumvolve DRUM, then operate plunger up and down for taking the ball
out from SHELTER. While you put the ball into SHELTER again, the
order of "each straight line with 4 same color ball" has been broken
to confused state.
Now, a challenge is ahead of you, you often successful in this line
but irregularly in that line, in a word, it always can not come to a
satisfactory arrangement of the whole lines fluently.
At the beginning, it is not easy to arrange the 5 straight lines
return to initial order in short time, so please try to solve this
puzzle from one line first, then completion of two, three, four even
to whole lines are returned to initial order finally.
At last, maybe you can analyse the mystery out, but maybe you can not
solve it today; however, how about tomorrow? one month later? one
year later? or ten years later you may still continued to challenge to
it singlely and unwilling.
@end(verbatim)
P.S. I @i(tried) to proofread this, but as you can see, it was difficult.
-------
Date: 14 August 1981 0111-EDT (Friday)
From: Dan Hoey at CMU-10A
To: Cube-Lovers at MIT-MC
Subject: Results of an exhaustive search to six quarter-twists
Message-Id: <14Aug81 011137 DH51@CMU-10A>
The first answer is that there are exactly 878,880 cube
positions at a distance of 6 quarter-twists from solved, and so
983,926 positions at 6qtw or less. These figures reflect a decrease
of 744 from the previously known upper bounds.
It turns out that the twelve-qtw identities reported by
Chris C. Worrell are complete, in a sense. The
only reservation here is that a fifth rule must be added to his
list of the ways in which ``a generator generates other
identities.'' This rule is substitution with shorter identities,
and it's not too surprising that it was left out, since the only
shorter identities are the ``trivial'' ones like XXXX=XYX'Y'=I,
where X and Y are opposite faces. In the case of the twelve-qtw
identities, this means that identities of the form aXXb and aX'X'b
generate each other.
The structure of the 12-qtw identities is clearer if we
write them in a transformed way:
I12-1 FR' F'R UF' U'F RU' R'U
I12-2 FR' F'R UF' F'L FL' U'F
I12-3 FR' F'R UF' UL' U'L FU'
The fifth rule is necessary so that I12-2 may generate the
identities
I12-2a FR' F'R UF FL FL' U'F and
I12-2b F'R' F'R UF FL FL' U'F'.
To see that this rule is necessary, it need only be observed that
inversion, rotation, reflection, and shifting all preserve the
number of clockwise/counterclockwise sign changes between
cyclically adjacent elements.
In what sense are the ``trivial'' identities trivial? I
have come to believe that they are trivial only because they are
short and simple enough that they are well-understood. The only
identities for which I can find any theoretical reasons for calling
trivial are the identities of the form XX'=I. In spite of the
simplicity of the ``trivial'' identities, their occurrence is one
of the major reasons that Alan Bawden and I were unable to show
earlier that I12-1-3 generated all identities of length 12. I fear
that the combination of 4-qtw and 12-qtw identities may turn out to
be a major headache when dealing with identities of length 14 and
16.
Date: 14 August 1981 03:12-EDT
From: Chris C. Worrell
Subject: CUBE-POLL@MIT-MC
To: CUBE-LOVERS at MIT-MC
The first CUBE-LOVERS poll has now begun. Send your answers to:
CUBE-POLL@MIT-MC
The questions:
1. Occupation (if student, undergrad or grad and major)
2. Age
3. Average solving time
4. How long it took you physically working with the cube in order to solve
it reliably.
5. How long have you been working with the cube (say since Jan. '81)
6. How many cubes do you own, and how many have died on you.
7. Solving methods:
A. Time-efficient, describe method in general.
B. Move-efficient, describe in more detail, including number of qtws
for each stage.
Include a maximum number of qtws for this method.
8. Transformations you have found or know of.
Include a description of what the transform does.
Include transforms you use in your method, such as top to middle slice
edge movers, and transforms which affect only one face.
The results of this question will be compiled into a dictionary of
transfors and may also aid people in their investigations of the cube.
9. Any specific intrests you have in the cube, such as applications to
group theory, or investigation of identities or whatever.
10. Any other intrests you might have, relating or not relating to puzzles.
Chris Worrell (ZILCH@MIT-MC)
Date: 17 August 1981 11:34-EDT
From: Allan C. Wechsler
Subject: 12q relations.
To: CUBE-LOVERS at MIT-AI
I12-1 FR' F'R UF' U'F RU' R'U
I12-2 FR' F'R UF' F'L FL' U'F
I12-3 FR' F'R UF' UL' U'L FU'
I can't believe I'm the first person to notice this:
Suppose we only know I12-1 and I12-2. Then we have
I12-1' U'RUR'F'U (FR'F'RUF')'
(I12-1')(I12-2) U'RUR'F'U (FR'F'RUF')' (FR'F'RUF') F'LFL'U'F
Reduce: U'RUR'F'U F'LFL'U'F
Conjugating by (U'RUR'F'U)', we get
F'LFL' U'FU'R UR'F'U
But this is just the RL mirror image of
FR'F'L UF'UL' U'LFU'
This is exactly I12-3. So there are really only two independent 12q
identities, and the third can be deduced from them.
Date: 19 Aug 1981 09:48 PDT
From: Lynn.ES at PARC-MAXC
Subject: Re: 10 sided "cube"
In-reply-to: KATZ's message of 4 Aug 1981 1123-PDT
To: Cube-Lovers at MIT-MC
cc: Lynn.es@PARC
I last week picked up one of the same octagonal cubes (Wonderful Puzzler
brand) as Alan R. Katz in his message. I might point out that it shares the same
workmanship and twistability (lack thereof) which Katz attributes to their brand
of standard cube.
After some experimenting, I found that the "parity error" involved is always a
pair of edges in reverse locations on the bottom. My solution algorithm is top,
equator, bottom [corner locations, edge locations, edge orientations, corner
orientations]. On an ordinary cube, one pair of edges reversed never happens
when you go to position edges. I suspect that if your algorithm does edge
locations before corners, then two corners in wrong locations would be the parity
error. Incidentally, the parity of Katz's cube is the reverse of mine, though the
center cubie colors are in the same relation to each other.
Edge orientation (I assume that is what Katz meant by "edges must be fliped")
parity errors are irrelevant to getting the cut-edge-colors rightly paritied. Edge
orientation parity errors happen because one of the cut-edges may have been
oriented wrongly. Or more precisely, an odd number of them reversed from the
way they were virginally (but they look right either direction). This is easily
cured by any of the "flip a pair of edges" macros.
The change in cube shape is actually helpful in solving, except for being
difficult to grab sometimes. Corners that need orientation really stand out. The
Greenwich meridian edges (assuming you solve it from the top down with the
octagonal faces left and right), being different shape, are instantly located.
The best part of the Wonderful Puzzler is the instructions, which I quote here:
THE CHALLENGE:
"ORIGINAL PUZZLER" presents not only a unique challenge but offers the
possibility of countless hours of relaxation. Your mental ingenuity may be tested
for a few hours -- many days -- several weeks -- or even a period of much
longer duration. If you can determine the key to unlock the knack for solving
the PUZZLER, the final trimph can be the psychological turning point in your
life. Mathematicians may be tested to the limit and cry over this one -- and you
may, too! You will gain a measure of satisfaction when you align one plane.
You will be delighted with the completion of two. You will be elated with the
completion of three or four! The completion of the fifth plane will quicken your
pulse!! -- and you will have scaled the peak once the last unit of the sixth plane
falls into place!!!!
PREPARATION & CAUTION:
*Spin the PUZZLER several times, as indicated on the cover, until all color units
are randomly distributed on each of six planes.
*Do NOT remove any color unit in the process of play.
*Initially, activate the random distribution GENTLY. With little use, this can be
accomplished easily and smoothly.
Patience and persistence will beat the Puzzler! Good Luck!
Challenge
Can you contend with more than 18'000'000'000 combination to reach a solution?
*end of quote*
I think the errors remaining above are all theirs. It is full of little gems: turning
point in your life, solving five planes (on the standard cube, which I believe
has the same instructions), 18 billion (better than Ideal's guess). The GENTLY
caution is valid; a neighbor kid blew mine apart, including a center cubie, by
ignoring this.
Date: 20 August 1981 02:46-EDT
From: Alan Bawden
Subject: Resending to the right place so that I can digest it later...
To: CUBE-LOVERS at MIT-MC
Date: 19 August 1981 23:22-EDT
From: Eric L. Flanzbaum
Subject: A small version if the cube ...
To: CUBE-LOVERS-REQUEST at MIT-MC
cc: ELF at MIT-MC
I don't know if this has been mentioned yet, but why
not mention it again? I just saw Rubik's cube being sold
on a keychain. It is about 1 1/2 inches and works/looks
like the real thing. They sell for about $2.50 which
is considerably less than the standard version. I live
out on the west coast, so I don't know if they sell
it back east.
Happy solving,
-Eric.
Date: 22 Aug 1981 1404-EDT
From: ROBG at MIT-DMS (Rob F. Griffiths)
To: Cube-Lovers at MIT-MC
Subject: Re: ELF's Small cube
Message-id: <[MIT-DMS].207641>
I have seen that one, and also one that is absolutely tiny. It
is the size of one of the cubies on the full sized Rubik's, and
is fully operational. They are really quite flexible and well
made, I don't know who manufactures them, but I will try to find
out.
-Rob.
Date: 22 August 1981 18:46-EDT
From: Chris C. Worrell
Subject: CUBE-POLL``MIT-MC
To: CUBE-LOVERS at MIT-MC
SO FAR I HAVE RECEIVE ONLY ABOUT 8 REPLIES, THIS IS HARDLY ENOUGH
TO DO STUDIES ON. PLEASE SEND MORE REPLIES. IF YOU NO LONGER
HAVE THE QUESTIONNAIRE SEND ME MAIL AND I WILL FORWARD IT TO YOU.
Date: 24 Aug 1981 1032-PDT
From: ISAACS at SRI-KL
Subject: recoloring the 10-sided cube
To: cube-lovers at MIT-MC
A nice way to recolor the 10 sided cube is to give a side a quarter twist,
so it looks sort of like a baseball, and then make the (fairly) obvious
4 groups of 9 faces each in four colors, and two in-between stripes of 3
facies each. Each of the 9-facie groups will have 2 triangular facies, and
3 slant rectangular facies, and 4 squares. I think its a fairly simple
variation to solve, but I just made it last night and have not worked with
it much yet.
By the way, I just got my first magic tetrahedron. This one came from
Japan, but says made in Hongkong by "World-wide"(?) copyright by Meffert.
Anyone know who he is? It seems very similar to the one invented by
Kristen Meier.
-------
Date: 25 Aug 1981 1016-PDT
From: ISAACS at SRI-KL
Subject: BAY AREA CUBE CONTEST
To: cube-lovers at MIT-MC
Date: 24 Aug 1981 1612-PDT
From: ISAACS at SRI-KL
Subject: CUBE CONTEST-BAY AREA
To: cube-lovers at MIT-MC
There will be a Rubics Cube contest at Games and Things, at the
Stanford Shopping Center this Saturday, Aug. 29, from 10:00 am to
4:00 pm. Speed contests, money prizes, a cube display (by me), etc.
Contestants are supposed to register at GAMES AND THINGS before 10:00
on Saturday. Address is 128 Stanford Shopping Center, Palo Alto, Ca.,
(415) 328-4331.
--- Stan Isaacs
-------
-------
Date: 25 Aug 1981 1019-PDT
From: ISAACS at SRI-KL
Subject: MORE TERMINOLOGY
To: CUBE-LOVERS at MIT-MC
More possible terms (from a new cube-solver who was a biology major):
Dorsal/Ventral for front/back, Port/Starboard for right/left (left/right?).
He doesn't have a consistant term for up/down.
-------
Date: 25 August 1981 19:04-EDT
From: Alan Bawden
Subject: Speed cubing
To: CUBE-LOVERS at MIT-MC
Does anyone have any idea what the world record for speed cube solving
really is? The only times I can find are:
1) In the Scientific American article Hofstadter mentions an
Englishman named Nicholas Hammond who averages "down to close to 30
seconds". Anyone have any more information on this? Like where
Hofstadter found out about this guy?
2) In the reports about the "Regional Cubing Championship" held here
in July the best time listed is 48.31 seconds, held by a 10 year old
named Jonathan Cheyer. (See PDL's message to Cube-Lovers dated 27 Jul
1981.)
3) I seem to remember reports that there were Hungarians who averaged
around 50 seconds. I thought I had read this in Singmaster, but I can't seem
to find it there.
4) The best time anyone will admit to on this list (as determined by
scanning the replys to ZILCH's poll) is 2 minutes. This time is
claimed by both Richard Pavelle (RP@MIT-MC) and Alan Katz (KATZ@ISIF).
Also Stan Isaacs (ISAACS@SRI-KL) claims that his two children take
about 1 1/4 minutes using essentially his methods.
5) Finally, Dan Pehoushek (JDP@SU-AI) tells me that a friend of his
frequently breaks the 30 second barrier. I should have thought to ask
for his name.
Anybody know of any more good speed cubists?
Date: 25 Aug 1981 2116-EDT
From: ELF at MIT-DMS (Eric L. Flanzbaum)
To: Cube-Lovers at MIT-MC
Subject: Speed Solving
Message-id: <[MIT-DMS].208572>
Hi Cube fans,
I have a couple of friends at school (oh about 4 or 5) who consistently
solve the cube (actually, they have races/contests) at a time of under
30-40 seconds. I don't know if this is really unusually fast, but as
from ALAN's previous message, it looks like they all rank in there.
By the way, these people are entering the 9th and 10th grade in the
fall.
Happy Solving,
-Eric L. Flanzbaum
ELF at MIT-AI
Date: 27 August 1981 18:39-EDT
From: Dennis L. Doughty
Subject: Speed cubing
To: CUBE-LOVERS at MIT-MC
cc: DUFTY at MIT-MC
My fastest time for the cube is 1 minute 17 seconds (everything worked out
correctly). My average time is 1:40-1:45. --Dennis
p.s. i'll answer the poll when I get the time.
Date: 28 August 1981 12:59-EDT
From: Robert H. Berman
To: CUBE-LOVERS at MIT-MC
You may be interested to see a cartoon about the cube on page 36 on
the August 31 issues of the New Yorker. Yes, the New Yorker.
--rhb
Date: 29 August 1981 07:54-EDT
From: Thomas L. Davenport
Subject: Cube Song
To: CUBE-LOVERS at MIT-MC
This past week PBS broadcast the latest "Mark Russell Comedy Special"
and in it he did a funny song about the cube. He even had one with
him on stage.
-Tom-
Date: 29 Aug 1981 1906-EDT
From: ROBG at MIT-DMS (Rob F. Griffiths)
To: cube-lovers at MIT-MC
Subject: English Whiz Kid
Message-id: <[MIT-DMS].208873>
From TIME: August 31,1981:
---------------------------
Along with diet books, cat books, and advisories on how to make
a profit from the coming apocalypse, there is a growing shelf
concerned solely with mastering that infuriating, six sided,
27-part boggler with 42.3 quintillion possible combinations known
as Rubik's Cube.. The latest entry: ''You Can Do The Cube''
(Penguin, $ 1.95) by Patrick Bossert, 13, a London schoolboy who
discovered the cube only this spring during a family ski vacation
in Switzerland. Within five days he had mastered the monster,
and later began selling his schoolmates a four-page, mimeographed
tip sheet for 45 cents. An alert editor at Penguin saw a copy and
persuaded the prodigy to turn pro. The 112 page result contains
3 dozen 'tricks' for solving the cube (using logic rather than
math), as well as a chapter on 'Cube Maintenance' (to loosen a
stiff cube, ''put a blob of Vaseline on the mechanism''). With
250,000 copies of the cubist's book in print, a Penguin executive
marvels: ''It's the biggest, runaway, immediate success we have
had since we published 'Lady Chatterley's Lover' in paperback.''
---------------------------
-Rob.
Date: 1 Sep 1981 1442-EDT
From: Bob Clements
Sender: CLEMENTS at BBNA
Subject: [Bob Clements : Rubik's cube sale]
To: Cube-lovers at MC
I didn't know of the cube-lovers list, but it was suggested to me that
I re-send this msg to cube-lovers. If this sale has already been
mentioned on this list, sorry for the repeat.
/Rcc
---------------
Mail-from: MIT-AI
Received-Date: 1-Sep-81 1343-EDT
Date: 1 Sep 1981 1221-EDT
From: Bob Clements
Sender: CLEMENTS at BBNA
Subject: Rubik's cube sale
To: Info-Micro at ai
I don't want to start a whole Rubik's discussion, but for those
in the Boston area who need to replace their worn Cubes, or get
their first one, the Caldor chain has them on special for $4.66
thru Saturday.
/Rcc
-------
---------------
-------
Date: 3 September 1981 15:40-EDT
From: Dennis L. Doughty
Subject: Practical use of the Rubik's cube/Speed cubing
To: CUBE-LOVERS at MIT-MC
cc: DUFTY at MIT-MC
This rush week, my fraternity extended a bid to a freshman by the name of
Larry Singer who generally solves the cube in 1:13 or thereabouts (my
average time is 1:40). All day Monday, our bids were playing such games as
"Pledge Pong" or "Pledge Pool." The idea here is that the bid challenges
an active to a game of pool or ping-pong, and if he loses, he pledges.
Well, Monday night, Larry challenged me to "Pledge Cube." One of the
brothers uniformly scrambled two cubes, and we were to compete in a
head-to-head competition. Well, we both were under considerable pressure,
naturally, and we both made several mistakes while solving the cube. I
won, but my winning time was 2:09. Larry and his close friend then both
pledged.
So now, no one in the house can tell me that there's no practical use for
the Rubik's cube.
--Dennis
Date: 5 September 1981 1446-EDT (Saturday)
From: Bob.Walker at CMU-10A
To: cube-lovers at mit-mc
Subject: Snake: **SPOILER**
Message-Id: <05Sep81 144605 BW80@CMU-10A>
I am new to the list, and in reading the archives, I found that
the transform for my favorite pattern was listed incorrectly.
Specifically, I refer to the Don Woods' message of 6 January which
listed transforms for the Snake, Worm, and Baseball (I think).
Anyway, the transform listed for the Snake was incorrect. What is
listed IS a pretty pattern, merely the wrong one. The blurb about
how to "hack" your way to the Snake, however, is correct. The
proper transform to achieve the snake (from Singmaster) is:
* * * S P O I L E R * * *
Snake: B R L' D' R R D R' L B' R R + U B B U' D R R D'
Date: 5 September 1981 01:08 edt
From: Greenberg.Symbolics at MIT-Multics
Subject: TV special
Friday night at 8 NBC Magazine will cover Rubik's cube. The coverage
will doubtless include footage of the July 25 Boston Area Cubeathon,
at which many MIT-Area cubists were present.
Date: 8 Sep 1981 0942-PDT
From: ISAACS at SRI-KL
Subject: misc
To: cube-lovers at MIT-MC
Some short notes on various cube stuff - will try to expand on some of it
later:
1. The Stanford Shopping Center/Games and Things cube contest of a couple of
saturdays ago: was won by Paul cunningham, 16 yrs old. There were about
40 entries; there were something like 8 rounds to get to a winner, double
elimination, each pair fighting it out with the best average of 3 cubes,
all cubes in a round scrambled exactly the same way. Best average-of-three
time was 56 seconds. Best time was 41 seconds. David Tabuchi, the Games
& Things speedster, has a best average-of-ten speed of 43 seconds! Brian
Robinson, with whom he works on the cube, has a best average-of-ten of 41
seconds. Davids fastest time was 24.98 seconds!!!
2. Cubes are multiplying like hotcakes. Not only changes in labeling, but
also changes in shape. I have seen or heard reports of about 8 shape
variations, and multiple size variations, from about 19 mm to about 60 mm.
And someone told me of a 12mm or so version. The corners have been cut off
in 3 different ways (The nicest cuts them halfway through the neighboring
edge, and uses 14 colors). The magic Tetrahedron is readily available now.
There is a build-it-yourself cube kit. There are still reports of the
elusive 4x4x4 cube, but no actual sightings as far as I know. By the
way, I collect puzzles, and am trying to find many of these cube variations.
If anyone knows where I can actually buy some, or would get me some, please
let me know. I will be happy to pay or trade for them.
3. The snake is not a cube, but it is a toy/puzzle/art object that should a
appeal to cube-lovers. It also comes in a variety of colors and sizes.
4. Also lots of new books and such. Such as:
a. Don Kolve, of Kirkland, Wash. He solves Top-middle-bottom(position
corners,twist corners, flip edges, position edges).
b. L.E.Hordern, of England. He does: bottom-middle-top ( position corners,
orient corners, orient edges, position edges).
c. Bridget Last, of Downham, England. She solves: Define face colors
("The easiest way of deciding which face is to be which colour is to
define the centre faces as being correct.") Then: position all corners,
orient all corners, position (with orientation) all edges.
d. Bob Easter, a friend in San Francisco, uses just one move to do
everything. The move is F R' F' R (an old friend). First he "walks"
the edge cubes to their proper corners, then rotates them into proper
order around the corner, then flips them 2 at a time to get them aligned.
Then he does similarly with the corners, using the same move, but
done 3 times to leave the edges alone. Lots of quarter twists, but
little memory. Perhaps the ultimate solution for people with strong
wrists.
e. Patrick Bussart, Puffin books (a division of Penguin Books). Patrick
has been in the news since he is only 12 or 13 years old; his solution
is very popular in England. He does top corners, top edges, bottom
corners, position rest of the edges, rotate rest of the edges.
5. I have been re-labeling cubes to make new puzzles, and would appreciate
suggestions. I have made tactile cubes of various types (that is, with
various materials). I'm trying to make one with 5 or 6 grades of sandpaper,
but find my touch cannot distinguish between the two middle grades. I
have made a magic square cube (each face is a 3x3 magic square; the
problem is to decide what the relative orientations and forms of the squares
should be; any suggestions?) and a magic cube cube (the center 14 cube is,
of course, invisible). And 2 word cubes - one has word squares (there are
three 3 letter words in each direction), and the other, six 3 word sentences
(FIX THE BOX, YES YOU CAN, FUN FOR ALL, etc. I haven't had these long
enough to know if they are "solvable" without trial and error; I think
that the magic square cube, especially, is difficult unless you have
the order in advance. So far, I have made 2 of them, the first had one
square marked, which makes it fairly easy (the other 5 squares are forced
by the first); in the second, I made sure each corner was unique, and the
edges as different as possible (2 have to be the same); but I haven't had
time to try it yet.
Enough - this message is too long.
-------
Date: 10 Sep 1981 1447-PDT
From: ISAACS at SRI-KL
Subject: cube query
To: cube-lovers at MIT-MC
The 2x2x2 cube is solved in the corner sub-group, ignoring the (non-existent)
edges. The so-called "Dinman Style" cube (probably meant to be "diamond")
has the corners cut off and everything stretched to make a somewhat distorted
rhombicuboctahedron (the six center facies are square; the corners are now
triangles, and the old edges are rectangles). Solving this involves only
positioning moves - all orientation (twisting) is invisible. Thus these
two cubes involve two "pure" subgroups. Can anyone design (by either cutting,
recoloring, or even inventing new mechanisms) cubes or pseudo-cubes which
only involve edge-type moves, or which only involve twisting, with
positioning ignored?
--- Stan Isaacs
-------
Date: 11 Sep 1981 0917-PDT
From: ISAACS at SRI-KL
Subject: half query answer
To: cube-lovers at MIT-MC
First of all, I was inaccurate about the "Dinman" style "cube" - it doesn't
only eliminate twisting, but also some positioning (ie, after the top and middle
are solved, the bottom is automatically solved). Perhaps by the judicious
adding of numbers to the facies, the pure position cube can be made. Also,
the answer to the edge-only subgroup is easy - just remove all the labels from
the corners, so they are all identically monochromatic. Is there a more
elegant solution?
--- Stan
-------
GENTRY@MIT-AI 09/11/81 20:43:40
To: CUBE-LOVERS at MIT-MC
It appears that due to time restrictions, the segment about the RUBIK's
cube on NBC Magazine has been postponed until next week. Check the listings
for your area to see when it will be televised.
Date: 15 Sep 1981 1553-PDT
From: ISAACS at SRI-KL
Subject: lower bounds
To: Hoey at CMU-10A
cc: cube-lovers at MIT-MC
[This message is being sent to Dan Hoey, and refers to his message of
9-Jan-81, subject: The Supergroup -- Part 2: at least 25 qtw and why]
Appended to this message is a longish message I recieved, which has
some good ideas to use. In particular, what about using your technique
on a 2x2x2 cube, or an (idealized) edge-only cube? And then comparing
it with his clculations for the 2x2x2.
I'm not sure without a 2x2x2 in front of me, but I think there are
only 2 distinct 1 qtw per set of opposite faces, and only one 2qtw move.
And that the period is only 2. Is that true? However, there should be
more low-number-of-twists identities.
I'm distrustful of the actual calculations in the message below, because
I don't see the 9 new configurations after only 1 twist. I think there are
only 6. Or am I missing something?
Also, Dan or someone else on the cube-lovers network: how about compiling
all the messages about lower bounds and identities (after a while) into
one file we can ftp and look at all together.
11-Sep-81 12:26:52-PDT,6785;000000000001
Mail-from: ARPAnet host BERKELEY rcvd at 11-Sep-81 1223-PDT
Date: 11 Sep 1981 11:43:07-PDT
From: ARPAVAX.sjk at Berkeley
To: isaacs@sri-kl
Subject: in case you haven't seen this ...
Article 16:
>From csvax:mhtsa!harpo!chico!esquire!psl Wed Sep 9 17:16:32 1981
Subject: Rubik's Cube
Newsgroups: net.games
Want to knoe how far away you can get from the solution on a Rubik's Cube?
A Simple Lower Bound
As everybody knows, the number of discrete configurations of the 3x3x3
Rubik's Cube is:
(8! * 12! * 3^8 * 2^12) / 12 = 4x10^19 = 43,252,003,274,489,856,000
One approach to a lower bound is to calculate the maximum possible number
of configurations you can reach with a particular number of moves and then
see how many moves you would have to make to reach the number above.
With no moves at all you get 1, the starting position. The first move gets
you 18, (any one of six faces turned one of three ways). The next move
gets you 18*15, (no point in turning the same face twice in a row), for a
total of 1+18+270 configurations reached after two moves. A table of these
values looks like:
---------possible configurations---------
moves new % max total
0 1 0.0% 1
1 18 0.0% 19
2 270 0.0% 289
3 4050 0.0% 4339
4 60750 0.0% 65089
5 911250 0.0% 976339
6 13668750 0.0% 14645089
7 205031250 0.0% 219676339
8 3075468750 0.0% 3295145089
9 46132031250 0.0% 49427176339
10 691980468750 0.0% 741407645089 Notice that
11 10379707031250 0.0% 11121114676339 not until 17
12 155695605468750 0.0% 166816720145089 moves has the
13 2335434082031250 0.0% 2502250802176339 total number
14 35031511230468750 0.1% 37533762032645089 of possible
15 525472668457031250 1.2% 563006430489676339 configurations
16 7882090026855468750 18.2% 8445096457345145089 exceeded the
17 118231350402832031250 273.4% 126676446860177176339 maximum.
So there is no possible way to reach some configurations in fewer than
17 moves. However, this analysis has assumed that each configuration
generated was a NEW one, but there are MANY cases where this will not
be so. A simple example is turning one face 180 degrees, the opposite
face 180 degrees, and then repeating those two moves -- four moves that
get us to an old, familiar configuration. If we factor out the sequences
that involve these opposite face identities the minimum number of moves
becomes 18. Needless to say there are still lots of useless move sequences,
but detecting them becomes a lot trickier.
A Rumored Upper Bound
Rumor has it that a computer program exists, (attributed to Thistlethwaite),
that provably will solve any Cube configuration in at most 41 moves.
Narrowing it Down
So the answer is somewhere between 18 and 41. How do you get further? One
way is to write a computer program that tries every sequence of moves until
it has generated every possible configuration at least once. That sounds
easy, and it is, but such a program would take a \\\L O N G/// time to run.
However, if we limit the problem a little by considering a Cube that is two
squares on a side (2x2x2), we have a chance of learning something.
2x2x2 Cube
By the same considerations stated above we can get a lower bound for the
2x2x2 Cube. There are 7! * 3^6 = 3,674,160 configurations and, since we
can limit ourselves to moving only three "orthogonal" sides of the 2x2x2
cube, on the n-th move you could reach 9 * 6^(n-1) new configurations thus
we find that with 8 moves you could reach at most 3,023,307 and with 9 you
could reach at most 18,139,851. (Note that this doesn't have the problem with
opposite side moves that the 3x3x3 cube has.)
Because the 2x2x2 cube is relatively simple we can actually run a program
to try all the possible move sequences and compare our bound with fact.
Listed below are the findings
------new configurations------- total configurations
moves -----possible---- ---actual--- ---possible --actual
number % number % number number
0 1 0.0% 1 0.0% 1 1
1 9 0.0% 9 0.0% 9 10
2 54 0.0% 54 0.0% 63 64
3 324 0.0% 321 0.0% 387 385
4 1944 0.0% 1847 0.0% 2331 2232
5 11664 0.3% 9992 0.3% 13995 12224
6 69984 1.9% 50136 1.4% 83979 62360
7 419904 11.4% 227536 6.2% 503883 289896
8 2519424 68.6% 870072 23.7% 3023307 1159968
9 15116544 411.4% 1887748 51.4% 18139851 3047716
10 90699264 2468.6% 623800 17.0% 108839115 3671516
11 544195584 14811.4% 2644 0.071% 653034699 3674160
Interestingly enough there are 2,644 configurations that require eleven
moves to reach a solution; this is less than one tenth of one percent of
the total configurations!
It's also interesting that it's better than a 50-50 bet that a randomly
ordered 2x2x2 cube can be solved in exactly nine moves, (it's not clear
how to turn this into a profitable bar bet, however).
Noticing that there are only 321 new configurations after three moves
instead of 324 leads us to guess that there are six non-trivial sequences
of six moves that end with the original configuration, (why?).
These results came from a C program running on a VAX 11/780 and even
though the 2x2x2 cube is simple compared to the 3x3x3 it took a lot of
time. The figures for 11 moves took over 51 hours of cpu time.
If you'd like to make a 2x2x2 cube with which to experiment you can simply
take all the little labels off a 3x3x3 cube except the ones on the corners
and then ignore the unlabeled cubes.
Here's one sequence that gets you to one of the 2,644 configurations:
f r f r f d2 f d- f d2 r2 f = rotate front face 90 degrees
r = rotate right face 90 degrees
d2 = rotate "down" face 180 degrees
d- = rotate "down" face 270 degrees
So Where's That Leave Us?
I just thought of a dandy way to get the answer for the 3x3x3 cube, but
the margins on this news item are a little too small for me to include it ...
-------
Date: 15 September 1981 21:54-EDT
From: Alan Bawden
Subject: Editor's note to the last message.
To: CUBE-LOVERS at MIT-MC
I will look into collecting all of the relevant messages on God's
number into one place. If you want to poke around in the archives
yourself (please be carefull, and don't delete them again) I will
remind you all that old cube-lovers mail is archived in the following
places:
MC:ALAN;CUBE MAIL0 ;oldest mail in foward order
MC:ALAN;CUBE MAIL1 ;next oldest mail in foward order
MC:ALAN;CUBE MAIL2 ;more of same
MC:ALAN;CUBE MAIL ;recent mail in reverse order
(I someone else wants to attempt the compilation, there is a better
chance it will get done. Let me know and I will be happy to lend a
hand.)
Some of the seeming inconsistencies in the message included by Isaacs
in his message are a result of the usual half versus quarter twist
screw. The reason the writer can see 9 configurations after a single twist
is because he has a different definition of a "single twist".
I also am not sure, but I also think that the counting argument given
here suffers from the some confusion Singmaster had when he computed a
lower bound of 17 htw. I think, in fact, that a lower bound of 19 htw
results if the argument is executed correctly (Singmaster corrected
himself about this by the fourth edition, I think). Someone with a
copy of Singmaster handy should look this up.
The 41 move count for Thistlethwaite's algorithm is probably a half
twist count given that it was reported by Singmaster.
Date: 16 September 1981 0003-EDT (Wednesday)
From: Dan Hoey at CMU-10A
To: ISAACS at SRI-KL, Cube-Lovers at MIT-MC
Subject: Re: lower bounds
In-Reply-To: Stan Isaacs's message of 15 Sep 81 17:53-EST and Alan Bawden's
message of 15 Sep 81 20:55-EST
Message-Id: <16Sep81 000353 DH51@CMU-10A>
Hi. I'm really pressed for time, but I'll drop a couple of
comments.
Alan pretty well said it--there are half-twisters and there
are quarter-twisters and the included message is one of the former.
I strongly favor the latter, since then all the moves are
equivalent, (M-conjugate, to you archive-readers). But Singmaster's
book, though in the other camp, is too good to ignore.
To extend the argument I gave on 9 January to the case
where quarter-twists and half-twists are counted equally (we call
such a move a `htw' whether it is quarter or half) let PH[n] be the
number of (3x3x3-cube) positions at exactly n htw from SOLVED. Then
PH[0] = 1
PH[1] <= 6*3*PH[0]
PH[2] <= 6*2*PH[1] + 9*3*PH[0]
PH[n] <= 6*2*PH[n-1] + 9*2*PH[n-2] for n > 2.
Solving yields the following upper bounds:
htw new total htw new total
0 1 1 10 2.447*10^11 2.646*10^11
1 18 19 11 3.267*10^12 3.531*10^12
2 243 262 12 4.360*10^13 4.713*10^13
3 3240 3502 13 5.820*10^14 6.292*10^14
4 43254 46756 14 7.769*10^15 8.398*10^15
5 577368 624124 15 1.037*10^17 1.121*10^17
6 7706988 8331112 16 1.385*10^18 1.497*10^18
7 102876480 111207592 17 1.848*10^19 1.998*10^19
8 1373243544 1484451136 18 2.467*10^20 2.667*10^20
9 18330699168 19815150304
At least 18 htw are required to reach all the 4.325*10^19
positions of the cube. This is the same argument that was used in
Singmaster's fifth edition, p. 34, and is the best I know. Lest ye
be tempted to pull the trick I did in the January message, remember
that half-twists are even permutations, so there is no assurance
that half the positions are an odd distance from SOLVED. This is
illustrated in the 2x2x2 case, where more than half of the
positions are at a particular odd distance.
And yes, all of Thistlethwaite's analysis seems to use the
half-twist metric. I am quite surprised, however, to hear the rumor
of 41 htw. As of Singmaster's fifth edition, the figure was 52 htw
``... but he hopes to get it down to 50 with a bit more computing
and he believes it may be reducible to 45 with a lot of
searching.'' If anyone has harder information on the situation, I
would like to hear it.
Well, back to real work. I saw a Rubikized tetrahedron in a
shop window earlier this evening; I'm not sure whether I'm relieved
or infuriated that the store was closed for the day.
Date: 21 Sep 1981 08:49 PDT
From: Eldridge.ES at PARC-MAXC
Subject: An incomplete solution
To: Cube-Lovers@MC
cc: XeroxCubeLovers^.pa
Reply-To: Eldridge
For months now I have been living in blissful ignorance thinking that I too
could solve "the cube". To my horror and dismay I have found that my solution
is not complete. There are some cubes on the market that have pictures of fruit
on the faces rather than solid colors. I wondered if the solution I use would get
the pictures on the faces all lined up in the proper direction. I found that it
didn't! The problem is that some of the center cubies do not line up in the same
direction as all the other cubies on the face. I am currently working on finding
some macros that rotate the center cubies without affecting the rest of the cube.
I would suggest that you might find one of these fruit cubes and try it. Or you
can do as I did and mark the faces of an original cube so that you can tell the
orientation of the cubies. Good Luck!
George
Date: 21 September 1981 15:50-EDT
From: Richard Pavelle