Date: 10 Jan 1981 13:19 PST
From: McKeeman at PARC-MAXC
Subject: Re: nomenclature
In-reply-to: DDYER's message of 9 Jan 1981 1648-PST
To: Dave Dyer , VaughanW.REFLECS at HI-Multics (Bill
Vaughan)
cc: cube-lovers at MIT-MC
I agree there are two problems:
1. Neat "programs" that allow the recording and carrying out of manipulations.
2. Neat "configurations" that allow the recording of the results of carrying out
manipulations.
In both cases uniqueness, transparanecy, conciseness and all other notational
goodies are appropriate.
Generalizing a bit on Dave's suggestion, how about:
Manipulation = Macro*
Macro = MacroName "=" Move*
Move = Move "*" Integer -- power
| Move ' -- inverse
| Move / Move -- conjugate
| MacroName
| Face (Near (Middle Far?)?)? -- hand moves
| "(" Move* ")" -- parenthesization
Face = F | U | R
Near = Middle = Far = 0 | 1 | 2 | 3
with considerate use of spaces and carriage control. Face signifies "temporarily
move that face into the right hand, do the moves, then move it back where it
came from". For compatibility, the digits count QTW clockwise (away).
The FLUBRD equivalences are:
R=R=R1=R10=R100
RR = R*2
RRR = R*3 = R'
L=R003
U=U
D=U003
F=F
B=F003
Slice=R01
AntiSlice=R103
I=F111
J=R111
X Y X' = Y/X
Note this has some chance of generalizing to other slice puzzles. (E.g. 4x4x4) It
also subsumes FLUBRD with a few appropriate Macro definitions like those
above.
------
For configurations the problem can be attacked by specifying which cubies go to
which cubies. Singmaster does some of this. The problem is to find a way to
specify patterns of change without just listing all the changes. There are
positional change, flipping and rotating to be accounted for.
For instance, we would like to say (in some much neater way)
EdgeFlip = For all X and Y in FLUBRD, edge XY goes to edge YX.
The corners each take one step on a Hamilton path.
Each corner is rotated 120o.
Each center exchanges with its opposit.
Each edge XY goes to UV where U is the opposit of X and V is the opposit of Y.
Etcetera, etcetera, etc.
BFUDRLy yours,
Bill
Date: 10 JAN 1981 1405-PST
From: WOODS at PARC-MAXC
Subject: Re: nomenclature
To: McKeeman, DDYER at USC-ISIB, VaughanW.REFLECS at HI-MULTICS
cc: cube-lovers at MIT-MC
In response to the message sent 10 Jan 1981 13:19 PST from McKeeman@PARC-MAXC
I object! Your proposed notation discriminates against lefthanded cubists!
-- Don.
-------
Date: 10 Jan 1981 14:16 PST
From: McKeeman at PARC-MAXC
Subject: Re: nomenclature (discrimination against lefties)
In-reply-to: WOODS' message of 10 JAN 1981 1404-PST
To: WOODS
cc: cube-lovers at MIT-MC
Don,
The discrimination is actually self-imposed. Merely rename your hands and you
will never know you were discriminated against.
Bill
Date: 12 Jan 1981 0913-PST
From: Isaacs at SRI-KL
Subject: Stanford Rubik's Cube Club
To: cube-lovers at MIT-MC
There is a newly formed Rubik's Cube Club, meeting at Stanford,
every Thursday, 7 p.m., Crother Memorial Hall, Conference Room.
For information, call Kersten (415)321-7725 or Paul 446-0729.
Open to all - beginners and experts. First meeting was 1/6/81.
Second will be 1/20.
-------
Date: 12 Jan 1981 0929-PST
From: Isaacs at SRI-KL
Subject: nomenclature
To: cube-lovers at MIT-MC
I gave my father a cube a year ago (for which my mother may never forgive
me), and he has been working with it ever since. He has developed his own
notation, based on Angevine: Basically, take a corner as an X-Y-Z
co-ordinate system, and call the planes (e.g.) X1, X2, and X3 (where
x2 is the slice). L is X1, R is X3, F is Y1, etc. + is a clockwise
rotation, looking at the -1 face (thus the X3,Y3, and Z3 faces rotate
backwards from bfudlr); - is CCW. Half twists are (on a typewriter) +/-
(he doesn't have a computer terminal).
Anyway, his feeling on Singmaster nomenclature (with wich I disagree)
are as follows:
"I do feel that Singmaster's limited cubist vocabulary impairs
communication of his knowledge and insights. His general use of only
six of the nine groups available for rotation is like using only twenty
of the twenty-six letters in our alphabet. With an unabridged dictionary
and a thesaurus practically anything could still be said, but not as
well as using the whole alphabet. ... Perhaps mathematicians just don't
care about communication with ordinary people."
My father is a lawyer.
-------
Date: 12 Jan 1981 10:05 PST
From: McKeeman at PARC-MAXC
Subject: Re: nomenclature
In-reply-to: Isaacs' message of 12 Jan 1981 0929-PST
To: Isaacs at SRI-KL
cc: cube-lovers at MIT-MC
Well, lawyers have themselves occasionally been subject to some criticism for
their "communication with ordinary people".
My dictionary says "Angevine" has something to do with the line of
Plantagenet Kings. I guess the connection is too subtle for me.
More constructively, the Isaacs senior notation seems 1-1 with FLUBRD except
that it has additional primitives for the slices. The macro facility some have
used fills that hole.
The real trick is to find notations with (lots of) formal properties reflecting cubik
realities. Partly that is a matter of notation design, but mostly it is a matter of
deeper understanding of the subject matter. I do not believe it is an accident
that great science and great notations have frequently come from the same hand.
Bill
Date: 12 Jan 1981 1353-PST
From: Isaacs at SRI-KL
Subject: Re: nomenclature
To: McKeeman at PARC-MAXC
cc: cube-lovers at MIT-MC, ISAACS
You'd have to ask him about his relationship to the Plantagenets (or
Anjous), but James Angevine (with an 'e') wrote out an early(?) solution
the the cube which was then sold by Logical Games, Inc, one of the first
distributers of (what they called) the Magic Cube. Singmasters first
published solution seemed dificult to communicate, so I sent the 'Angevine
Solution' to my father.
Logical Games, Inc, incedentally, is currently manufacturing the cube in
the U.S.A., in white plastic with (I think) a slightly more pleasent color
scheme.
You're right that it's essentially the BFUDLR + slice, plus the
different use of CW and CCW on the face furthest away.
I hope to see you at the Rubik's Cube Club meeting. (Actually, I may
have to miss this weeks - maybe I'll see you there the 22nd.)
--- Stan
-------
Date: 15 January 1981 18:30 cst
From: VaughanW.REFLECS at HI-Multics (Bill Vaughan)
Subject: Weird Algorithm - spoiler warning?
To: Cube-Lovers at MIT-MC
On going through the old mail, I was a little surprised that
nobody uses the same algorithm that I do to solve the cube. But
since mine isn't terribly efficient, that's not much of a wonder.
Anyway, here it is.
1. Do bottom edges. Honest to god. I put all the bottom edges on
top by random dithering, then for each one, turn it so its
attached side facie abuts the color-matching face cubie, then
rotate that face 180o. That gets the bottom edges right, but
random hacking is almost as easy...
2. Do middle edges. (Getting colors right) I only use two moves
here. FR'F'R and R'FRF'. I pick a cubie that's on a top edge but
belongs on a middle edge - put its side facie adjacent to the
matching color face cubie (deja vu) and use FR'F'R if it has to
rotate right-and-down, or R'FRF' if it has to rotate
left-and-down.
3. Get top edges in correct places. Essentially as in Singmaster,
but I use only one of two moves. Align top edges so either: all
are OK (skip rest of this step) or one is OK and 3 are wrong. (if
that's impossible, use one of this step's moves at random and
restart step - guaranteed to work.) Now either FURU'R' or FRUR'U'
can be used to get everything OK.
4. Flip top edges as required. I use two different moves for this
according to whether adjacent or opposite edges need to be
flipped. Opposite: let Q = "turn body-slicing slice 1 qtw
clockwise as seen from right". Then 4(QU) 4(UQ) flips FU and BU.
Adjacent: FR'F'R.RU'R'U.UF'U'F flips RF and UF - you must reorient
the cube to do this on two U edges. (I like this move because of
its symmetry and - somehow - completeness. It also rotates the
corner cubies adjacent to the edge cubies that it flips.)
5. Get the corners right. Here I have some fun, but the basic
moves are: 3(FR'F'R) = (LFU,RBU) (RFU,FRD); 3(FRF'R') which also
does a double interchange - tho' it's asymmetrical and I don't use
it much; and B'FR'F'RB which stirs 3 of the top 4 CW or CCW - I
never remember because I just use it twice for the "wrong"
direction.
6. Tumble any corners that need it. Usually not many because of
the nice color flipping properties of 3(FR'F'R) -try it. My
tumbling move is a monotwist 2(FR'F'R).L.2(R'FRF').L' -- or
replace the L and L' with LL if necessary - sometimes it's nastier
and you have to do it twice.
I've never counted worst-case moves. The algorithm is based almost
entirely on Singmaster's Y commutator, and once you get that into
your finger bones, you hardly ever make a mistake. On the other
hand, this algorithm is bad enough it hardly deserves a spoiler
warning.
Bill q
Date: 15 January 1981 19:13 cst
From: VaughanW.REFLECS at HI-Multics (Bill Vaughan)
Subject: "Swirl Patterns"
To: Cube-Lovers at MIT-MC
I've been recently investigating a set of patterns that I call
Swirl patterns for lack of a better name. In a Swirl, each face
looks like one of these:
X X X X X X
X Y Z (Left-hand swirl) Z Y X (Right-hand swirl)
Z Z Z Z Z Z
where X and Z are complementary colors, and Y is something else.
I've classified them roughly into 6 classes, based on handedness
of swirl and relative alignment of faces.
If you look at the 3 faces adjacent to a corner, they may have the
same handedness, or they may have mixed handedness. In addition,
two adjacent faces may have parallel or perpendicular swirls.
(Parallel swirls have their XYZ columns parallel; perpendicular
swirls have thir XYZ columns perpendicular.)
Here are my 6 classes; there are another 6 which are mirror images
of these, but I don't count them. Nor do I care (at the moment)
about color pairings - though I know they are important - or about
the colors of the face cubies, which probably aren't important.
1. Same handedness. Two of the three faces have parallel swirls.
2. Same handedness. All three faces have mutually perpendicular
swirls.
3. Mixed handedness. The two same-handed faces are parallel, with
thir XYZ columns in contact (i.e. forming a belt around the cube).
4. Mixed handedness. The two same-handed faces are perpendicular;
the opposite-handed face is perpendicular to both.
5. Mixed handedness. The two same-handed faces are perpendicular;
the opposite-handed face is parallel to one.
6. Mixed handedness. The two same-handed faces are parallel, with
their XYZ columns pointing towards the third face.
Four of these classes are in the antislice group and are a short
distance (8 qtw) away from SOLVED. They are classes 1, 3, 5 and 6.
They are also the antislice group's analogues of the slice group's
"6-spot" or "twelve-square" patterns.
What got me started on this is a problem that I still have. One
day while playing aimlessly in the antislice group (I thought I
had remained in it) I ran across a class 2 Swirl, which was (a)
quite pretty (when looked at from the correct corner it looks
like a pinwheel) and (b) a bear to solve. (Clearly I thought it
was one of the "easy" Swirls.)
Having solved it, I wanted to get back to it and found I didn't
know how. I tried solving to it and came up with an impossibility
- that's how I know the color arrangements must be important - and
I haven't found it in my searches yet - nor have I found class 4.
Questions: what's the fastest way to get to a class 2 Swirl? What
color arrangements are permissible? Is it really in the antislice
group? (I now believe not.) Is any class 4 Swirl achievable? How
quickly? Is there anything else interesting about Swirls?
I'm still playing with these - will give more data as I get it.
Bill
Date: 16 January 1981 12:09 cst
From: VaughanW.REFLECS at HI-Multics (Bill Vaughan)
Subject: more on Swirl patterns: the Pinwheel
To: Cube-Lovers at MIT-MC
I now have a class 2 swirl that looks like this:
uud
uld
udd
bbfrrrffbrrr
buflfrfdblbr
bfflllfbblll
uud
urd
udd
It's pretty, and looks like a pinwheel from 4 of the corners, so i
call it a Pinwheel.
My current algorithm to get from SOLVED to Pinwheel is 84 qtw and
not worth publishing - I expect to reduce that substantially in
the near future.
Bill
Date: 01/16/81 1322-EDT
From: PLUMMER at LL
Subject: Ad in Popular Science
To: Cube-lovers at MIT-MC
Seems like a guy will tell you how to solve the cube for only $5.
Check the classified ads in current Popular Science! --Bill
-------
Date: 20 Jan 1981 1632-PST
From: Isaacs at SRI-KL
Subject: Rubiks Cube Club meeting
To: cube-lovers at MIT-MC
The next meeting of the Stanford University Rubiks Cube Club (SURCC??)
will be thursday, Jan. 22.
Meyer Library
Room 145
7:30 Puzzling with the cube, introduction to cube solving algorithms
8:00 Discussion on how to design and build new 3-D puzzles (magic
tetrahedron, etc.)
8:30 Cube theory and Pretty Patterns. Until 9:30 or so.
Further information: Kersten, (415)321-7725.
Also see article in Stanford Daily. See you all there.
-- Stan
-------
Date: 21 January 1981 1246-EST (Wednesday)
From: Guy.Steele at CMU-10A
To: Isaacs at SRI-KL
Subject: Re: Rubiks Cube Club meeting
CC: cube-lovers at MIT-MC
In-Reply-To: Isaacs@SRI-KL's message of 20 Jan 81 19:32-EST
Message-Id: <21Jan81 124647 GS70@CMU-10A>
Stanford University Cube Kludge Society??
(Sorry, just kidding.)
--Guy
Date: 21 Jan 1981 10:07 PST
From: McKeeman at PARC-MAXC
Subject: Re: Rubiks Cube Club meeting
In-reply-to: Guy.Steele's message of 21 January 1981 1246-EST (Wednesday),
<21Jan81 124647 GS70@CMU-10A>
To: Guy.Steele at CMU-10A
cc: Isaacs at SRI-KL, cube-lovers at MIT-MC
No. Actually
Stanford University Rubik Environment For University Nuts.
Bill
Date: 22 January 1981 0010-EST (Thursday)
From: Dan Hoey at CMU-10A, James Saxe at CMU-10A
To: Cube-Lovers at MIT-MC
Subject: Correction to "Symmetry and Local Maxima"
Reply-To: Dan Hoey at CMU-10A
Message-Id: <22Jan81 001000 DH51@CMU-10A>
In our message "Symmetry and Local Maxima" (14 December
1980 1916-EST) we examined local maxima both in the Rubik group and
in the Supergroup. David C. Plummer has discovered a flaw in our
argument for the Supergroup, which we now correct.
Plummer has previously noted (30 DEC 1980 0109-EST) that
the T-symmetric position GIRDLE CUBIES EXCHANGED, depicted near the
end of section 4, is an odd distance from SOLVED. This is also true
of the composition of GIRDLE CUBIES EXCHANGED with GIRDLE EDGES
FLIPPED, ALL EDGES FLIPPED, PONS ASINORUM, or any combination of
the three, for a total of eight positions. In addition, there are
four different T groups, each corresponding to a choice of opposite
corners of the cube. Thus 32 of the 72 positions with Q-transitive
symmetry groups are an odd distance from SOLVED.
The discussion of the Supergroup in S&LM noted that the
only face-center orientations which yield Q-transitive symmetry
groups are the home orientation and all face centers twisted 180o
(called NOON in Hoey's message of 7 January 1981 1615-EST). Any
position with either of these face center orientations must be an
even distance from SOLVED, so that any reachable position which is
T-symmetric in the Supergroup must be an even distance from SOLVED.
In our earlier note, we erroneously calculated the number
of Supergroup positions with Q-transitive symmetry groups by simply
doubling the number of such positions in the Rubik group to allow
for the two allowable face-center orientations. What we failed to
notice--until Plummer pointed it out--is that neither of the
allowable face-center orientations can occur in conjunction with an
odd position.
The corrected count of known Supergroup local maxima is
determined by counting the 40 *even* symmetric positions, multiplying by
two, and subtracting 1 for the identity, yielding 79. As Plummer notes,
this is surprisingly close to the number of known local maxima in the
Rubik group, which stands at 71. The number of known local maxima
modulo M-conjugacy is 25 for the Rubik group and 35 ( = 2*(26-8) - 1 )
for the Supergroup.
Date: 21 Jan 1981 14:42:58-PST
From: microsoft!zibo at Berkeley
Gentlepeople:
I have moved and would like to have my CUBE-LOVERS mail correctly sent:
My former destination was: ZBIKOWSKI@MARKET. I now reside in:
CSVAX.MICROSOFT!ZIBO. Could you make the necessary changes? Gracias.
Date: 27 January 1981 0102-EST (Tuesday)
From: Jim Saxe, Dan Hoey
To: Cube-Lovers at mit-mc
Subject: Pretty Patterns and Solutions
Sender: Dan Hoey at CMU-10A
Reply-To: Dan Hoey at CMU-10A
Message-Id: <27Jan81 010221 DH51@CMU-10A>
We are disappointed at Chris C. Worrell's use of the term
"Baseball" for the position known in the literature as the "Worm".
Worrell's term propagates the apparently popular misconception that
baseballs are covered with three-lobed pieces of leather. The
position which *we* call "Baseball" reflects the construction much
more accurately:
D D D
U U U
F F F
U U U L R B R B L B L R
F F F L R B R B L B L R
D D D L R B R B L B L R
F F F
D D D
U U U
Currently, our best process for this position is 34 qtw. The
corners are fixed with FRLUDB, two edge four-cycles are inserted in
the middle, and a Spratt wrench is conjugated inside that:
FRL (RRLL UUDD F' U (F' LUD' BUD' RUD' FUD' F) UDD RRLL F) UDB.
The class of patterns which Bill Vaughan calls "Swirl
Patterns" (15 January 1981 19:13 cst) are also known as "6-2L"
patterns in Singmaster, and the particular one he calls the
"Pinwheel" (on 16 January 1981 12:09 cst) is an M-conjugate of the
AC-symmetric "Twelve-L's" mentioned in our message on Symmetry and
Local Maxima (14 December 1980 1916-EST, Section 6). [Incidentally,
the diagram he displays in the message of 16 January is in error;
the left and right face centers have been swapped. This is made
less obvious by the unusual orientation of the cube in that
diagram.] We have found a totally magical 12 qtw process for the
Pinwheel: FB LR F'B' U'D' LR UD.
Vaughan's definition of Swirl Patterns seems unduly
restrictive to us on one count: he requires the two L's on each
face to be of "complementary" (evidently meaning opposite) colors.
This is not necessary for an L pattern. According to our analysis,
however, at least two of the faces of any L pattern must have L's
of opposite colors, and five is easily seen to be impossible. We
know of no patterns having three or four such pairs. But there are
several with two pairs. Our favorite example is a relative of the
Baseball which we name for Linda Lue Leiserson, who has the
appropriate initials.
D D D
F U D
F F F
U U U L B B R L L B R R
D F U L R B R B L B L R
D D D L L B R R L B B R
F F F
U D F
U U U
We have a 24 qtw process for Linda Lue's L:
F'BB L (B U'LR' F'LR' D'LR' B'LR' B') R UD B'FF
This has a Spratt wrench, conjugated by B', embedded in magic.
David C. Plummer (3 SEP 1980 2123-EDT) reported that it is
possible for each of the six faces of the cube to show a capital
"T". Our analysis indicates that there are two sorts of T patterns:
D U D D D U
D U D U U U
U U U D D U
L L L F F F R R R L R R F F F R L L
R L R B F B L R L L L L B F B R R R
R L R B F B L R L L R R B F B R L L
D D D D U U
U D U D D D
U D U D U U
B B B B B B
F B F F B F
F B F F B F
Tanya's T Plummer's T
Tanya's T is named for Tanya Sienko (who inspired the problem) and
for euphony. Plummer's T is named for Plummer's Cross (which has
the same symmetry group) and for homophony. There are 24
M-conjugates of Tanya's T, while Plummer's T has 8 M-conjugates. By
adapting a process due to David C. Plummer, we have developed a
16-qtw process for Tanya's T: (FF UU)^3 (UU LR')^2. The first part
swaps two pairs of edge cubies, and the second part is magic. We
have found a 28-qtw process for Plummer's T, which is entirely
magical: FF UD' F'B' RR F'B U'D RL FF RL' UD' RL FF R'L U'D'.
A position which is not so visually striking, but which is
important in the symmetry theory we have discussed earlier, is "All
Corners Twisted":
B U B
U U U
F U F
U L U L F R U R U R B L
L L L F F F R R R B B B
D L D L F R D R D R B L
F D F
D D D
B D B
This can be achieved in 30 qtw with FLU (LRRFFB')^4 U'L'F'.
The process is a conjugated from a 24 qtw process invented by
Thistlethwaite. Unfortunately, Thistlethwaite's process twists the
wrong corners, and no cancellation can be performed in the
conjugation. If any process can be found which twists four corners
clockwise and four counterclockwise, leaving the rest of the cube
fixed, then any such pattern can be made by adding at most 6 qtw.
Date: 29 JAN 1981 0344-EST
From: BSG at MIT-AI (Bernard S. Greenberg)
Mailed-by: BSG @ MIT-Multics
Subject: Lisp Machine Cubesys Improvements
To: CUBE-LOVERS at MIT-MC
Largely due to the newfound coincidence of relentless Lisp Machine
hacking with my job, I have added a slew of features to Lisp Machine
Cubesys, viz., all kinds of New Window System and flavor hacking. It is
now completely mouse-oriented, all moves are made by mousing menu items,
OR by mousing at cube-sides on the display (either display) to turn
faces, left mouse button for left (ccw) turn, etc.! To find out more
about it, load it in the usual way, (load "bsg;cubpkg >"), invoke it
in the usual way ((cube)), and type the HELP key (or mouse the HELP
menu item). Have fun, -bsg
Date: 1 February 1981 0539-EST (Sunday)
From: Dan Hoey at CMU-10A
To: Cube-lovers at MIT-MC
Subject: Algorithm for finding cube group orders
Message-Id: <01Feb81 053933 DH51@CMU-10A>
David C. Plummer (31 Dec 1980 1210-EST) gave a preliminary
analysis of the 5x5x5 cube. I complete it here. Let a move consist
of twisting any of the six faces, at a depth of 1 or 2. It will be
necessary to consider the two depths as distinct; M1P will refer to
the number of depth 1 moves (mod 2), while M2P will refer to the
number of depth 2 moves (mod two). It is important to realize that
the two parities vary independently. The tabs on each face are
assigned types
C L E R C
R D A D L
E A X A E
L D A D R
C R E L C
as in Plummer's analysis.
Let COP ("C" Orientation Parity) and CPP ("C" Permutation)
parity be defined as before. As before, COP=0 (mod 3). We must be
explicit about the CPP this time: Since either kind of move is an
odd permutation of the "C" faces, CPP=M1P+M2P.
As in the 4x4x4 case, "R"'s may be ignored and "L"'s have
no orientation. The permutation parity (LPP) is important, however.
Depth 1 moves are an even permutation of the "L"'s (two 4-cycles),
so they do not affect the LPP, but Depth 2 moves are an odd
permutation of the "L"'s (three 4-cycles). Therefore LPP=M2P. Note
that while LPP and CPP may vary independently, they together
determine both M1P(=LPP+CPP) and M2P(=LPP).
The "E" faces act as in the 3x3x3 case, with orientation
and permutation parity. Orientation changes on four "E"'s with
every move, so EOP=0 (mod 2). Permutation parity changes with every
move, so EPP=M1P+M2P. This has already been determined by CPP, so
only half of the "E" permutations are possible.
Every move is an odd permutation of the "D" faces, so
DPP=M1P+M2P. Since M1P+M2P=CPP is determined, only half of the "D"
face permutations are possible.
Moves work differently on "A" faces depending on depth:
Depth 1 moves are odd permutations of the "A"'s, and depth 2 moves
are even. Thus APP=M1P, which is determined by CPP+LPP, so only
half of the "A" permutations are possible.
Finally, the "X" faces have orientation which changes on
every move, so XOP=M1P+M2P, and only half of the "X" orientations
are possible.
Thus there are 96 orbits, corresponding to COP (mod 3) and
EOP, EPP+CPP, DPP+CPP, APP+CPP+LPP, and XOP+CPP (mod 2). The basic
combinatoric is as Plummer described:
8! C Permutations
3^8 C Orientations
24! L Permutations
1 R Permutation
12! E Permutations
2^12 E Orientations
24! D Permutations
24! A Permutations
4^6 X Orientations
which when multiplied together and divided by 96 yields about
5.289*10^93. [This differs from Plummer's result by a factor of
4096 because (4^6/2) he didn't count X Orientations, and (2) he did
not realize that LPP and CPP are independent.] My implementation of
Furst's algorithm claims that all of these are reachable.
To count the number of reachable color patterns, divide this note
that there are by (4!)^6/2 invisible D permutations, (4!)^6/2
invisible A permutations, and 4^6/2 invisible X orientations that
satisfy the invariants. While there are pairs of L/R edges that
look the same, they cannot be interchanged, for that would entail
putting an L tab into an R position. So there are 2.829*10^74
different color patterns achievable.
----------------------------------------------------------------
Date: 1 February 1981 0612-EST (Sunday)
From: Dan Hoey at CMU-10A
To: Cube-lovers at MIT-MC
Subject: Algorithm for computing cube group orders
Message-Id: <01Feb81 061255 DH51@CMU-10A>
Oops... you have just received part 3. This is part 1....
This note is in somewhat delayed response to the note by
David C. Plummer (31 DEC 1980 1115-EST) regarding the 5x5x5 Rubik
cube, and some related ideas. In that note he tried to calculate
the size of that cube's Rubik group, but left several of the values
open to conjecture. I will complete the answer, and answer a few
others that haven't been addressed here.
Computing the size of a Rubik group is a special case of
computing the size of a permutation group, given generators for
that group. The technique we have already seen in these pages is in
two parts. The first part seems relatively easy: certain invariants
must be observed in the generators, such as "Corner Orientation
Parity" and "Total Permutation Parity." [In this general setting,
such invariants as "Colortabs on the same cubie move together" must
also be considered.] It may take some thought to dig out the
invariants, but once you have seen them demonstrated for Rubik's
Cube, you have an idea of what to look for. The second part is the
devil: it must be demonstrated that every permutation satisfying
those invariants is actually generated. This involves developing a
solution method for the puzzle. Given the days or weeks (or
eternity) it takes most people to develop such a method--with cube
in hand!--it is hardly surprising that few answers have been
developed.
Well, the second part is no longer a hard problem. The
answer lies in a paper by Merrick Furst, John Hopcroft, and Eugene
Luks, entitled "Polynomial-Time Algorithms for Permutation Groups,"
which was presented at the 21st Annual Symposium on Foundations of
Computer Science, October 1980. Among the results is an algorithm
which takes as input a set of permutations on n letters, and
reports the size of the group G which is generated by those
permutations.
The algorithm operates by decomposing G into a tower of
groups I=G[0], G[1], ..., G[n]=G, where G[i] contains those
permutations p in G for which p(k) = k whenever i < k <= n. The
index of G[i-1] in G[i] is developed explicitly by the algorithm;
in fact, a representative g[i,j] of every coset of G[i-1] in G[i]
is exhibited. These coset representatives generate G; in fact,
every element of G is representable as a product of the form
(g[1,j1])(g[2,j2])...(g[n,jn]). For this reason the coset
representatives are called "strong generators" for G. There is a
good deal of structure that can be learned from the strong
generators, in addition to the size of G.
I have coded this algorithm in Pascal, and offer the
program for the use of anyone who needs to find group orders. The
relevant files are on CMU-10A, from which other sites may FTP
without an account. The relevant files are
all:group.pas[c410dh51] The source
all:rubik.gen[c410dh51] A sample input -- the supergroup
all:rubik.lst[c410dh51] Sample output.
Of course, CMU Pascal is probably slightly different from yours,
and OS-dependent stuff like filenames is likely to be wrong. I'll
be glad to help out in cases of transportability problems. The
other problem you may run into is resource availability. The
running time of the algorithm is proportional to (nm)^2, where m is
the total number of strong generators; the supergroup (n=72, m=279)
takes 639 cpu seconds on a KL-10, and bigger problems grow rapidly.
The program also requires 47000+47m words.
It might seem that the problem has been answered, but I
find that simply knowing the size of a group is not very
satisfying. There doesn't seem to be a better way of demonstrating
lower bounds, but the upper bounds that come from invariants are
much more elegant than a simple numerical answer. Unfortunately, I
know of no mechanical way of finding the invariants. Furthermore,
using group theory does not help much when we ignore the
Supergroup. Consider the 4x4x4 cube. If we are only concerned with
the color pattern on the cube, then a twist may or may not affect
the four face centers--it depends on whether they are the same
color or not.
In summary, the algorithm has inverted the hard and easy
parts of cube analysis. The size of the group is now easy to
determine, making invariant-finding the hard part. Further, the
algorithm works on the Supergroup, making counting distinct color
patterns the part which requires further analysis. Two messages
follow, supplying these answers for the 4x4x4 and 5x5x5 cubes.
Date: 1 February 1981 0651-EST (Sunday)
From: Dan Hoey at CMU-10A
To: Cube-lovers at MIT-MC
Subject: Analysis of the 4x4x4 cube
Message-Id: <01Feb81 065108 DH51@CMU-10A>
The first problem for the 4x4x4 cube is in eliminating
positions that arise from whole-cube moves. This was done with the
3x3x3 by keeping the face-center positions fixed, but there are no
face centers on the 4x4x4--or there are, but they don't maintain a
fixed orientation relative to each other. I standardize the spatial
orientation by keeping the DBR corner in a fixed position and
orientation. A move then consists of twisting one, two, or three
layers parallel to the U, F, or L face. Thus F3' is equivalent to
twisting the B face. I will refer to the number of layers twisted
as the "depth" of the move.
Following David C. Plummer's notation (31 DEC 1980 1210-EST),
organize each face of the cube as
C L R C
R X X L
L X X R
C R L C.
I will assume familiarity with David Vanderschel's analysis of the
3x3x3 case, which was presented in his message of 6 August 1980.
"C" faces act as they do in the 3x3x3 case, except that one
of them does not move. Corner Orientation Parity (COP) is preserved
and Corner Permutation Parity (CPP) changed by every quarter-twist.
Depending on the depth, a quarter twist can permute the "L"
faces in an odd or an even permutation. Also, "L" faces do not
change orientation (or move to "R" positions). Every "R" face is
determined by the "L" face (on an adjacent side of the cube) with
which it shares a cubie. Thus the arguments for EOP and EPP do not
apply.
Every quarter-twist is an odd permutation of the "X" faces:
either one, three, or five four-cycles, depending on the depth.
Letting XPP be the permutation parity of the "X" faces, the Total
Permutation Parity TPP=XPP+CPP (mod 2) is preserved by every
quarter-twist.
Thus the 4x4x4 cube group has at least six orbits,
according to COP (mod 3) and TPP (mod 2). The basic upper bound of
7! Corner Permutations
3^7 Corner Orientations
24! L Permutations (which determine the R permutations), and
24! X Permutations,
divided by six, yields an upper bound (of about 7.072*10^53). I have
run Furst's algorithm on the problem, and my program claims that all
these positions are reachable.
To calculate the number of reachable color patterns, note
that there are 4! permutations of each quadruple of "X" faces which
are indistinguishable. However, the TPP constrains the XPP so as to
reduce this by a factor of two. Dividing 7.072*10^53 by (4!)^6/2
yields 7.401*10^45.
[At this point, you may find it instructive to view the
message before last, which analyzes the 5x5x5 cube in the context
of this message and the one immediately preceding. I regret the
accidental disorder. These three are all for now, although I have
results on tetrahedra, octahedra, and a dodecahedron which I am in
the process of writing up.]
Date: 6 Feb 1981 at 1330-CST
From: korner at UTEXAS-11
Subject: cube lube
To: cube-lovers at mit-mc
After trying almost all the cube lubricants suggested (with
the notable exceptions of the plastic eating varieties)- I would like
to suggest that a local maxima seems to be silicon gel (of the sort
used to lubricate SCUBA O rings or food processors- not the spray, the
gel).
To use this stuff, one must disassemble the cube. As long as it's
apart, take a fine flat file to the cubies and remove any seams from the
molding process and any imperfections from the glue job (cement beads or
protruding plates). Cubus hungarius finishes well with just a file,
cubus americanus (the white one) may need work with wet fine sandpaper
to restore a smooth surface after filing. If you're really fanatic,
adjust the screw tension (ala Singmaster). Clean off the debris and apply
liberal coats of the gel to all tab faces. Reassemble the cube and enjoy-
one handed cubing not guaranteed but definitely possible.
-KMK
-------
Date: 9 FEB 1981 2345-EST
From: JURGEN at MIT-MC (Jonathan David Callas)
Subject: True Stories of Cubism
To: CUBE-LOVERS at MIT-MC
I was at the Hirshhorn (Smithsonian Modern Art Museum) last Sunday to see the
exhibit of Avant-garde Russians, and lo, I saw in the museum shop what could
only be Cubus Albus! I played with it
for awhile (I solved the top 2 tiers before my girlfriend said "That's enough!
You can do that at home!") and it worked more smoothly than my C. Americanus!
So now, I guess, not only is the Cube a source of mathematical inspiration,
but an objet d'art as well.
Happy Cubing,
Jurgen at MIT-MC
Date: 11 FEB 1981 1600-EST
From: RP at MIT-MC (Richard Pavelle)
To: CUBE-LOVERS at MIT-MC
The March issue of Scientific American is out. Guess what is on the
cover as well as in the interior?
Date: 12 Feb 1981 0816-PST
Sender: OLE at DARCOM-KA
Subject: The England Scene
From: Ole at DARCOM-KA (Ole J. Jacobsen)
To: Cube-Lovers at MIT-MC
Message-ID: <[DARCOM-KA]12-Feb-81 08:16:37.OLE>
Yes, cubes are indeed very big here in England due mostly to the fact
that they have been featured on television several times recently.
About 3 weeks ago a kid solved a cube in 37 secs on the Saturday mor-
ning BBC1 show "Multicoloured Swap Shop" (very appropriate name). In
a follow up a group of people challenged him, but "only" managed it
in 57 secs. Nothing was however said about local maxima etc, so it
wasn't a very scientific exercise. The recordholder's solution se-
quence was shown in slow motion (his hands still seemed to move very
fast) and as far as I could determine he uses Kertezs's Algorithm,
i.e layer-by-layer, but with some clever shortcuts rather than just
using the macros blindly. At the moment cubes are impossible to get
but we are hoping for a new shipment to arrive soon. A cube club
will probably be formed here at Newcastle University, Newcastle upon
Tyne and I would be surprised if other universities won't be doing
the same.
By the way, has anyone ever tried turning cube when the temperature
is 5-6 degrees C? I have, because that is the temperature my room is
at when I come home at night. English houses are VERY cold.
Ole
Date: 16 February 1981 1229-EST (Monday)
From: Guy.Steele at CMU-10A
To: bug-lispm at MIT-AI, cube-lovers at MIT-MC
Subject: Scientific American
Message-Id: <16Feb81 122922 GS70@CMU-10A>
Congratulations to cubemeisters, LISP Machinists, and Symbolicists
alike for making *Scientific American*.
Now that the LISP Machine has been used to serve the cause of
cubing, has any thought been given to the converse? For example,
perhaps a mouse/joystick-like device could be built based on
cube technology?
Also, anyone thought about the limiting case of odd-shaped
polyhedra: the continuous cube (or, Rubik's sphere)? There are
three possible places to introduce continuity. For a given
twist, one must choose an axis, choose a depth of slice,
and choose an angle of twist. For the cube all three are
quantized. What are the geometric/topological properties
of an object where some subset of these three choices are
given a continuous domain? (I haven't the mathematics undert
my belt to attack this problem -- sorry.)
--Guy
Date: 16 February 1981 2327-EST (Monday)
From: Jim Saxe, Dan Hoey at CMU-10A
To: Cube-Lovers at MIT-MC
Subject: Four colors suffice
CC: Mary Shaw at CMU-10A, Paul Hilfinger at CMU-10A, Bill Wulf at CMU-10A,
Dorothea Haken at CMU-10A
Sender: Dan Hoey at CMU-10A
Reply-To: Dan Hoey at CMU-10A
Message-Id: <16Feb81 232721 DH51@CMU-10A>
Douglas Hofstadter, in the Metamagical Themas column in
Scientific American this month, shows two alternate ways of
coloring a cube. Both suffer from two drawbacks: They fail to
distinguish all cube positions, and they use more than six colors.
This seems inefficient to us, since there is a coloring of the cube
which distinguishes all elements of the Supergroup and uses
only four colors (and which, like Hofstadter's colorings and
the standard coloring, satisfies the restriction that every
whole-cube move is a color permutation, as discussed in point 2
below).
Our coloring, called the Tartan, is formed by assigning the
colors blue, green, red, and yellow to the four pairs of antipodal
corners of the cube. Thus for each face of the cube, the four
corners of the face are assigned four different colors. We use the
term ``plaid'' to denote such an assignment of colors to the
corners of a square. To color the cube, divide each facelet of each
cubie into four squares, and color the squares so all facelets on a
side of the cube display the plaid associated with that face. The
result is shown below, with the initial assignment of colors to
corners in lower case.
(r)---------------(y)
| R Y R Y R Y |
| B G B G B G |
| |
| R Y R Y R Y |
| B G B G B G |
| |
| R Y R Y R Y |
| B G B G B G |
(r)---------------(b)---------------(g)---------------(y)
| R B R B R B | B G B G B G | G Y G Y G Y |
| G Y G Y G Y | Y R Y R Y R | R B R B R B |
| | | |
| R B R B R B | B G B G B G | G Y G Y G Y |
| G Y G Y G Y | Y R Y R Y R | R B R B R B |
| | | |
| R B R B R B | B G B G B G | G Y G Y G Y |
| G Y G Y G Y | Y R Y R Y R | R B R B R B |
(g)---------------(y)---------------(r)---------------(b)
| Y R Y R Y R |
| G B G B G B |
| |
| Y R Y R Y R |
| G B G B G B |
| |
| Y R Y R Y R |
| G B G B G B |
(g)---------------(b)
| G B G B G B |
| R Y R Y R Y |
| |
| G B G B G B |
| R Y R Y R Y |
| |
| G B G B G B |
| R Y R Y R Y |
(r)---------------(y)
To understand the importance of the Tartan, there are
several points to consider:
1. By reading off the four colors of a plaid in clockwise
order, starting at an arbitrary point, we obtain four permutations
of the four colors. Quadruples read from different faces are
disjoint, so all 24 permutations of the four colors appear on the
Tartan, once each.
2. Every motion in the group C of whole-cube rotations is a
permutation of the pairs of antipodal corners, and so corresponds to
a recoloring of the Tartan. Some restriction of this sort is
necessary to prevent us from simply drawing a different
black-and-white picture on each facelet and calling that a
two-coloring.
3. Point 2 implies that C is isomorphic to a subgroup of
S4, the group of permutations on the four colors. But both C and S4
have 24 elements, so C is isomorphic to S4 itself (a fact
well-known to crystallographers).
4. Since every color permutation is realizable by a
whole-cube move, there is only one Tartan (up to whole-cube moves).
This is why we use colors as labels, rather than some FLUBRDoid
positional scheme. [The actual choice of colors and the name
``Tartan'' arise from the DoD Ironman project.]
5. Every reflection of the Tartan is color-equivalent to a
rotation. In particular, the identity is color-equivalent to a
reflection through the center of the cube. If you were to lend your
Tartan to someone who ran it through a looking-glass, you could not
discover the fact except by removing the face-center caps and
examining the screw threads!
We have constructed a Tartan from a Rubik's cube and
colored tape. Due to the similar appearance of the plaids, it takes
us several times as long to solve the Tartan as it takes to solve
Rubik's cube.
Our search for pretty patterns has not been particularly
rewarding. Part of the reason seems to be that the cube's
appearance is strongly constrained by the Tartan's coloring. On
Rubik's cube one may make a particular face pattern (e.g. orange T
on white background) using any of several identically colored
facelets. On the Tartan, however, the plaid on any facelet of a
cubie, together with the orientation of the plaid relative to the
cubie, determines the plaid and orientation of the other facelet(s)
of the cubie.
The one nice pattern we have is in fact the conceptual
precursor to the Tartan. It is Pons Asinorum (FFBBUUDDLLRR) applied
to the position shown in the diagram above. In this position, the
plaids of adjacent facelets line up with each other to display the
same arrangement of plaids, magnified by a factor of two. Each face
looks like the following, for some assignment of colors to the
numbers 1 through 4:
(1)---------------(2)
| 1 2 2 1 1 2 |
| 4 3 3 4 4 3 |
| |
| 4 3 3 4 4 3 |
| 1 2 2 1 1 2 |
| |
| 1 2 2 1 1 2 |
| 4 3 3 4 4 3 |
(4)---------------(3)
Date: 17 Feb 1981 07:54:00-PST
From: microsoft!zibo at Berkeley
Is it possible??? In the SciAm article they mentioned 4x4x4 cubes...
Has anyone seen them??
Date: 17 Feb 1981 10:49 PST
From: McKeeman at PARC-MAXC
Subject: Re: 4x4x4 cube
In-reply-to: Your message of 17 Feb 1981 07:54:00-PST
To: microsoft!zibo at Berkeley
cc: cube-lovers at MIT-MC
Zibo,
All things are possible in the computer. My undertanding is that a 4x4x4 is
being built, but I have not heard that it is yet complete.
The real mind-bender is the continuous Rubik Sphere. Take a sphere, slice it
arbitrarily many times. Now each slice makes a plane of rotation and gives a
degree of freedom. The Rubik Cube is a special case for which there is a
mechanical implementation.
Bill
Date: 17 Feb 1981 1445-PST (Tuesday)
From: Mike at UCLA-SECURITY (Michael Urban)
Subject: Rubik's Sphere
To: cube-lovers at mit-mc
While it shouldn't be TOO hard to program an arbitrarily-fine
simulation of a Rubik sphere, one wonders how it's colored.
do the latitute/longitude coordinates on the sphere correspond
to hue/intensity? While you can settle for any coordinate
labelling of points, different coloring schemes will dictate
what constitutes a "pretty" pattern, yes?
Mike
-------
Date: 17 February 1981 18:03 cst
From: VaughanW at HI-Multics (Bill Vaughan)
Subject: Rubik's Sphere
Sender: VaughanW.REFLECS at HI-Multics
To: Mike at UCLA-Security, cube-lovers at MIT-MC
Do you color a Rubik's Sphere discretely or continuously? And if
continuously, it's probably too easy to solve - seems as though
the discontinuities would show you where to turn it; and by always
turning along a discontinuity until it vanishes, you can get to
SOLVED by what seems to be God's Algorithm.
Bill
Date: 17 Feb 1981 16:12 PST
From: McKeeman at PARC-MAXC
Subject: Re: Rubik's Sphere
In-reply-to: Mike's message of 17 Feb 1981 1445-PST (Tuesday)
To: Mike at UCLA-SECURITY (Michael Urban)
cc: cube-lovers at mit-mc (I wish Hofstadter were on the net)
Mike,
Are you proposing a truly continuous Rubik sphere with an infinite, nay
uncountable, number of slicings with continuously varying hue to distinguish
"slices"?
Such cubes could differ in the "function" that connects the motion of
"neighboring slices". We could have linear, quadratic, and even
hyperexponential axes of rotation. Then giving the cube a spin about each of its
(many) axes, we would have a continuously shifting pattern of color. Maybe
would should leak this idea to George Lucas for the visuals of StarWars III? Or
maybe one of the LISP machine folks can whip up a simulation overnite?
Bill
Date: 17 Feb 1981 1622-PST
From: Steve Saunders
Subject: Rubik-like sphere
To: Cube-Lovers at MIT-MC
And if you color it continuously, why not have continuous moves, too? For
instance, a smooth twist about an axis (like twisting a rubber ball that's
glued to sticks at its poles -- carries meridians into spirals), or a smooth
bending (like pushing one of those poles sideways while holding the other
fixed -- makes parallels not parallel). I suspect that the groups resulting
from some sets of smooth motions would be very simple, but some might have
interesting interactions. A problem with all this smoothness (a feature?) is
that it would enable approximate solutions, iterative converging infinite
"solutions", and disputes about whether SOLVED has in fact been reached --
none of which occur with the real Rubik's.
Steve
-------
Date: 17 Feb 1981 1716-PST (Tuesday)
From: Lauren at UCLA-SECURITY (Lauren Weinstein)
Subject: Sphere carrying case
To: CUBE-LOVERS at MC
I know where I'm going to keep my Rubik's Sphere when I get one:
inside my Klein Bottle!
--Lauren--
-------
Date: 17 Feb 1981 18:00 PST
From: McKeeman at PARC-MAXC
Subject: Re: Rubik's Sphere
In-reply-to: VaughanW.REFLECS's message of 17 February 1981 18:03 cst
To: VaughanW at HI-Multics (Bill Vaughan)
cc: Mike at UCLA-Security, cube-lovers at MIT-MC, SAUNDERS at USC-ISIB
Bill,
Well, My idea is that the continuous slicing is constrained by some function so
that you basically only have one degree of freedom per axis. The trick is to
scramble on several axes, and then try to get back. If the coloring is continuous
before any twisting, then it is always continuous. (I think)
Another Bill
Date: 20 February 1981 20:17 est
From: Greenberg.Symbolics at MIT-Multics
Subject: A lighter note
To: CUBE-HACKERS at MIT-AI
in this increasingly hirsute forum:
A man called us at Symbolics today, having seen our name in the
Scientific American article. He was having trouble getting
cubes in the Chicago area, and wanted to know if we could sell
him some.... (a true story)...
Date: 20 February 1981 20:19 est
From: Greenberg.Symbolics at MIT-Multics
Subject: A lighter note
To: cube-hackers at MIT-MC
in this increasingly hirsute forum:
A man called us at Symbolics today, having seen our name in the
Scientific American article. He was having trouble getting
cubes in the Chicago area, and wanted to know if we could sell
him some.... (a true story)...
Date: 20 FEB 1981 2108-EST
From: RP at MIT-MC (Richard Pavelle)
Subject: Rubik
To: CUBE-HACKERS at MIT-MC
When speaking to the editor of Scientific American yesterday, the subject
of the cube came up. I mentioned that Rubik did not get dollar much less
a forint for his effort. Guess what! A subscription to Sci. Am. is on the
way to him (lifetime I suppose).
Date: 21 February 1981 00:14-EST
From: Ed Schwalenberg
Subject: A lighter note
To: Greenberg.Symbolics at MIT-MULTICS
cc: CUBE-HACKERS at MIT-AI
Isn't this the C-Machine that you all are working on?
Date: 03/05/81 0839-EDT
From: PLUMMER at LL
Subject: another article
To: CUBE-LOVERS at MIT-AI
Check today's Wall Street Journal: front page, center. --Bill
-------
DAN@MIT-ML 03/05/81 20:58:59 Re: Rubiks Cube info needed
To: cube-lovers at MIT-AI
I have a few questions which you may be able to help me with...
1. Could you please add me to this mailing list
2. I am looking for a rubiks cube solver to play with on my microcomputer,
and would like to know if such a program exists. Would prefer Pascal
or "C" (as I most readily hack these), but Lisp, et.c would be fine.
3. Is there an archive of "Cube" info, back letters, documents,
bibliographies, etc. lying around on one of the ITS machines?
Thanks - Dan
Date: 6 MAR 1981 0849-EST
From: JURGEN at MIT-MC (Jonathan David Callas)
Subject: Cube Solver
To: DAN at MIT-ML
CC: CUBE-LOVERS at MIT-MC
I have a program written in pascal that won't *SOLVE* the cube but
will manipulate it. It will also find the order of a given move.
It was written by Tom Davis (of this list) and modified by me.
It should run on any USCD system with no hassles, and very minor
ones for any other sort of pascal. Since Tom was giving the program out before,
I shall assume that there is no problem with distributing this version.
If you (or anyone else) wants acopy, write me (Jurgen at MC) & I'll send
you a copy.
-- Happy cubing,
-- Jurgen at MC
Date: 7 Mar 1981 0224-PST
From: Alan R. Katz
Subject: how about...
To: cube-lovers at MIT-MC
cc: katz at USC-ISIF
How about a Braile cube (with dots instead of colors) for the blind,
or so one could solve it with both eyes closed??? (dont ask me why
you would want to solve it with both eyes closed).
Alan
-------
Date: 8 Mar 1981 1834-EST
From: JURGEN at MIT-DMS (Jonathan David Callas)
To: KATZ at USC-ISIF, CUBE-LOVERS at MIT-MC
Subject: Braille Cube
Message-id: <[MIT-DMS].189113>
I'm sure that it's all ready been done. (Blind people are very clever that way)
There are the equivalents of dymo label-makers that print in Braille, and any
random sighted person could label a cube (even randomized). I'd bet that it would very very slow, though. The eyes have a much greater informational bandwidth than the fingers.
-Jurgen at MC
Date: 9 MAR 1981 0855-EST
From: JURGEN at MIT-MC (Jonathan David Callas)
To: CUBE-LOVERS at MIT-MC
I have sent out copies of the cube program I mentioned earlier to (I think)
everyone who asked for it. If you didn't get it, or it was munged, or
you would like it, I saved a copy of the msg in:
DM:USERS1;JURGEN CUBE
--Happy Cubing
--Jon
Date: 9 Mar 1981 10:02 PST
From: McKeeman at PARC-MAXC
Subject: Re: how about a Braille cube...
In-reply-to: KATZ's message of 7 Mar 1981 0224-PST
To: Alan R. Katz
cc: cube-lovers at MIT-MC
Wonderful idea! Probably even fundable by some gov't agency for the
handicapped. As to why do it with your eyes closed, that was in some sense the
original intent of the cube: spatial visualization. Besides, it confuses me to look
at during a macro.
Bill
Date: 9 Mar 1981 at 1721-CST
From: korner at UTEXAS-11
Subject: edge cubie rotation
To: cube-lovers at mit-mc
does anyone have a nifty edge cubie rotation algorithm that doesn't
do a di flip in the process. I'm getting tired of
f r 3(F R R F) R 3(U R R U) F
There must be something better- I just haven't found it.
-Kim Korner
-------
Date: 10 MAR 1981 0556-EST
From: ACW at MIT-AI (Allan C. Wechsler)
Subject: edge cubie rotation
To: korner at UTEXAS-11
CC: CUBE-LOVERS at MIT-AI
My basic triple-edge tool is FFRL'UUR'L. It rotates three edges that
all lie in one equator. Manipulation hint: move that equator instead
of doing RL' and R'L.
Something like Kim's tool can be obtained by setting up with RL'U and
finishing with U'LR'. All together: RL'U FF RL' UU R'L U'LR'.
---Wechsler
Date: 10 Mar 1981 1910-PST
From: CSL.JHC.DAVIS at SU-SCORE
Subject: Edge Cubie Rotation
To: korner at UTEXAS-11
cc: cube-lovers at MIT-AI
I have been using an even shorter tool to do Kim Korner's
transformation. In the Befuddler notation, it is:
R' L B L' R D D R' L B L' R
It is much easier to do than this notation makes it seem. I
think of it as pushing a center cubie down to the bottom,
turning it off to the side, and bringing back the old top.
Then I move it around to the other side of the bottom, and
go back down to pick it up.
If you begin the transformation with RR LL instead of R' L, and
end it with RR LL instead of L' R, it does the same thing
except with no flipping of center cubies.
-- Tom Davis
PS. As in Wechler's tool, think of center-slice moves.
-------
Date: 12 MAR 1981 2317-EST
From: ATTILA at MIT-MC (Sean N Levy)
Sent-by: ATTIL0 at MIT-MC
Subject: Re: Rubik's Sphere carrying case
To: CUBE-LOVERS at MIT-MC
CC: Lauren at UCLA-SECURITY
Instead of putting it in a klien bottle, how about using one of
Escher's impossible boxes (decorated with an Escher on the outside,
of course...)
-- Attila
Date: Monday, 16 March 1981 19:28-EST
From: Pat O'Donnell
To: cube-lovers at mc
Subject: other orbits
Has anyone investigated what kinds of patterns exist in the other 11
orbits?
Date: 18 March 1981 22:34-EST
From: Alan Bawden
To: CUBE-HACKERS at MIT-MC
Check out this week's TIME magazine. (The "Living" section, I believe.)
ISRAEL@MIT-AI 03/20/81 15:55:21 Re: two-person games using the cube
To: CUBE-LOVERS at MIT-AI
Folks,
I was examining my cube the other day and I noticed that each side
looks like a tic-tac-toe board and I realized that we've never considered
the idea of two-person games using the cube. Here are some games that I've
come up with. Some of these may be trivial and uninteresting (i.e. obvious
wins for the first or second player) and some may be too easy to draw with,
but I'll throw them out anyway.
The first game I thought of was Rubik's tic-tac-toe. This is just regular
tic-tac-toe with a twist (pun intended). Each person takes turns first
writing his symbol on one of the 54 facelets on the cube. After doing
that he twists one face and passes the cube to his opponent. There are a
number of different variations of this game.
1) The first person to win any side of the cube wins. This seems to be
a very easy game so to make it more interesting we add the rule
that for a person to win, he must do it before executing a twist to
the cube.
2) To win, a person must win a majority of the faces on the cube. This
game has the interesting property that if the cube is full (a draw
in normal tic-tac-toe) twists can continually be made until a win is
reached, both people agree on a draw, or some arbitrary upper limit
on the number of moves beyond a full cube is passed.
3) One person must fill up all nine facelets of any face with his symbol.
This game may be too difficult to win.
4) Each person has pattern of X's, O's, and don't cares which his opponent
doesn't know and has to get one face to look like that pattern.
Each of these games can be modified by adding restrictions on the twist such
as; a) only quarter turns CW and CCW are allowed; b) a player cannot turn the
same face his opponent just turned; c) a player cannot turn the same face that
he turned last turn; d) if a player made a quarter turn last turn he must make
a half turn this turn and vice versa; or any combinations of the above
restrictions or others. Does anyone know of good erasable writing utensil to
use on your cubes or have a metal cube that can be used for these games
with magnetic X's and O's Another version of these games could be played
without writing on your cube by allocating one, two or three colors to each
person and starting from a randomized cube, try to play any of the above games
with each turn being taken by twisting a side and using the players set of
colors as his symbol.
- Bruce
^_
Date: 21 MAR 1981 1454-EST
From: LSH at MIT-MC (Lars S. Hornfeldt)
To: CUBE-LOVERS at MIT-MC, RP at MIT-MC
What are the best CUBE-times nowadays?
A young guy Kimmo Eriksson in Stockholm yesterday solved 10 (ten)
cubes in a series, with an AVERAGE of 52 seconds,
and with individual times varying between 47 sec and 61 sec.
-lsh
Date: 22 March 1981 0829-EST (Sunday)
From: Dan Hoey at CMU-10A
To: Cube-Lovers at mit-mc
Subject: No short relations and a new local maximum
Message-Id: <22Mar81 082919 DH51@CMU-10A>
Well, the gigabyte (well, 300Mb) came in, and brute force is
having its day. I have a little program that generates all positions
accessible from a given position in a given number of quarter-twists.
With the increased storage available here, I was able to run it to five
quarter-twists.
The first important fact to emerge is that there are exactly
105046 different positions at a distance of at most 5 qtw from START.
This has two consequences to the argument given in my message on the
Supergroup, part 2 (9 January 1981 0629-EST). Note that the results
here pertain to the usual group of the cube, rather than the
Supergroup, since the program does not keep track of face-center
orientations.
The first consequence is that there are exactly 93840 positions
exactly 5 qtw from START. The message cited above proved the
inequality P[5] <= 93840; this is now known to be an equality.
The second consequence is that there are no relations
(sequences that lead back to START) of length 10, with the exception of
those that follow from the relations FFFF = FBF'B' = I (and their
M-conjugates). This is because relations of length 10 would reduce
P[5], which is not the case. There are, however, relations of length
12; the only known ones are FR'F'R UF'U'F RU'R'U [given in Singmaster]
and its M-conjugates.
These results can be extended to the Supergroup, by noting that
the set of observed positions places a lower bound on the number of
Supergroup positions at a distance of 5 qtw, while the upper bound
given in the cited message relies on the relations FFFF = FBF'B' = I,
which are relations in the Supergroup.
A particular result which may be of greater interest to readers
of this list concerns the relation between symmetry and local maxima.
In our message on the subject (14 December 1980 1916-EST) Jim Saxe and
I mentioned that the six-spot pattern is not a local maximum, as
verified by computer. [The same program was used, but only four-qtw
searches were needed.]
With five-qtw searches, it became possible to check another
conjecture, using an approach that Jim suggested. The four-spot
pattern
U U U
U U U
U U U
R R R B B B L L L F F F
R L R B F B L R L F B F
R R R B B B L L L F F F
D D D
D D D
D D D
is solvable in twelve qtw, either by (FFBB)(UD')(LLRR)(UD') or by its
inverse, (DU')(LLRR)(DU')(FFBB). It is immediate that a twelve qtw
path from this pattern to START can begin with a twist of any face in
either direction. The program was used to verify that there are no ten
qtw paths. (It generated the set of positions attainable at most five
qtw from START and the set of positions obtainable from the four-spot
in at most five qtw, and verified that the intersection of the two sets
is empty.) Thus the four-spot is exactly twelve qtw from START and all
its neighbors are exactly eleven qtw from START, proving that the
four-spot is a local maximum. (Worried that there might be an eleven
qtw solution to the four-spot? Send me a note.)
This is the first example of a local maximum which cannot be
shown to be a local maximum on the basis of its symmetry. To be more
precise, let us define a "Q-symmetric" position to be a position whose
symmetry group is Q-transitive. This extends the terminology developed
in "Symmetry and Local Maxima". In that message, we showed that all
Q-symmetric positions, except the identity, are local maxima. Until
now, these were the only local maxima known. The four-spot, however,
is not Q-symmetric; the position obtained by twisting the U or D face
of the four-spot is not M-conjugate to the position obtained by
twisting any of the other faces. This lays to rest the old speculation
that one might find all local maxima, and thereby bound the maximum
distance from START, by examining Q-symmetric positions.
Date: 28 MAR 1981 1259-EST
From: DCP at MIT-MC (David C. Plummer)
Subject: New toy (long message, but read it anyway!!)
To: CUBE-LOVERS at MIT-MC
Tanya Sienko is visiting me, and she says that the cube is the
craze of Japan. She also presented me with a new toy, given to
her by some Japanese. (I don't know if is in this counrty --
yet.)
The thing is shaped like a barrel mounted on a supporting
structure. The barrel can move one UNIT up or down in the
structure. Around the circumference of the barrel there are five
equally distributed columns. Two of the columns have four rows,
and three of them have five. The ones with five have a plunger on
the associated part of both the top and bottom (or left and
right) parts of the supporting structure. Two plungers are next
to each other, and the third is opposite their midpoint. There
are 23 balls in the device: four each of green, yellow, blue,
red, orange (one for each column) and three black balls. (in a
minute you will see where these black balls go). The barrel is
divided into four parts. The left- and right-most parts are fixed
with respect to the supporting structure. Each has three cavities
either to hold a ball or one of the plungers. The barrel moves,
so either the left has balls in the cavity and the right has the
plungers, or vice versa. The middle two sections of the barrel
have two cavities in each row, and these rotate around the
circumference, taking balls with them.
I have been trying to say left and right, because I think the
corect way to thing of this devices is as follows: Hold it
horizontally, with the barrel centered in the supporting
structure. This means that each plunger is half way into its
cavity. A MOVE consists of moving the barrel one half unit right
or left, then moving one of the rotating middle sections forward
or backward one unit, and then returning the barrel to center
position. This creates four generators: move barrel [left,right],
then move middle section-[left,right] forward (or backward, which
is the inverse). Visually:
| | | |
A A A A A B B B B B
\ \ / /
A A A A B B B B
/ / \ \
A A A A A B B B B B
\ \ / /
A A A A B B B B
/ / \ \
A A A A A B B B B B
| | | |
| | | |
C C C C C D D D D D
\ \ / /
C C C C D D D D
/ / \ \
C C C C C D D D D D
\ \ / /
C C C C D D D D
/ / \ \
C C C C C D D D D D
| | | |
Where A is move barrel left , move left section
B is move barrel right, move left section
C left , right
D right, right
The top and bottom of these drawings are connected, cavities
(filled with the balls) move along the lines. All balls move in
the same direction the same number of units (i.e., the middle
sections are rigid). I hope this is a good enough description, if
not send me mail and I will send an addendum.
The object, so I hear, is to get each column (row in these
pictures) a single color, and if there are five slots (of which
there are three), the fifth has a black ball in it, when the
barrel is pushed all the way to one side, the plungers take up
three of the outside-barrel-sections, and the black balls take up
the opposite three. from a symmetric point of view, I think it
would be more general to SOLVE it so that the black ball is in
the middle of the five balls (this may not be solvable though)..
If we ignore the obvoius left-right symmetry of the above
pictures, the first assumption of the combinatorics of this beast
is simply P(23;4,4,4,4,4,3)=numbers of ways to permute 4 balls of
each of 5 colors and 3 balls of another color=
23!
------------------- = 541111756185000 = 541 trillion
4! 4! 4! 4! 4! 3!
Until I have played with it for a while, I can't even guess on
how many orbits there are. Perhaps only one -- I don't know.
Super-groups come in a few classes:
(1) Each non-black ball gets a second label (1-4)
giving size 23!/3! = 4.3*10^21
(2) Each black gets a second label (1-3)
giving size 23!/(4!)^5 = 3.25*10^15
(3) (1) and (2), all balls distinct
giving size 23! = 25.8*10^21
If anybody sees one in this county, please let me know. Tanya
believes they are only in Japan at the moment. She has donated
the one I have seen to me/SIPB, so people at MIT and area are
free to come to 39-200. PLEASE BE CAREFUL with it. It is plastic
and it looks breakable -- especially the outer part of the
supporting structure looks like it dould break. I think a better
construction would be to have them be plates which are attached
to the axis with screws. This might lead to a temptation to
disassemble, which may be epsilon below breakage.
Date: 28 March 1981 16:13-EST
From: Carl W. Hoffman
Subject: Also from Japan ...
To: CUBE-LOVERS at MIT-MC
Cubes of different sizes and colors. There is one 3 centimeters on an edge
sitting in the SIPB office.
Date: 31 Mar 1981 2133-PST
From: Gary R. Martins
Subject: B E W A R E !!
To: cube-lovers at MIT-MC
cc: gary at RAND-AI
Bought a new cube today. One of Ideal's "Rubik's Cube"s. Same
price as first cube, bought about a month ago. Packaging looks
same. Ditto cube, except that the center-white face has
"Rubik's Cube (tm)" printed on it in various fonts. Also, closer
inspection of the package shows that a stick-on stripe acknowledges
manufacture in Hong Kong.
The cube itself is INFERIOR in various ways. I'd recommend you
not buy them, unless the vendor will offer you a refund.
The worst and most obvious feature of this cube is that is seems
to have NO lubricant in it. The faces seem more vulnerable to
fingernail damage etc. and the colors and materials seem shoddier.
The cube has a flimsy feel to it, and seems poorly finished in general.
Anybody else notice this, or have I just caught a lemon ?
Gary
-------
Date: 1 APR 1981 0104-PST
From: MAXION at PARC-MAXC
Subject: Re: B E W A R E !!
To: gary at RAND-AI, cube-lovers at MIT-MC
cc: MAXION
In response to the message sent 31 Mar 1981 2133-PST from gary@RAND-AI
I had the exact same experience. The first one was wonderful; the
second (just as you described) was awful. Same packaging, same story,
same observations as yours.
Roy
-------
ZEMON@MIT-AI 04/01/81 07:45:10 Re: B E W A R E !!
To: CUBE-LOVERS at MIT-AI
I have one of the offending cubes -- yes, it \is/ falling apart.
The colored faces have developed crinkles, holes and some are even
peeling off. This is after only 4 weeks of use.
Taking the cube apart and sprinkling its insides liberally
with baby powder will effectively lubricate it (although it will smell
for a while) and make it essentially noiseless, I have heard.
-Landon-
Date: 1 Apr 1981 1459-PST
From: Gary R. Martins
Subject: Cube Lube
To: cube-lovers at MIT-MC
cc: gary at RAND-AI
Is there a consensus on the best lube for one's cube ? White
lithium grease, silicon grease, silicon spray, and baby powder
have all been mentioned. Anybody know what's really 'best' ?
Gary
-------
Date: 2 Apr 1981 0132-CST
From: Clive Dawson
Subject: Re: Cube Lube
To: gary at RAND-AI, cube-lovers at MIT-MC
In-Reply-To: Your message of 1-Apr-81 2203-CST
I suspect that "best" in this case is probably a matter of
personal opinion...Also note that a lot depends on trimming,
filing, sanding, etc.
Besides the lubes mentioned by Gary, I can also recommend dry graphite
powder (I used "Mr. Zip Extra Fine Graphite") which gave me very good
results on my cube. Then I finally got a chance to examine Kim
Korner's cube (Korner@UTEXAS) and must admit his is much much better.
He used silicon gel, of the sort used to lubricate "o" rings in Scuba
equipment. See his message to Cube-lovers of 6-Feb-81 for more
information. About the only shortcoming I noticed was a very slight
"slimy" feeling to the cube which I'm sure will wear off with time...
By the way, on the subject of the declining quality of Ideal Toy's
version of the cube-- I too was surprised when I examined one
which was bought last month by a friend of mine. The first thing
I noticed was that the some of the interior faces of each cubie
were missing. My first reaction was that they'd found a way to
skimp on plastic; then I thought that maybe it was a way to
cut down on internal friction. Judging from some of the other
recent reports, it sounds like my first hunch was correct. Another
annoying characteristic was the shoddy work in attaching the
colored faces. Most were not only crooked, but also liberally
sprinkled with air bubbles throughout.
Happy cubing,
Clive
-------
Date: 2 Apr 1981 1723-PST
From: Hopper at OFFICE
Subject: Re: B E W A R E !!
To: gary at RAND-AI, cube-lovers at MIT-MC
cc: hopper at OFFICE
I've bought cubes recently as follows:
1-FEB (approx) , Ideal's with "Rubik's CUBE tm" on the center white cubie.
Pakaged in cardboard and cellophane. Hollow edge and corner cubies. Squeekie,
but fine after lube with graphite. No problems with facies. Good size
tolerances--very smooth operation after lube.
20-FEB (approx) , bought 3 that looked identical to the pevious one, except
they were packaged in cylindrical plastic packages.
Two of the three turned out the same as the one purchased 1-FEB, except that
size tolerences were very poor and operation was very rough, even after
lubrication. No problems with the faces.
The third cube was packaged the same and has the same "Rubik's CUBE tm" on
the center white cubie, but is quite different. Although the cubies are
hollow, it is heavier then the others. The plastic isn't so squeeky and
seems more like earlier cubes from last year. The edge cubies have casting
ridges visible through the middle of the faces like the early, early cube I
got before last summer. The shoulders on the edge cubies were flat so the
action was very rough. The corner cubies were loose. Filing down the
inside surfaces of the edge cubies cured the loose corners, and rounding
their shoulders made the action quite acceptable. LASTLY, ONE (just ONE!)
of the red facies is inferior, darker in color, and crinkling!
20-MARCH (approx) , bought a cube with no acknowledged manufacturer, with
cylindrical plastic pakage very similar to Ideal's, labeled "Made in Taiwan".
Cubies are hollow very much like the third one from the 20-FEB batch, but the
corner cubies have covers glued in the openings of the inside surfaces. The
quality of the facies seems good, but it may be too soon to tell. The orange
facies are not brilliant like Ideal's--more of a peach color. The size
tolerances seem quite good compared to Ideal's recent cubes. Biggest drawback
(and possible overriding factor) is the plastic seems much softer than Ideals's
and lubrication (at least with graphite--I haven't tried other recommended
lubes such as silica gel) doesn't seem to make it any easier to turn. It
remains quite stiff. Also, some of the centers were screwed in much to tightly.
I'd be interested to hear any other experiences with recently-bought cubes.
I'm curious about their availability in the Bay area and elsewhere. --Dave--
-------
Date: 2 Apr 1981 2325-PST
From: Gary R. Martins
Subject: New Yorker
To: cube-lovers at MIT-MC
cc: gary at RAND-AI
Current issue of 'New Yorker' magazine has some cubic discussion
in the opening 'Talk of the Town' section. Also mentions Marvin
Minsky! P. 29, March 30, 1981 issue.
Gary
-------
Date: 3 April 1981 0500-est
From: Allan C. Wechsler
Subject: Magic barrel.
To: CUBE-LOVERS at AI
Haal yawm! I am a barrel-solver this day!
---Wechsler
Date: 3 Apr 1981 0750-PST
Sender: OLE at DARCOM-KA
Subject: Cube lube (yet again)
From: Ole at DARCOM-KA (Ole J. Jacobsen)
To: Cube-lovers at MIT-MC
Message-ID: <[DARCOM-KA] 3-Apr-81 07:50:47.OLE>
The following is a collection of thoughs and experiences on
cube lubrication. It only applies to the Ideal cubes which
are the only ones I have played with, but should have some
applicability to other brands.
Getting a smooth turning cube seems to me to be a combination
of the right lubrication with the spring/screw tension. If the
screws are too lose, the cube will turn easily, but frequently
jam since the lose cubies tend to get in each other's way. On
the other hand a very high tension without any lubrication would
mean a very stiff and fast wearing cube. My solution is simply to
use candle wax. I have taken my cube apart and rubbed each cubie
with a standard candle. (My friend from the Chemical Eng. Dept.
says parafine wax would be even better. This is what we used to
rub on our skis back in Norway before all the fancy ski-waxes
became available. I don't know how easy it is to get these days.)
I also "fine-tune" the cube by adjusting the screws on each face.
A couple of strips of double-sided tape stops the caps from acci-
dentally falling out during use. You may need to take the cube
apart a couple of times after the initial lubrication to allow
superfluous wax to fall out. I also found that turning a newly
waxed cube under a hot tap seems to make the wax settle nicely.
This lubrication has the advantage of not (seemingly) coming out
on your hands or otherwise disappear,- one treatment will last
you very long indeed. The only slight problem is that the cube
needs some "warming up" when it has been left idle for some time
especially in cold places. (Ref. my earlier message) But a couple
of minutes of random twisting produces a smooth and silent cube.
Good luck
OLE
Date: 4 Apr 1981 1727-EST
From: JURGEN at MIT-DMS (Jonathan David Callas)
To: Cube-lovers at MIT-MC
Subject: Cube preferences
Message-id: <[MIT-DMS].192744>
I have two cubes, a C. Americanus ("Rubik's Cube") which I bought last summer,
And a C. Albus that I got from Logical Games in Haymarket Va. The white
cube came lubricated with something resembling musician's cork grease, and
has not needed to be lubricated. The Rubik's cube has never been lubricated
either, but hasn't seemed to need it. Iprefer the white cube to the black one
for some nebulous reason. It is not nearly as smooth-turning as the black
one, but in a perverse way, I like that. It seems to be better built, but I
can't substantiate that with facts, that's just gut-feeling. I *DO* like
the fact that they are uncommon, and now that people don't go "Ooh, what's
*THAT*" when they see the cube, and now people do get amazed at the sight
of the white-faced cube. Now that it seems that Ideal is going for the bucks (a
friend of mine has also gotten one of the cheap cubes, but I thought it was
my imagination. I guess now there's real reasons for getting the white cubes.
--Happy cubing,
--Jurgen
Date: 6 APR 1981 1501-EST
From: DCP at MIT-MC (David C. Plummer)
Subject: Japan frob revisted (180+ lines)
To: CUBE-LOVERS at MIT-MC
This is a long overdue re-explanation of the Japanese frob. Hoey
and Saxe at CMU gave several comments and suggestions.
PART I -- Try again
===================
Take a hollow cylinder (like a doubly unlidded coffe can), cut it
open and unravel it. We now have something like
xxxxxxxxx
| |
b | where the cut was made along the x
o t and top and bottom are where the
t o lids used to be
t p
o |
m |
| |
xxxxxxxxx
The supporting structure corresponds roughly to the lids of the
can.
LL " ' "" ' " RR
LLLLLL" B ' B "" B ' B " B RRRRRR
LL " ' "" ' " RR
LL " ' "" ' " RR
LL " ' "" ' " RR
LL " B ' B "" B ' B " RR
LL " ' "" ' " RR
LL " ' "" ' " RR
LL " ' "" ' " RR
LLLLLL" B ' B "" B ' B " B RRRRRR
LL " ' "" ' " RR
LL " ' "" ' " RR
LL " ' "" ' " RR
LL " B ' B "" B ' B " RR
LL " ' "" ' " RR
LL " ' "" ' " RR
LL " ' "" ' " RR
LLLLLL" B ' B "" B ' B " B RRRRRR
LL " ' "" ' " RR
LL " ' "" ' " RR
(In the three dimensional case, the top and bottom of this
picture are connected together.)
L is the left part of the supporting structure, and R is the
right. They are firmly connected to each other, and are therefore
fixed in space with respect to each other. They are really
circular, but this is a view of the outside. B are the balls. The
balls can move left or right by being PLUNGED. The only allowed
plunge in the above diagram is to move the supporting structure
to the left (or equivalently the barrel to the right). With
respect to the barrel, only the balls in the first, third and
fifth rows (refered to as columns in previous message) are
affected. The diagram would now look like
LLL B B B B B RRR
L B B B B R
LLL B B B B B RRR
L B B B B R
LLL B B B B B RRR
Balls move vertically by by moving either of the two " ' "
sections vertically, and the balls within that section stay fixed
in space with respect to each other and the section, but not
fixed with respect to the other balls (within the context of one
turn) or the supporting structure.
Thus the moves are:
PLUNGE RIGHT or LEFT, whichever is appropriate
(or move barrel LEFT or RIGHT)
and MOVE LEFT or RIGHT section UP or DOWN
My suggestion was that in the between move state, the barrel was
centered in the plungers, so the PLUNGE move is HALF-PLUNGE LEFT
or RIGHT (both of which are apporpriate), then do the vertical
move, then UN-HALF-PLUNGE.
PART II -- Hoey's comments to my original message
=================================================
[Hoey 28 March 1981 1500-EST]
Is the move you designate by
| |
A A A A A
\ \
A A A A
/ /
A A A A A
\ \
A A A A
/ /
A A A A A
| |
really the permutation that takes
| | | |
1 2 3 4 5 6 7 3 4 5
\ \ \ \
6 7 8 9 10 11 8 9
/ / / /
10 11 12 13 14 to 15 16 12 13 14
\ \ \ \
15 16 17 18 19 20 17 18
/ / / /
19 20 21 22 23 1 2 21 22 23
| | | | ?
Do you mean to imply that moves of the form
| | | |
X X X X X X X X X X
\ \ / /
X X X X X X X X
\ \ \ \
X X X X X or X X X X X
/ / / /
X X X X X X X X
/ / \ \
X X X X X X X X X X
| | | |
(whatever they mean) are prohibited (as primitives, at least) by
the construction of the barrel?
[Both answers are YES]
PART III -- Comments later that night
=====================================
[In response to the updated description
Hoey 20 March 1981 1836-EST]
First, it should be made clear that in (either) plunged
position, the two " ' " sections rotate freely; i. e. it is not
necessary to plunge in between. For instance, one could solve by
counting plunges, but not rotations. Jim suggested that it might
be "neater theoretically", but I think it smells of the half-twist
metric.
Second, the inclusion of permutation diagrams will make the
puzzle clear to anyone who doesn't understand the mechanics.
Something like I gave in the last message, but with all
permutations given, the note that "\|/" are only comments, and the
description of the goal: move Black to 5,14,23, and make the sets
1-4, 6-9, 10-13, 15-18, and 19-22 each a solid color.
I ran this through the Furst/Hopcroft/Luks algorithm, and
found that in the Supergroup (all balls distinct) you get the
alternating group on 23 balls: all even permutations. Thus if any
two balls are indistinguishable, you can get all configurations.
Saxe remarks that there is only fourfold symmetry: Reflection
left-to-right and up-to-down. Their composition is in fact
achievable: turn the whole puzzle upside down, while continuing to
face the front of it. Strangely enough, this is an ODD permutation:
it takes you to the other orbit!
[Hoey 28 March 1981 2143-EST Subject: Simpler and harder toy]
Try taping the center two rings together. Thus A is
always performed with C, and B with D. The same set of
permutations is achievable!
[I assume the proof is an enumeration of states by the above
algorithm.]
PART IV -- Developments by Alan Bawden (ALAN@MC), Allan Wechsler
(ACW@AI) and myself.
================================================================
Alan Bawden sat down patiently one night (Tuesday March 31 1981,
I think) and discovered the necessary TOOL (or concept)
(singular !!) that is needed to solve the toy. I will not give a
spoiler here. Getting most of it is rather easy. The last few
balls take a little extra work. Alan told me the concept, and the
next day I successully solved it. Alan solved it later that day,
and soon Allan Wechsler solved it a few days later (signified by
his yelp to this mailing list). The three of us solve it slightly
differently (s)o like the cube, there are personal sovling
styles). We now solve it reliably, including the last few balls.
Happy what-ever-ing...
Date: 8 Apr 1981 16:07 EST
From: Marshall.WBST at PARC-MAXC
Subject: Please add me to the distribution list
To: Cube-Lovers at MIT-MC
cc: Marshall.WBST
Please add my name to the rubik's cube distribution list. I have a copy of
Kertesz' solution but am interested in better solutions and/or insights into the
underlying group.
Thank you
--Sidney (Marshall.WBST at PARC-MAXC)
Date: 18 April 1981 08:52-EST
From: Lars S. Hornfeldt
To: CUBE-LOVERS at MIT-MC, gary at RAND-AI
Kimmo Eriksson is 14 years old (a good age for cubism),
and in his series of 10 consecutive cubes, the average time was
52 sec, and average number of moves was 95, varying between
70 to 120 (half-turns and slices counted as one move).
He uses 5 macros with uncountable longer variants.
The longest of the macros are 11 moves.
-lsh
Date: 20 Apr 1981 0906-PST
From: Isaacs at SRI-KL
Subject: (Response to message)
To: LSH at MIT-MC
cc: ISAACS, cube-lovers at MIT-MC
Who and where is Kimmo Eriksson? Where was this timing done? What are
his macros? In what order does he solve it? The same way each time? etc.
---Stan Isaacs
-------
CMB@MIT-ML 04/23/81 13:04:49
To: cube-lovers at MIT-MC
From the Boston Globe:
Abbie Hoffman, the former Yippie leader who managed to escape the toils of
the law for seven years by living incognito in upstate New York, has finally
gone to prison, but not for having been a fugitive. Hoffman surrendered in
New York yesterday to begin a three-year prison term for selling cocaine and
jumping bail. In the photo, he is shown being frisked. In his left hand he
holds a magic cube puzzle, which he said he will solve in prison. In his
right hand he holds a copy of the book, "Fire in the Minds of Men," that had a
bookmark looking suspiciously like a hacksaw blade. This was whisked away from
him. Hoffman denied the props, including the hacksaw, were a publicity stunt.
Date: 23 Apr 1981 1105-PST
From: Gary R. Martins
To: CMB at MIT-ML
cc: cube-lovers at MIT-MC, gary at RAND-AI
In-Reply-To: Your message of 23-Apr-81 1304-PST
He should have offered those narcs a snort of vodka !
G
-------
Date: Sunday, 26 April 1981 10:54-EDT
From: Pat O'Donnell
To: Cube-Lovers at MC
cc: PAO at MIT-EECS
May issue of Reader's Digest has a (very) short article on the cube.
It includes a claim for a French fellow solving the cube in an average
of 32 seconds. The article contains almost no technical
information--mostly historical.
Date: 27 Apr 1981 0923-PDT
From: Isaacs at SRI-KL
Subject: cubes, barrels, and stuff
To: cube-lovers at MIT-AI
I was just at the fourth international puzzle party in L.A. and saw
several offshoots of the cube. The barrel, previously mentioned in this
digest, is the best. It is called "The Ten Billion Puzzle" (I think).
(Note to people in the Palo Alto area - come to the Rubiks Cube Club And
Other Puzzles at Stanford on Thursday night if you want to see a couple.)
Also there were two small cubes, about 2/3 size, one from Japan, and
the other from (I think) Taiwan. There was a 2x2 version (about half the
size of Rubiks), with things like hearts, stars, etc on it. There was also
a Rubik type, but with figures instead of colors.
The Missing Link is now out from Ideal, and should be easily findable
(as of this week). But, though they treat it as a follow-up on the cube,
it is MUCH simpler, and closer in principle to a sliding block puzzle.
Nice, but simple.
There were also several other types of cylinders, but mostly related
to the Missing Link, or to Instant Insanity type, rather than cube type.
By the way, Jerry Slocum, puzzle collector extraordinaire and the
puzzle party host, thinks the magic cube will have a real impact on
society - that it will lead to a resurgance of interest in puzzles in
general, and in thinking-type games. Let us hope he is right. (Send
a puzzle to your congressman - make him think!)
--- Stan Isaacs
-------
Date: 27 April 1981 12:15 cdt
From: VaughanW.REFLECS at HI-Multics
Subject: Re: cubes, barrels, and stuff
To: Isaacs at SRI-KL
cc: cube-lovers at MIT-AI
In-Reply-To: Msg of 04/27/81 11:23 from Isaacs
*nothing* could make my congressman think!
Date: 27 Apr 1981 1322-EDT
From: IC.RAG at MIT-EECS
Subject: Re: Re: cubes, barrels, and stuff
To: VaughanW.REFLECS at HI-MULTICS, Isaacs at SRI-KL
cc: cube-lovers at MIT-AI
In-Reply-To: Your message of 27-Apr-81 1315-EDT
*NOTHING* could make me think of my congressman!
-------
Date: 27 Apr 1981 1139-PDT
From: Gary R. Martins
Subject: Re: Re: cubes, barrels, and stuff
To: VaughanW.REFLECS at HI-MULTICS
cc: Isaacs at SRI-KL, cube-lovers at MIT-AI, gary at RAND-AI
In-Reply-To: Your message of 27-Apr-81 1215-PDT
Try *M* *O* *N* *E* *Y* !! Worked wonders for the
FBI !
G
-------
Date: 28 Apr 1981 0233-PDT
From: Peter D. Henry
Subject: mailing list add request
To: cube-lovers at MIT-MC
please add me to the mailing list... thanks
Peter D. Henry
PDH@sail
Date: 28 April 1981 08:34-EST
From: David C. Plummer
Subject: mailing list add request
To: PDH at SU-AI
cc: CUBE-LOVERS at MIT-MC
Done.
Date: 29 April 1981 1334-EDT (Wednesday)
From: Guy.Steele at CMU-10A
To: cube-lovers at MIT-MC
Subject: New member for mailing list
Message-Id: <29Apr81 133430 GS70@CMU-10A>
Please add Paul.Haley @ CMUA to the cube-lovers mailing list?
Date: 6 May 1981 2030-EDT
From: ROBG at MIT-DMS (Rob F. Griffiths)
To: cube-lovers at MIT-MC
Message-id: <[MIT-DMS].196841>
While visiting the local gaming shop today, I heard a rumour
about a pending lawsuit between (I think) The Original maker
of Rubik's cube and Ideal.. Anyone know anything about this?
-Rob.
Date: 8 May 1981 1036-PDT
From: Isaacs at SRI-KL
Subject: Non-twisting corner moves
To: cube-lovers at MIT-MC
Does anyone know good move sequences for exchanging a pair or corner
cubies on a face without twisting? (Of course, a pair of edges will have
to exchange also.) Or of cycling 3 corners without twisting? I'm looking
for the "shortest" sequence, and the "easiest to remember" sequence.
Most of the moves I've seen are long and complicated.
--- Stan Isaacs
-------
Date: 8 May 1981 14:30-EDT
From: David C. Plummer
Subject: Non-twisting corner moves
To: Isaacs at SRI-KL
cc: CUBE-LOVERS at MIT-MC
How about
L' [(R' DD R) U (R' DD R) U'] L
for moving the top three corners around perserving the top color.
Or,
(R' D' R) U' (R' D R) U
For moving three front pieces around, preserving a couple colors.
12 and 8 moves is pretty short...
Date: 9 May 1981 08:47-EDT
From: Lars S. Hornfeldt
Sender: LSH0 at MIT-MC
To: CUBE-LOVERS at MIT-MC, isaacs at SRI-KL
Kimmo Eriksson is a 14 year old computer fan who lives in Stockholm.
As mentioned, he solved a series of 10 cubes in 47-61 s, average 52,
using 70-120 moves, average 95 (half-turns and slices counted as one).
* He always starts with the WHOLE yellow layer (regardless of ini-
tial state, probably because the regularity allows faster reflexes)
* Then the middle layer (betw. yellow and white)
* Then all top-corners into place, then into correct orientation.
* Finally turn and move the top-edges (requires 0-3 macro-moves).
He keeps strictly to this scheme, but uses a large set of macros,
that are different longer varities of the following basic five:
For middle: RUR'U'F'U'F
Move corner: RU'L'UR'U'LU
Turn corner: RUR'URU2R'U2
Move edge: MU2M'UMU2M'UMU2M' (M moves the Mid-line of the Bottom
Move and turn edge:MUM'U2MUM' up Front, ie = LR' )
For timing, he starts a stopwatch, grabs the cube, solves it
- while watching (easy) the watch during the last macro in order to
read off the time exactly as the last macro is completed.
After re-mixing the cube, the procedure is repeated (10 times).
-lsh
Date: 10 May 1981 1258-EDT
From: Jerry Agin
Subject: Counting moves
To: Cube-Lovers at MIT-MC
Frequently when I play with the cube, I try to solve it in as few
moves as possible. I find this to be more intellectually challenging
than going for speed. Does any one else do this? I'd be interested
in comparing notes. Presently it takes me between 70 and 100 quarter-
twists, provided I don't make gross errors. (My guess is that if I
were counting slices and half-twists as one, the number would be
between 50 and 70.)
-------
Date: 15 May 1981 0456-PDT
Sender: OLE at DARCOM-KA
Subject: New pseudo-cube
From: Ole at DARCOM-KA (Ole J. Jacobsen)
To: Cube-lovers at MIT-MC
Cc: Oyvind
Message-ID: <[DARCOM-KA]15-May-81 04:56:31.OLE>
Pardon me if this object has been described before, but I don't remember
seeing it. While browsing for cubes in a local store here in Newcastle
the other day, I came accross a new "cube" which I shall try to describe
and I invite you all to think of a good name for it.
First of all let me point out that this new toy is really just a Rubik's
Cube with some modification with respect to coloring and construction.
Imagine taking your normal cube and making 4 vertical slices along the
corner diagonals. Your top and bottom faces would now look like:
---
/ \
I I
\ /
---
(sorry about poor ratio, but I hope you get the idea) Now recolor the new
faces and voila, your new toy is complete. The construction has the following
consequences: 1. The object is no longer symmetrical, U and D faces
are different from L R F and B.
2. The "corners" have only got TWO colors, but act as
corners of the Rubik's Cube, the mechanics is identical.
3. Four new "edges" which I will call wedges have appeared
in the middle layer. These have only ONE color, but as
you will discover when using your edge moves: the orient-
ation matters. Edges and wedges may be interchanged.
I will now describe the coloring of my particular cube, note that there are 10
different colors. The U face is blue and the D face is white. Then starting at
the 6'o clock edge column (i.e 1/3 of the F face) we have: GOLD(e),ORANGE(w),
RED(e),PURPLE(w),YELLOW(e),PINK(w),GREEN(e),LIGHT BLUE(w). (Where e=edge and
w=wedge colums respectively). I chose this particular orientation because it
makes red=left and green=right which is nice. Note however that this "cube"
may be reassembled in various legal patterns since the edge column/wedge column
neighbouring properties are not forced. This further complicates solving since
there in no way of knowing which sequence the "corners" and edges go in layer1
unless you have a map. Once this is known, solving is straight forward, but
as said the wedges will confuse you. Qestion: Is there some way of deter-
mining the parity etc, such that the object may be solved without a map
of layer one? I invite comments from Jim and Dan. If it is the case
that these pseudo-cubes (how about Rubik's Drum) are not available in
the US, I can send one or two samples. The drums are made in Taiwan and
are not as well finished or as smooth turning as Ideals cubes. Random
twisting produces very strange shapes and the Cruxi Plummeri et Cristmani
are simply out of this world.
OK, thats it. Hope this made sense, but this thing is more difficult to
describe that its predecessor,- so forgive me if I haven't succeeded.
Cheers OLE
Date: 15 May 1981 19:04 edt
From: Greenberg.Symbolics at MIT-Multics
Subject: Re: New pseudo-cube
To: Ole at DARCOM-KA (Ole J. Jacobsen)
cc: Cube-lovers at MIT-MC, Oyvind at DARCOM-KA
In-Reply-To: Message of 15 May 1981 07:56 edt from Ole J. Jacobsen
I have seen this thing under
the name "space shuttle" - in a cylinder like rubiks cubes used to
come in. One way to determine the color layou is to make
an assumption (as I did) and proceed to solve-- if you lose, you
get the "impossible" single-pair-swap. Then swap two edges
of your assumption and try again. Incidentally, the plural
of crux is cruces, not cruxi.
Date: 18 May 1981 1046-EDT (Monday)
From: Dan Hoey at CMU-10A
To: Cube-Lovers at MIT-MC
Subject: Drum info
Message-Id: <18May81 104634 DH51@CMU-10A>
Most of this note is pretty straightforward application of
the known cube properties, but if you want to know about the
drum....
The drum shows everything you see on a regular cube except
the orientation of the four truncated edges, or wedges. Because the
(invisible) edge parity is preserved, each visible position of the
drum corresponds to 2^4/2 cube positions. Thus there are
5.406x10^18 drum positions.
To count the number of solutions, note that as in the
normal cube, the face centers force each edge to its home position
and orientation. In addition, each corner has a facelet that says
whether it is top or bottom and fixes the corner's orientation.
This means that solved positions are obtained from each other by
permuting top corners, bottom corners, and wedges. But the three
cubies on a diagonal face must match, and so the three permutations
are the same. Only even permutations are achievable in this way
(since the cube of an odd permutation is odd) and there are 4!/2=12
of these. One easy process that goes from one solved position to
another is FF RR FF BB RR BB.
I asked Ole Jacobsen what he meant when he said of the
wedges that "as you will discover when using your edge moves: the
orientation matters." It turns out this is because he solves by
layers: top-middle-bottom, and doesn't know which way to orient the
edges in the middle so that the edges on the bottom will have the
right parity. There are several ways out of that problem; one is to
turn the drum sideways and solve left-middle-right.
The problem of solving without knowing the order of the
wedges is trickier. Solving sideways is one method: do the left
side any way; on the right side there two possibilities, one of
which will work. (This is Bernie Greenberg's suggestion, modified
so you don't need to memorize the whole map.)
One interesting thing to do with a drum is to turn it into
baseball. Using colored tape and disassembly, change the colors and
positions so that the wedges appear in the UF, DF, BL, and BR
positions when the colors match. On a baseball, there are only two
solved positions.
Date: 18 May 1981 17:59-EDT
From: Richard Pavelle