From dik@cwi.nl Thu May 28 12:33:49 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA02289; Thu, 28 May 92 12:33:49 EDT
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA03118 (5.65b/2.10/CWI-Amsterdam); Thu, 28 May 1992 17:18:31 +0200
Received: by boring.cwi.nl
id AA00307 (5.65b/2.10/CWI-Amsterdam); Thu, 28 May 1992 15:00:49 +0200
Date: Thu, 28 May 1992 15:00:49 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9205281300.AA00307.dik@boring.cwi.nl>
To: cube-lovers@life.ai.mit.edu
Subject: Corrected calculations are now done.
After an initial false start I have now calculated the path-lengths in
phase 1 of Kociemba's algorithm. The figures are as follows:
path configurations
0: 1
1: 4
2: 50
3: 592
4: 7156
5: 87236
6: 1043817
7: 12070278
8: 124946368
9: 821605960
10: 1199128738
11: 58202444
12: 476
The figure 50 for path length 2 is easily verified by hand. I have a list
with information about the configurations requiring a path-length of 12
(actually the paths leading to such a configurations). As should be true
for each minimal path in phase 1, all paths start and terminate with a
quarter turn of F, R, B or L.
Some details.
Phase 1 of the algorithm brings the cube in the subgroup generated by
[F^2, R^2, B^2, L^2, U, D]. There are in this case 2,217,093,120
(2048 * 2187 * 495) cosets. This can be (and has been) reduced largely
by observing symmetries. In this case rotating the complete cube
along the UD axis by a quarter turn, rotating the cube along the RL
axis by a half turn and mirroring through the FRBL plane reveal
equivalent cosets. Although it is possible to remove *all* cosets that
are equivalent to some canonical coset this was not done. The removal
has only been done for the twists of corner cubes, reducing the factor
2187 to 168, and reducing the number of configurations to be handled to
170,311,680. For each configuration a minimal path was calculated.
This was done starting with an absolute minimum found through the coordinate
axis and through the 2-dimensional coordinate spaces. When a path of that
length was not found the path length was increased and a new attempt was made.
This was done until a path was found. All searches were exhaustive. On
average paths were searched for 3 different lengths (519,177,716 attempts
for 170,311,679 configurations).
The computations were done on a farm of workstations where each workstation
got a portion of the flip dimension (2048 cases of 83,160 configurations).
Computation time for one portion was from 1 to 2 hours (1.5 on average), so
the total computation was about 3000 hours. On a system with enough
memory (50 MByte) it would have taken only 1 hour (this based on experiments
with the corner cubes-only part). It could also have been with a single
processor and a 50 MByte file, in that case CP time would also be about 1
hour, but the I/O time would exceed the 3000 hours very much.
Using this result and the result by Hans Kloosterman the diameter of the
cube group is at most 37. I conjecture the maximal path length in phase 2
of Kociemba's algorithm is 16, although the requirements on computer time
cq. memory do inhibit calculations at this moment (only in memory would be
feasible, but that requires 500 to 1000 MByte and computation time would be
about one day). This figure of 16 would reduce the upperbound of the groups
diameter to 28.
dik
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
dik@cwi.nl
From @mitvma.mit.edu:hans@freyr.research.ptt.nl Fri May 29 13:21:10 1992
Return-Path: <@mitvma.mit.edu:hans@freyr.research.ptt.nl>
Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) id AA09766; Fri, 29 May 92 13:21:10 EDT
Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2)
with BSMTP id 3176; Fri, 29 May 92 13:22:18 EDT
Received: from research.ptt.nl by MITVMA.MIT.EDU (Mailer R2.08 R208004) with
BSMTP id 8330; Fri, 29 May 92 13:22:17 EDT
Received: from dnlunx.research.ptt.nl (DNLUNX) by research.ptt.nl (PMDF #12085)
id <01GKLFC81PO0DQGQ2Q@research.ptt.nl>; Fri, 29 May 1992 19:21 +0100
Received: by gefjon.dnl (4.1/SMI-4.1) id AA03306; Fri, 29 May 92 19:20:23 +0200
Date: Fri, 29 May 92 19:20:22 MET DST
From: J.M.Kloosterman@research.ptt.nl (Hans Kloosterman)
Subject: Lower-bound Kociemba's algorithm
To: cube-lovers@life.ai.mit.edu
Message-Id: <9205291720.AA03306@gefjon.dnl>
X-Envelope-To: cube-lovers@life.ai.mit.edu
X-Mailer: ELM [version 2.3 PL11]
Dik Winter writes:
> Using this result and the result by Hans Kloosterman the diameter of the
> cube group is at most 37. I conjecture the maximal path length in phase 2
> of Kociemba's algorithm is 16, although the requirements on computer time
> cq. memory do inhibit calculations at this moment (only in memory would be
> feasible, but that requires 500 to 1000 MByte and computation time would be
> about one day). This figure of 16 would reduce the upperbound of the groups
> diameter to 28.
Unfortunately Dik's conjecture for phase 2 is too optimistic.
Recall the maximum distances of the 4 stages of my algorithm:
1. 7 moves within the group <R, L, F, B, U,D>
2. 10 moves within the group <R, L, F2,B2,U,D>
3. 8 moves within the group <R2,L2,F2,B2,U,D>
4. 18 moves within the group <R2,L2,F2,B2,U,D>
(Stage 3 and 4 together requires at most 25 moves.)
These number of moves are minimal and cannot be improved within their
group of moves. (Stage 2 can also not be improved using all moves.)
From this we may conclude that the maximum path length in stage 2 of
the algorithm of Kociemba is at least 18 moves.
Taking the results of Dik Winter for stage 1 into account, the lower-bound
for the mximum of Kociemba's algorithm becomes 30 moves.
Hans Kloosterman
From dik@cwi.nl Fri May 29 20:32:28 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA23171; Fri, 29 May 92 20:32:28 EDT
Received: from steenbok.cwi.nl by charon.cwi.nl with SMTP
id AA17356 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:32:23 +0200
Received: by steenbok.cwi.nl
id AA01086 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:32:22 +0200
Date: Sat, 30 May 1992 02:32:22 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9205300032.AA01086.dik@steenbok.cwi.nl>
To: J.M.Kloosterman@research.ptt.nl, cube-lovers@life.ai.mit.edu
Subject: Re: Lower-bound Kociemba's algorithm
(About my conjecture of 16 moves for phase 2:)
> Unfortunately Dik's conjecture for phase 2 is too optimistic.
> Recall the maximum distances of the 4 stages of my algorithm:
> 1. 7 moves within the group <R, L, F, B, U,D>
> 2. 10 moves within the group <R, L, F2,B2,U,D>
> 3. 8 moves within the group <R2,L2,F2,B2,U,D>
> 4. 18 moves within the group <R2,L2,F2,B2,U,D>
> (Stage 3 and 4 together requires at most 25 moves.)
> These number of moves are minimal and cannot be improved within their
> group of moves.
Did you (since your article) do an exhaustive search? In your article you
mentioned that you had 6 positions that still do require 18 moves. And you
mention that you doubted that there would be 17 move solvers. Have you
proven since then that it can not be done in less than 18? If not, send me
your positions and I will try.
I have currently a program running that tries all phase 4 positions. It
is possible to reduce the number of searches from 3,981,312 (the article
contains a typo here) to 428,544 by observing equivalent positions (as
I did mention in a previous article (*)). Assuming my conjecture of 16 the
complete calculations would take about 1000 to 1500 hours (%). Not
unprecedented ;-). (There must be a reason that I am a member of the CWI
research group on large scale computing.) There are now only two machines
munching at the problem, but there would be no problem to start up a few more
again. I just did it to see what happens.
dik
--
* The equivalent positions are found by rotation of the complete cube along
the UD axis for a quarter turn, along the RL axis through a half turn and
mirroring along the FRBL plane. When looking at one dimension only this
reduces the number from 40320 to 2768. Restricting to Hans's initial positions
in phase 4, this reduces the number from 576 to 62. So the count becomes:
62 * 576 * 24 / 2
in stead of
576 * 576 * 24 / 2 (= ((4!)^5) / 2).
--
% I found that an exhaustive search upto 16 moves takes about 10 seconds.
Increasing to 17 would up the time to 110 seconds. So if you mail me
the situations for which you do not yet have less than 18 moves I will
have an attempt at them.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
dik@cwi.nl
From dik@cwi.nl Fri May 29 20:44:04 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA23423; Fri, 29 May 92 20:44:04 EDT
Received: from steenbok.cwi.nl by charon.cwi.nl with SMTP
id AA17413 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:44:02 +0200
Received: by steenbok.cwi.nl
id AA01102 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:44:01 +0200
Date: Sat, 30 May 1992 02:44:01 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9205300044.AA01102.dik@steenbok.cwi.nl>
To: J.M.Kloosterman@research.ptt.nl, cube-lovers@life.ai.mit.edu
Subject: Re: Lower-bound Kociemba's algorithm
As an afterthough, it would be interesting if it is possible to reduce the
number of moves in your fourth phase. The main difference between your
algorithm and Kociemba's is that yours is deterministic. Kociemba's
algorithm performs quite a number of searches before finding the optimal
solution. And even than it is not known whether the solution is indeed
optimal, longer searches might reveal better solutions. Your algorithm
gives an upper bound to the number of moves, and the solution is reached
in limited time. Kociemba's algorithm is in theory unlimited in time.
My experience is that it is best to limit the first phase in Kociemba's
algorithm to 13 moves. But that is only because of time constraints.
dik
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
dik@cwi.nl
From @mitvma.mit.edu:hans@freyr.research.ptt.nl Sat May 30 14:26:53 1992
Return-Path: <@mitvma.mit.edu:hans@freyr.research.ptt.nl>
Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) id AA07529; Sat, 30 May 92 14:26:53 EDT
Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2)
with BSMTP id 6142; Sat, 30 May 92 14:27:59 EDT
Received: from research.ptt.nl by MITVMA.MIT.EDU (Mailer R2.08 R208004) with
BSMTP id 1345; Sat, 30 May 92 14:27:58 EDT
Received: from dnlunx.research.ptt.nl (DNLUNX) by research.ptt.nl (PMDF #12085)
id <01GKMUCOZXDCDQGRT4@research.ptt.nl>; Sat, 30 May 1992 19:41 +0100
Received: by gefjon.dnl (4.1/SMI-4.1) id AA03556; Sat, 30 May 92 19:40:37 +0200
Date: Sat, 30 May 92 19:40:36 MET DST
From: J.M.Kloosterman@research.ptt.nl (Hans Kloosterman)
Subject: Re: Lower-bound Kociemba's algorithm
In-Reply-To: <9205300044.AA01102.dik@steenbok.cwi.nl>; from "Dik.Winter@CWI.NL"
at May 30, 92 2:44 am
To: Dik.Winter@cwi.nl
Cc: cube-lovers@life.ai.mit.edu
Message-Id: <9205301740.AA03556@gefjon.dnl>
X-Envelope-To: cube-lovers@life.ai.mit.edu
X-Mailer: ELM [version 2.3 PL11]
> Did you (since your article) do an exhaustive search? In your article you
> mentioned that you had 6 positions that still do require 18 moves. And you
> mention that you doubted that there would be 17 move solvers. Have you
> proven since then that it can not be done in less than 18? If not, send me
> your positions and I will try.
I have done an exhaustive search and none of the 6 situations of 18 moves
could be reduced to 17 moves (within the group of <R2,L2,F2,B2,U,D>).
For the case you want to verify, one of them is:
L2 U R2 B2 U2 B2 L2 D2 L2 F2 D' L2 B2 F2 L2 F2 U' D
Hans Kloosterman
From dik@cwi.nl Sat May 30 18:12:35 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA10599; Sat, 30 May 92 18:12:35 EDT
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA26864 (5.65b/2.10/CWI-Amsterdam); Sun, 31 May 1992 00:12:31 +0200
Received: by boring.cwi.nl
id AA02915 (5.65b/2.10/CWI-Amsterdam); Sun, 31 May 1992 00:12:30 +0200
Date: Sun, 31 May 1992 00:12:30 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9205302212.AA02915.dik@boring.cwi.nl>
To: J.M.Kloosterman@research.ptt.nl
Subject: Re: Lower-bound Kociemba's algorithm
Cc: cube-lovers@life.ai.mit.edu
> I have done an exhaustive search and none of the 6 situations of 18 moves
> could be reduced to 17 moves (within the group of <R2,L2,F2,B2,U,D>).
> For the case you want to verify, one of them is:
>
> L2 U R2 B2 U2 B2 L2 D2 L2 F2 D' L2 B2 F2 L2 F2 U' D
>
Of course I verified it ;-). This one does indeed kill Kociemba's algorithm.
On a fast processor (65 MHz SPARC) with a larger limit database than Kociemba
is using himself (the database is about 5 MByte for the second phase), it took
3 hours 15 minutes to find a minimal solution. Of 18 moves.
From dik@cwi.nl Mon Jun 8 16:40:43 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA20905; Mon, 8 Jun 92 16:40:43 EDT
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA09954 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:40:42 +0200
Received: by boring.cwi.nl
id AA12825 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:40:41 +0200
Date: Mon, 8 Jun 1992 22:40:41 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9206082040.AA12825.dik@boring.cwi.nl>
To: cube-lovers@life.ai.mit.edu
Subject: How big is the Magic Domino? (part2, data)
2x2x2 cube
q+h loc max q only loc max
0: 1 - 1 -
1: 9 - 6 -
2: 54 - 27 -
3: 321 - 120 -
4: 1847 11 534 -
5: 9992 8 2256 -
6: 50136 96 8969 -
7: 227536 904 33058 16
8: 870072 13212 114149 53
9: 1887748 413392 360508 260
10: 623800 604516 930588 1460
11: 2644 2644 1350852 34088
12: 782536 402260
13: 90280 88636
14: 276 276
3x3x3 corners only
q+h loc max q only loc max
0: 1 - 1 -
1: 18 - 12 -
2: 243 - 114 -
3: 2874 6 924 -
4: 28000 88 6539 -
5: 205416 792 39528 -
6: 1168516 15304 199926 114
7: 5402628 46068 806136 600
8: 20776176 325680 2761740 17916
9: 45391616 9757376 8656152 10200
10: 15139616 14665856 22334112 35040
11: 64736 64736 32420448 818112
12: 18780864 9654240
13: 2166720 2127264
14: 6624 6624
magic domino, 1 solution
q+h loc max q only loc max
0: 1 - 1 -
1: 10 - 8 -
2: 67 - 48 -
3: 420 - 260 -
4: 2335 - 1330 -
5: 12260 - 6556 -
6: 61038 3 31301 -
7: 291004 12 144392 -
8: 1327429 793 638407 2
9: 5821374 6170 2709620 64
10: 24141784 87202 10873023 1261
11: 89480354 990826 39768668 15728
12: 262907144 13212972 124815946 214530
13: 485409604 91824956 296531984 2741192
14: 508704668 161596512 460831364 23949864
15: 232904952 175407548 435219080 72423024
16: 14508468 13668852 215035460 91647012
17: 129376 128592 38469576 35228568
18: 112 112 624320 618368
19: 1056 1056
magic domino, 4 solutions
q+h loc max q only loc max
0: 4 - 4 -
1: 28 - 24 -
2: 136 - 108 -
3: 672 - 480 -
4: 3228 - 2116 -
5: 15072 - 9120 -
6: 69000 - 39188 -
7: 310784 92 166408 -
8: 1369220 1052 691508 56
9: 5888676 8656 2812496 192
10: 24209988 92284 11015008 1860
11: 89458152 976008 39837904 16104
12: 262772436 13124304 124673780 202940
13: 485358148 91776620 296336800 2667824
14: 508703948 161595792 460769708 23896632
15: 232904952 175407548 435217336 72421280
16: 14508468 13668852 215035460 91647012
17: 129376 128592 38469576 35228568
18: 112 112 624320 618368
19: 1056 1056
magic domino, 8 solutions
q+h loc max q only loc max
0: 8 - 8 -
1: 56 - 48 -
2: 272 - 216 -
3: 1344 - 960 -
4: 6456 - 4232 -
5: 30144 - 18240 -
6: 138000 - 78376 -
7: 621568 184 332816 -
8: 2732664 3096 1383016 112
9: 11649816 28672 5612576 384
10: 46553800 331960 21772432 4584
11: 158726064 3909520 75752384 47792
12: 377277280 46692640 208971608 783864
13: 507933248 129847936 388348544 11790688
14: 414571632 181149888 466373488 54544928
15: 102181280 86967456 334811104 78445984
16: 3271456 3221680 114248208 79836432
17: 7312 7312 7974528 7869280
18: 19616 19616
From dik@cwi.nl Mon Jun 8 16:38:55 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA20835; Mon, 8 Jun 92 16:38:55 EDT
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA09864 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:38:52 +0200
Received: by boring.cwi.nl
id AA12813 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:38:51 +0200
Date: Mon, 8 Jun 1992 22:38:51 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9206082038.AA12813.dik@boring.cwi.nl>
To: cube-lovers@life.ai.mit.edu
Subject: How big is the Magic Domino?
Having done a number of calculations on maximal distances I thought about
getting at newer pastures. The Magic Domino. The results follow in the
next mailing, this post discusses a bit about the information found there.
I added to the next mail also the previous results for the 2x2x2 and the
corners of the 3x3x3 together with some additional results not presented
previously. There are for each puzzle five columns. The first one
enumerates the number of moves, the next two give the results if both
quarter and half turns are accepted as moves, the last two give the
information if only quarter turns are accepted (of course, on the Domino
this distinction is there only for the U and D faces, the others know
half turns only). For each case there are two columns, the first giving the
number of positions requiring the stated number of moves, the second
column gives the number of local maxima (i.e. each move brings you closer
to a solution).
There are three tables for the Domino. The one you want to pick depends
on how you view the puzzle. The first view is that there is only one
solution with on top 1 to 3 running from left back to right back. The
second view is that rotation of the puzzle makes different configurations
indistinguishable, so the total number of configuration is (8!)^2 / 4.
An alternative way to look at it is that there are 4 solutions. One the
standard solution, the others obtained by rotating the domino along the
UD axis. The distinction between the two views is only a factor of four
in the number of configurations for the different path-lengths. Finally,
we can view as a solution the configuration with on top 1 to 3 running from
right back to left back in stead of the other way around. Actually this
solution is not worse than the other, because, if we turn over a solved
Domino we go from one to the other. This view can also be expressed by saying
there are 8 solutions. I give results for all three cases. The numbers upto
(and including) path-length 2 have been checked by hand.
Some remarkable observations.
When we compare the tables for 1 solution and those for 4 solutions we see
that for short path-lengths the number of configurations is multiplied by
4. On the other hand, for long path-lengths the number of configurations
is equal! We can say that rotation of the Domino has only a short range
effect.
On the other hand, if we compare both with the 8-solutions tables we see that
the latter allows shorter solutions in general, so mirroring has a long
range effect.
Each of the 6 calculations on the Magic Domino took 2 to 2.5 hours on one
processor of an SGI 4D-420S. The program is completely memory bound (and
the cache does not help). It needs at least 31 MByte of core (and must be
resident) otherwise you will get no results at all in reasonable time. I
tried it on the 32 MByte FPS; while it will happily give results initially
at some stage it will not longer run. Not only that it will not walk either,
and also not crawl. It is just sitting there paging in and paging out (a
phenomenon known as page thrashing). I found that the program would get
less than 0.005 % of the CPU on an otherwise unloaded machine.
The program would enable me to write a 27901440 byte file that would assist
in an optimal solver for the Domino.
dik
From dik@cwi.nl Mon Jun 8 20:48:48 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA28679; Mon, 8 Jun 92 20:48:48 EDT
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA26132 (5.65b/2.10/CWI-Amsterdam); Tue, 9 Jun 1992 02:48:46 +0200
Received: by boring.cwi.nl
id AA13958 (5.65b/2.10/CWI-Amsterdam); Tue, 9 Jun 1992 02:48:46 +0200
Date: Tue, 9 Jun 1992 02:48:46 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9206090048.AA13958.dik@boring.cwi.nl>
To: cube-lovers@life.ai.mit.edu
Subject: Magic Domino part3
Considering my previous fiasco, I have now incorporated the changes needed to
make the magic domino calculating program working into the program that
calculated the corners on a 3x3x3 cube. The results still match, which
gives me confidence that the algorithms are correct indeed. Moreover,
it reduced the time to do the 3x3x3 corner calculations to 8 minutes.
dik
From wft@math.canterbury.ac.nz Wed Jun 10 00:37:06 1992
Return-Path: <wft@math.canterbury.ac.nz>
Received: from CANTVA.CANTERBURY.AC.NZ by life.ai.mit.edu (4.1/AI-4.10) id AA08570; Wed, 10 Jun 92 00:37:06 EDT
Received: from math.canterbury.ac.nz by csc.canterbury.ac.nz (PMDF #12052) id
<01GL2137V0CW90NKNZ@csc.canterbury.ac.nz>; Wed, 10 Jun 1992 16:36 +1200
Received: by math.canterbury.ac.nz (4.1/SMI-4.1) id AA27010; Wed,
10 Jun 92 16:36:33 NZS
Date: Wed, 10 Jun 92 16:36:33 NZS
From: wft@math.canterbury.ac.nz (Bill Taylor)
Subject: Name query.
To: Cube-Lovers@life.ai.mit.edu
Message-Id: <9206100436.AA27010@math.canterbury.ac.nz>
X-Envelope-To: Cube-Lovers@life.AI.MIT.EDU
Can anyone tell me:-
Why is the "Pons Asinorum" pattern so called ?
---------------------------------------------------------------------
Bill Taylor wft@math.canterbury.ac.nz
Artificial intelligence beats real stupidity.
---------------------------------------------------------------------
From ronnie@cisco.com Wed Jun 10 15:12:01 1992
Return-Path: <ronnie@cisco.com>
Received: from wolf.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA01827; Wed, 10 Jun 92 15:12:01 EDT
Received: from lager.cisco.com by wolf.cisco.com with TCP; Wed, 10 Jun 92 12:11:36 -0700
Message-Id: <9206101911.AA01765@wolf.cisco.com>
To: wft@math.canterbury.ac.nz (Bill Taylor)
Cc: Cube-Lovers@life.ai.mit.edu
Subject: Re: Name query.
From: ronnie@cisco.com
Date: Wed, 10 Jun 92 12:11:35 PDT
Sender: ronnie@cisco.com
> Can anyone tell me:-
>
> Why is the "Pons Asinorum" pattern so called ?
Pons Asinorum is Latin for "Asses' Bridge," and is the name of the
proposition that the base angles of an isoceles triangle are equal.
It is more generally any test of ability imposed upon the inexperienced
or ignorant.
Ronnie (who has xwebster)
From GOET@rcl.wau.nl Thu Jun 11 02:46:21 1992
Return-Path: <GOET@rcl.wau.nl>
Received: from NET.WAU.NL by life.ai.mit.edu (4.1/AI-4.10) id AA18013; Thu, 11 Jun 92 02:46:21 EDT
Received: from RVD.WAU.NL by NET.WAU.NL (PMDF #12413) id
<01GL2YYMEHS0001ZSZ@NET.WAU.NL>; Thu, 11 Jun 1992 08:46 GMT +01:00
Received: from RCL.WAU.NL by RCL.WAU.NL (PMDF #12413) id
<01GL2YSCT30W9ED95T@RCL.WAU.NL>; Thu, 11 Jun 1992 08:41 GMT +01:00
Date: Thu, 11 Jun 1992 08:41 GMT +01:00
From: "Kees Goet - Landbouwuniversiteit, Afd. I&D" <Kees.Goet@rcl.wau.nl>
Subject: Unsubscribe
To: cube-lovers@life.ai.mit.edu
Message-Id: <01GL2YSCT30W9ED95T@RCL.WAU.NL>
X-Envelope-To: cube-lovers@life.ai.mit.edu
X-Vms-To: IN%"cube-lovers@life.ai.mit.edu"
X-Vms-Cc: GOET
Could somebody please remove me from this list.
Thanks in advance.
Kees Goet.
From STEFANO@agrclu.st.it Thu Jun 11 10:14:42 1992
Return-Path: <STEFANO@agrclu.st.it>
Received: from pol88a (pol88a.polito.it) by life.ai.mit.edu (4.1/AI-4.10) id AA25505; Thu, 11 Jun 92 10:14:42 EDT
Received: from AG-IN by POLITO.IT (PMDF #12666) id
<01GL3B97N2G095MMGN@POLITO.IT>; Thu, 11 Jun 1992 14:38 GMT+1
Date: Thu, 11 Jun 92 14:36 CET
From: STEFANO BONACINA <STEFANO@agrclu.st.it>
Subject: Signoff
To: cube-lovers@life.ai.mit.edu
Message-Id: <40E32DA5DC3F00DAC9@agr04.ST.IT>
X-Organization: SGS-THOMSON Microelectronics
X-Envelope-To: cube-lovers@life.ai.mit.edu
X-Vms-To: IN%"cube-lovers@life.ai.mit.edu"
Could anyone tell me how can I unsubscribe from this mailing list?
Thanks in advance.
Stefano
From alan@ai.mit.edu Thu Jun 11 15:20:31 1992
Return-Path: <alan@ai.mit.edu>
Received: from august (august.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA09200; Thu, 11 Jun 92 15:20:31 EDT
Received: by august (4.1/AI-4.10) id AA04098; Thu, 11 Jun 92 15:21:19 EDT
Date: Thu, 11 Jun 92 15:21:19 EDT
Message-Id: <9206111921.AA04098@august>
From: Alan Bawden <Cube-Lovers-Request@ai.mit.edu>
Sender: Cube-Lovers-Request@ai.mit.edu
To: Kees.Goet@rcl.wau.nl, STEFANO@agrclu.st.it
Cc: Cube-Lovers@ai.mit.edu
In-Reply-To: "Kees Goet - Landbouwuniversiteit, Afd. I&D"'s message of Thu, 11 Jun 1992 08:41 GMT +01:00 <01GL2YSCT30W9ED95T@RCL.WAU.NL>
Subject: Please do not bother everybody with administrative requests
Date: Thu, 11 Jun 92 14:36 CET
From: STEFANO BONACINA <STEFANO@agrclu.st.it>
Could anyone tell me how can I unsubscribe from this mailing list?
Thanks in advance.
Stefano
Date: Thu, 11 Jun 1992 08:41 GMT +01:00
From: "Kees Goet - Landbouwuniversiteit, Afd. I&D" <Kees.Goet@rcl.wau.nl>
Could somebody please remove me from this list.
Thanks in advance.
Kees Goet.
As everyone is informed when they subscribe, administrative requests should
be directed to Cube-Lovers-Request@AI.MIT.EDU (me). Even if you lost my
original greeting message, the "-Request" suffix is a sufficiently
widespread convention for mailing lists that you should have tried it
first, before bothering the entire mailing list.
STEFANO@agrclu.st.it, I have removed you.
Kees.Goet@rcl.wau.nl, I will be sending you separate mail about your
subscription.
- Alan
From gls@think.com Thu Jun 11 17:11:55 1992
Received: from mail.think.com (Mail1.Think.COM) by life.ai.mit.edu (4.1/AI-4.10) id AA12924; Thu, 11 Jun 92 17:11:55 EDT
Return-Path: <gls@Think.COM>
Received: from Strident.Think.COM by mail.think.com; Thu, 11 Jun 92 16:52:32 -0400
From: Guy Steele <gls@think.com>
Received: by strident.think.com (4.1/Think-1.2)
id AA18593; Thu, 11 Jun 92 16:52:31 EDT
Date: Thu, 11 Jun 92 16:52:31 EDT
Message-Id: <9206112052.AA18593@strident.think.com>
To: ronnie@cisco.com
Cc: wft@math.canterbury.ac.nz, Cube-Lovers@life.ai.mit.edu
In-Reply-To: ronnie@cisco.com's message of Wed, 10 Jun 92 12:11:35 PDT <9206101911.AA01765@wolf.cisco.com>
Subject: Name query.
From: ronnie@cisco.com
Date: Wed, 10 Jun 92 12:11:35 PDT
> Can anyone tell me:-
>
> Why is the "Pons Asinorum" pattern so called ?
Pons Asinorum is Latin for "Asses' Bridge," and is the name of the
proposition that the base angles of an isoceles triangle are equal.
It is more generally any test of ability imposed upon the inexperienced
or ignorant.
The term also carries the connotation that the test is in fact
of the simplest and most elementary kind. If you can't prove
the Pons Asinorum of geometry, then you don't know even the most
elementary concept of geometry--i.e., as a geometer, you know as
much as a donkey. And if you cannot form the Pons Asinorum
pattern, you sure don't know much about cubing.
--Guy Steele
From ACW@riverside.scrc.symbolics.com Thu Jun 11 17:44:29 1992
Return-Path: <ACW@riverside.scrc.symbolics.com>
Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA13947; Thu, 11 Jun 92 17:44:29 EDT
Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 834944; 11 Jun 1992 17:39:46-0400
Date: Thu, 11 Jun 1992 17:39-0400
From: Allan C. Wechsler <ACW@riverside.scrc.symbolics.com>
Subject: Name query.
To: gls@think.com, ronnie@cisco.com
Cc: wft@math.canterbury.ac.nz, Cube-Lovers@life.ai.mit.edu
In-Reply-To: <9206112052.AA18593@strident.think.com>
Message-Id: <19920611213942.5.ACW@PALLANDO.SCRC.Symbolics.COM>
Date: Thu, 11 Jun 1992 16:52 EDT
From: Guy Steele <gls@think.com>
From: ronnie@cisco.com
Date: Wed, 10 Jun 92 12:11:35 PDT
> Can anyone tell me:-
>
> Why is the "Pons Asinorum" pattern so called ?
Pons Asinorum is Latin for "Asses' Bridge," and is the name of the
proposition that the base angles of an isoceles triangle are equal.
It is more generally any test of ability imposed upon the inexperienced
or ignorant.
The term also carries the connotation that the test is in fact
of the simplest and most elementary kind. If you can't prove
the Pons Asinorum of geometry, then you don't know even the most
elementary concept of geometry--i.e., as a geometer, you know as
much as a donkey. And if you cannot form the Pons Asinorum
pattern, you sure don't know much about cubing.
--Guy Steele
I think the metaphorical leap from geometry to cubing was probably made
by Bernie Greenberg, in whatever year it was that Hofstadter did his Sci
Am column. Hofstadter came to MIT to talk to a bunch of cubers,
gathering material for his article. I was in the group and my name is
mentioned in the article -- the only time I have ever gotten my name
into Sci Am.
"Pons Asinorum" has a lot of Bernie's style about it -- casual use of
Latin, whimsical metaphor, fondness for naming things. He had a bunch
of cube operators with Latin names, and also some wacky English ones. I
remember the Spratt Wrench (F R'L D R'L B R'L U R'L) which flips four
edges and was what everyone used before monoflips were discovered.
Bernie also had things with names like the Lesser Hammer of the Right
and the Greater Hammer of the Right; his "patter" was fabulous. I
regret not having a videotape of Bernie solving the cube in, say, 1978.
(I hope I've got the year right.)
While I'm reminiscing, I should confess that my standard corner operator
is still the same as it was then: (FUR)^5, which exchanges two corners,
leaves the rest of the corners alone, and fucks the edges completely.
(Prudes, do not hassle me. This has been a technical term in cubing
around MIT since The Beginning.) Because of this property of "furry
five", I have to home and orient all the corners first, before I touch
the edges. It's the kind of quirky algorithm you don't see among
younger cubers, because everybody these days learns how to solve the
thing from a book. In the Beginning, there were no books, and I proudly
state that I solved the cube from scratch, by brainpower. Later I
discovered that there were easier ways to do things than (FR)^105! I
had pages and pages covered with little cube diagrams with arrows
showing how the stickers were permuted by a particular sequence.
I'm interested in hearing other reminiscences from people who actually
solved the cube -- you're disqualified if you learned how to solve it
from somebody else, or from a book.
From dik@cwi.nl Thu Jun 11 18:46:53 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA16508; Thu, 11 Jun 92 18:46:53 EDT
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA05569 (5.65b/2.10/CWI-Amsterdam); Fri, 12 Jun 1992 00:46:44 +0200
Received: by boring.cwi.nl
id AA22860 (5.65b/2.10/CWI-Amsterdam); Fri, 12 Jun 1992 00:46:42 +0200
Date: Fri, 12 Jun 1992 00:46:42 +0200
From: Dik.Winter@cwi.nl
Message-Id: <9206112246.AA22860.dik@boring.cwi.nl>
To: cube-lovers@life.ai.mit.edu
Subject: Re: Name query. Actually reminiscences.
> I
> regret not having a videotape of Bernie solving the cube in, say, 1978.
> (I hope I've got the year right.)
I think it must be later. The cube was put on the market in Hungary in 1977
and first exported in 1980. Although earlier examples were privately
exported I presume.
> While I'm reminiscing, I should confess that my standard corner operator
> is still the same as it was then: (FUR)^5, which exchanges two corners,
> leaves the rest of the corners alone, and fucks the edges completely.
Happens to me also. I still use operators I found myself in favour of
(shorter) processes found later in books. I remember them better!
> I'm interested in hearing other reminiscences from people who actually
> solved the cube -- you're disqualified if you learned how to solve it
> from somebody else, or from a book.
I got one for my birthday in 1981 (yes, I was late). By the end of the party
it was completely scrambled. One long night and a long day afterwards had me
solve the cube. Although at that moment I had not completely lined up
procedures to do it. Later I more or less procedurized it.
Much stranger was my first encounter with Square 1. As all puzzles it was
scrambled within minutes after I brought it home. I tried to solve it, but
for some reason I did not yet see how to bring it back in the shape of a cube.
The next day when I came home from work it was in the shape of a cube. It
appears that my 8 year old daughter had done that! Solving the remainder was
fairly simple.
dik
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
dik@cwi.nl
From sjfc!ggww@cci632.cci.com Sun Jun 14 22:54:44 1992
Return-Path: <sjfc!ggww@cci632.cci.com>
Received: from uu.psi.com by life.ai.mit.edu (4.1/AI-4.10) id AA27747; Sun, 14 Jun 92 22:54:44 EDT
Received: from sjfc.UUCP by uu.psi.com (5.65b/4.1.031792-PSI/PSINet)
id AA14756; Sun, 14 Jun 92 22:36:33 -0400
Received: by cci632.cci.com (5.54/5.17)
id AA07588; Sun, 14 Jun 92 22:03:16 EDT
Received: by sjfc.UUCP (5.51/4.7)
id AA00173; Sun, 14 Jun 92 21:37:14 EDT
Date: Sun, 14 Jun 92 21:37:14 EDT
From: sjfc!ggww@cci632.cci.com (Gerry Wildenberg)
Message-Id: <9206150137.AA00173@sjfc.UUCP>
To: cube-lovers@life.ai.mit.edu
Subject: Remove me from this list.
Please remove me from the mailing list.
Gerry Wildenberg ggww@sjfc.uucp
St. John Fisher College sjfc!ggww@cci.com
Rochester, NY 14618 ggww@sjfc.edu (New, may not yet work.)
From ACW@riverside.scrc.symbolics.com Tue Jun 16 16:12:32 1992
Return-Path: <ACW@riverside.scrc.symbolics.com>
Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA28765; Tue, 16 Jun 92 16:12:32 EDT
Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 838315; 16 Jun 1992 16:14:52-0400
Date: Tue, 16 Jun 1992 16:13-0400
From: Allan C. Wechsler <ACW@riverside.scrc.symbolics.com>
Subject: Re: Name query. Actually reminiscences.
To: Dik.Winter@cwi.nl, cube-lovers@life.ai.mit.edu
In-Reply-To: <9206112246.AA22860.dik@boring.cwi.nl>
Message-Id: <19920616201302.7.ACW@PALLANDO.SCRC.Symbolics.COM>
Date: Thu, 11 Jun 1992 18:46 EDT
From: Dik.Winter@cwi.nl
Much stranger was my first encounter with Square 1. As all puzzles it was
scrambled within minutes after I brought it home. I tried to solve it, but
for some reason I did not yet see how to bring it back in the shape of a cube.
The next day when I came home from work it was in the shape of a cube. It
appears that my 8 year old daughter had done that! Solving the remainder was
fairly simple.
Our four-year-old managed to assemble our Snafooz into a cube once. No
one else has been able to do it, and he can't duplicate his success.
(He can't even read.)
From wft@math.canterbury.ac.nz Fri Jun 19 03:07:51 1992
Return-Path: <wft@math.canterbury.ac.nz>
Received: from CANTVA.CANTERBURY.AC.NZ by life.ai.mit.edu (4.1/AI-4.10) id AA19564; Fri, 19 Jun 92 03:07:51 EDT
Received: from math.canterbury.ac.nz by csc.canterbury.ac.nz (PMDF #12052) id
<01GLEQZFO09S9X3ZMD@csc.canterbury.ac.nz>; Fri, 19 Jun 1992 19:07 +1200
Received: by math.canterbury.ac.nz (4.1/SMI-4.1) id AA12708; Fri,
19 Jun 92 19:07:30 NZS
Date: Fri, 19 Jun 92 19:07:30 NZS
From: wft@math.canterbury.ac.nz (Bill Taylor)
Subject: reminiscences
To: Cube-Lovers@life.ai.mit.edu
Cc: wft@math.canterbury.ac.nz
Message-Id: <9206190707.AA12708@math.canterbury.ac.nz>
X-Envelope-To: Cube-Lovers@AI.AI.MIT.EDU
Allan C. Wechsler asks for general reminiscences from people who solved
the cube. It's just as well hardly anyone's replied, or the list would be
swamped with boring anecdotes ! So maybe I'll add an anecdote or two of
my own.
Dik.Winter@cwi.nl writes
> > While I'm reminiscing, I should confess that my standard corner operator
> > is still the same as it was then: (FUR)^5, which exchanges two corners,
> > leaves the rest of the corners alone, and fucks the edges completely.
>
>Happens to me also. I still use operators I found myself in favour of
>(shorter) processes found later in books. I remember them better!
Very true. This reminds me of what I read in (I think) the math games
column of Scientific American, about mid-to-late 80's. The cubing craze had
largely passed, and someone who had been an addict, but hadn't touched it
for some years, had occasion to try it again. He realized with horror, that he
couldn't remember a single thing! However, as he began to fiddle with the
cube rather disconsolately, he found himself automatically doing the right
things. "I couldn't remember how to do it, but my fingers could !!", he said.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~
This was my experience too, a few years ago. It's quite uncanny, like
starting to ride a bike again after decades of not doing; only more so.
That's what comes of sticking loyally to your first halfway-decent discoveries
on cube-solving. From madly over-addictive use, they become burned into
your brain tissue. My own pet method has always been to put all the edges
right first, using only common sense (except maybe at the very end some
slight method needed); then put the corners right using the basic
"8-fold way"..... R~ U L U~ R U L~ U~ . We found this eightfold way quite
easy to remember, the face order is very natural, and the sequence of
clockwise vs anti-clockwise turns, i.e. ACCA CCAA, seems somehow like a
sonnet rhyming scheme (now burnt irrevocably into my finger-moving cortex).
This eightfold way is just a commutator of a face move and (a commutator of
two face moves)); so it turns out to be group-theoretically natural, as
commutators do "as little as possible". The eight-fold way can also be
viewed in a natural geometric light, as just a standard 3-permutation of
corners, similaritied away from one another to avoid interference. (Don't
know what the standard technical terms for all this are, sorry; it's probably
old hat to most readers here.) Viewing it this way, one can quickly re-create
several (8-fold) variants, and some 10-fold ones, all of the same type, and
all variously useful. By similarities, one can usually put a corner into a
more useful spot, so as to get two corners done at a time, with one 8-fold.
ENOUGH; of teaching grandmothers to suck eggs. I was going to reminisce. Not
many people seem to do the cube this way, that is, edges first. It was shown
to me by my late colleague Brent Wilson (the other of the "we" refered to
above). At first it seemed a little unnatural, but once you get used to it,
it seems super efficient. I suppose everyone feels that way about their own
methods. The particular 8-fold mentioned above was my own invention, so I've
always had a soft spot for it.
Brent and I both started out on the cube the same way, which is
I suppose standard. We spent some little time learning to do the base. Then we
spent some considerably longer time learning to do the middle layer. We found
later that we had both expected the same thing:- that when the middle and
base layers were all successfully done, the top layer would automatically
have to be right !! So of course, we were both temporarily devastated when
it turned out otherwise; and we both realized that we were in the presence
of a mighty puzzle, and were in for some great fun. So we went ahead and
discovered all the usual group-theoretic things, one by one, over the months.
I have anothger reminiscence to tell about my colleague. I once read of
someone, (J.H.Conway ?), who was alleged to do the cube behind his back !
Well Brent practiced this trick also (unaware of anyone else having
done it, if indeed it was done the same way, even). He invented the method
after having discovered the only all-commuting position, i.e. with all
edges flipped, corners all correct. He perfected a smooth method of doing
this behind his back. The trick is, of course, merely to have a pre-prepared
cube in this position. It doesn't QUITE look random, but if you ADD to it
a couple of random twists, it now looks totally random; at first (and second)
glance. He would show this "random" cube to us, let us hold it (very briefly!),
then take it and do the "all-flips" behind his back. Keeping up a continuous
patter, as he brought it back he would be saying "...so there's only a couple
of twists to go", and then as it appeared he would do the last two twists by
sight, without hesitation. As the two "randomizing" twists commute with
the other position, he didn't have to memorize them; indeed he could even
let the audience do them ! Of course this would mean he would have to have
the pure "flipped" pattern to start with, which was easier to detect, alas.
Well, one time, he was to give a talk to some school kids. He wanted to do
the cube behind his back, as a piece-de-resistance. He decided to train
himself up into being able to undo FOUR random twists by sight. He duly did
this. Then when the talk came around, he had a cube prepared in "all-flip"
position, with two twists added, to make it look quite random. Then, when
the highlight of the talk came around, he would display it to the class,
let one or two handle it briefly, to agree it was just another muddled up
cube. Then, HE WOULD EVEN ASK two members of the audience to add an extra
random twist each (just to prove the cube wasn't in a prepared position!)
Then he would do the all-flips operation behind his back, keeping up his
patter. He expected to be able to handle undoing the four random flips left
over, by sight, as he was completing his patter.
When the great event came along, everything went perfectly, without a hitch.
BUT, amazingly, by a 144-to-1 chance, the two flips that the audience added
exactly undid the two that he had put on himself ! So when he brought it
from behind his back, it was already perfectly done. Without batting an
eyelid, he brought his patter to a halt then and there. Needless to say, the
kids were even more staggered than they would have been otherwise. He
resisted all imploring entreaties to tell them how it was done (like all
good conjurers); and I don't thimk he ever did the trick again!
By great good luck, however, I have a vieotape of him doing this trick, from
the demo itself. So if any of you are ever in New Zealand, you can
look me up, and ask to see this amazing event !
Like Allan Wechsler, I would be delighted to hear anyone else's reminiscences,
or cube anecdotes generally. There must be tons, so, don't be shy!
Cheers, Bill Taylor.
From reid@math.berkeley.edu Sun Jun 21 13:11:03 1992
Return-Path: <reid@math.berkeley.edu>
Received: from math.berkeley.edu by life.ai.mit.edu (4.1/AI-4.10) id AA17637; Sun, 21 Jun 92 13:11:03 EDT
Received: from phnom-penh.berkeley.edu.berkeley.edu by math.berkeley.edu (4.1/1.33(math))
id AA08266; Sun, 21 Jun 92 10:11:00 PDT
Date: Sun, 21 Jun 92 10:11:00 PDT
From: reid@math.berkeley.edu (michael reid)
Message-Id: <9206211711.AA08266@math.berkeley.edu>
To: Cube-Lovers@life.ai.mit.edu
Subject: Re: reminiscences
another call for reminiscences ...
) From: sjfc!ggww@cci632.cci.com (Gerry Wildenberg)
)
) Please remove me from the mailing list.
yeah, remember back in those days when people were actually asking to
be ADDED to the mailing list?
:-)
btw, administrivia should be sent to "cube-lovers-request@ai.mit.edu".
thank you for your cooperation. ^^^^^^^
>From: wft@math.canterbury.ac.nz (Bill Taylor)
> This eightfold way is just a commutator of a face move and (a commutator of
> two face moves)); so it turns out to be group-theoretically natural, as
> commutators do "as little as possible".
here's the way i describe this. if sigma is a permutation on n symbols,
(say 1, 2, 3, ... , n), define the "support" of sigma to be those
integers which are NOT fixed by sigma. if tau is another permutation
on the same set, such that supp(sigma) and supp(tau) are disjoint,
then sigma and tau commute (i.e. the commutator is the identity).
if supp(sigma) intersect supp(tau) has just one element, then the
commutator is a three-cycle. as a rule of thumb, the smaller the
intersection of the supports, the smaller the support of the commutator.
in bill's example, ( R~ U L U~ R U L~ U~ ) the two permutations are
"R" and "U L U~", which only affect one corner in common. (actually,
to consider the cube as a permutation group, each corner is really
3 objects, one for each orientation.) but the analogy works well.
this idea is also helpful for creating three-cycles of corner-edge pairs
as well. on the 5x5x5 cube, you can make three-cycles of large blocks.
in fact, a larger cube is probably a better visual aid for understanding/
explaining this concept. another good commutator to try is with the
two sequences "B1 D2 B3" and "R1 U2 R3", which affect two corners
in common. (this is a fairly well-known maneuver.)
> "I couldn't remember how to do it, but my fingers could !!", he said.
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~
> This was my experience too, a few years ago. It's quite uncanny, like
> starting to ride a bike again after decades of not doing; only more so.
i've also experienced the exact same thing. it seems as though my hands
remember a short sequence by which i've just conjugated some maneuver,
which helps at the end of the conjugation. also in these commutators,
after i've done A and working on B, it feels as though my hands are
anxious to undo A, almost as if the cube is spring loaded and will just
snap back.
[lame story about a friend pulling some hoax deleted]
actually, i think i FINALLY understand the story. the idea is that not
only is "brent" going to solve the cube behind his back, but he's also
going to do it WITHOUT first looking at it. actually, the story isn't
nearly as lame as i first thought, after realizing this.
when i visited mit in 1984(?), i saw joe killian do the real trick.
i certainly would have complained if i hadn't been allowed to scramble
the cube exactly as i wished. in fact, i may have even insisted that
he use MY cube (not too sure, though), just to be certain that the
surface hasn't been textured in any way. it was quite impressive.
about 5 minutes of studying, then behind the back without peeking.
he said that all it took was a good system of remembering where all
the pieces are. but i don't know what his system was.
by the way, bill, this "brent" wouldn't happen to be your friend who can
do the cube in 0.87 seconds, would he? :-)
and speaking of tall tales, let's see if anyone can top this one:
back in the days when i was into speed, er, speed CUBING, i'd solve
the cube maybe 200 times every day. for some reason, i got into the
habit of scrambling it behind my back (probably from listening to too
many complaints like "you're just watching all the moves you're doing!"
yeah, such a good complaint deserves such a fine solution.) well,
anyway, one time i stopped scrambling it, and as usual, i get 15
second to study it (standard racing rules). however, much to my
surprise, the cube was quite UNscrambled! how could this possibly be?
well, the only explanation is certainly that after scrambling the cube
thousands of times, my hands began to get into a rhythm (maybe even a rut).
they'd just do the same sequence over and over again. depending upon
my concentraion level, i'd find that sometimes i needed to make a
conscious effort to vary the sequence. in fact, at least once i got
a pattern that i'd previously seen: it was 4 dots with 6 corners twisted
(hence has order 6). so it's not too implausible.
like bill, and unlike dik, i spent quite some time struggling with
the cube before i finally solved it; probably about 6 or 7 weeks.
in fact for some time, probably about 2 weeks, i was convinced that
it couldn't be done, except by very dumb luck (as in story above).
of course, in those days, i hadn't heard of cube-lovers, hadn't even
seen a computer, didn't know the furst-hopcroft-luks algorithm, hadn't
even heard of anyone who could solve it ... but i was just a high school
freshman (age 14) at the time. i didn't even know what a group was!
this was shortly before the big craze started here in the u.s. (late 1980).
at school, some friends and i talked about it, but the main questions were:
how was it made, and how many combinations did it REALLY have? i was truly
convinced that trying to solve it would be futile. there was also a
shortage of them at the time, so i didn't get one until xmas. in fact,
i remember the tv commercial that ideal put out. they didn't even make
it clear that it actually turned in all possible directions! we had all
sorts of ridiculous diagrams and ideas of cables and magnets, but none of
them quite worked. and how could it turn in all directions? i heard of a
bookstore somewhere that had one on display (but were otherwise sold out),
so i went to see it. i remember spending a few minutes twisting it to
find an axis that wouldn't turn! in fact, i could keep turning the same
face in the same direction, around and around and around ... and the
cables inside never got caught!
sometimes i'm amazed at just how stupid i can be when i try ...
anyway, the story about how i finally figured out how to solve it isn't
nearly as interesting. after i first heard about people that could do it,
i started to work on it more seriously. the key ideas were: get all
the corners, (here was something that you could do, and then still do
more without destroying what's already done. but this was hard and
usually took more dumb luck and/or persistence.) then two opposite layers.
(again, the middle slice still can turn, even with half turns on the
sides F, B, R, L.) it took several days to flip the last two edges on
the middle layer. (i just kept picking a different pair of opposite
layers to solve and stumbled across a U layer monoflip in the process.
of course, it took months before i realized what was actually happening.)
also figuring out how to take it apart (and finally seeing how it was made)
was helpful, 'cause then i could experiment easily.
well, i've droned on long enough. anyone else got any interesting stories?
mike
From STEVENS@macc.wisc.edu Tue Jun 23 07:59:22 1992
Return-Path: <STEVENS@macc.wisc.edu>
Received: from vms3.macc.wisc.edu by life.ai.mit.edu (4.1/AI-4.10) id AA05161; Tue, 23 Jun 92 07:59:22 EDT
Received: from VMSmail by vms3.macc.wisc.edu; Mon, 22 Jun 92 09:19 CDT
Message-Id: <22062209193919@vms3.macc.wisc.edu>
Date: Mon, 22 Jun 92 09:19 CDT
From: PAul STevens - MACC - 2-9618 <STEVENS@macc.wisc.edu>
Subject: Re: reminiscences
To: CUBE-LOVERS@life.ai.mit.edu
X-Vms-To: IN%"cube-lovers@life.ai.mit.edu",STEVENS
reid@math.berkeley.edu writes:
>well, i've droned on long enough. anyone else got any interesting stories?
I think my solution may be considered cheating; but I was pretty proud of
it. I had almost decided to give up on the thing. But I had just
designed and built an 8080 'computer'. It had 2k bytes of 2102's and
had to be programmed with binary switches and whenever the program
clobbered itself the entire program had to be re-entered in binary.
So I wrote a program to look for combinations of moves that left most
of the cube alone and only moved a few cubelets. I studied the best of
these at great length and managed to combine some into 'better' moves,
eventually finding some that moved only three or so cubelets. These
were then combined into a solution. A rather god-awful solution I
think. But my fingers learned the moves and I have never abandoned them
for fear of becoming totally confused. The same ugly solution has been
passed on for at least one generation and perhaps will persist for
hundreds of years.
I still don't know what a group or commutator or ... is except what I
have deduced from reading mail from this group. I get the front face
corners exactly right, the back corners in the proper position, and then
the back corners rotated properly.
Finally the edges go where they belong one at a time, first on the
front and back and finally the four on the sides/top/bottom.
I have noticed a lack of discussion of cubes that have pictures on
them such that the entire cube can be right except that a single
center can be upside-down. I have also painted a 4x4x4 so that the
center 4 squares on each face have to be in the proper position. Every
time I solve this cube I have to rediscover how it is done. My fingers
refuse to learn it for me.
Behind the back? You gotta be kidding!
PAul
From MONET01@mizzou1.missouri.edu Tue Jun 23 13:31:40 1992
Return-Path: <MONET01@mizzou1.missouri.edu>
Received: from MIZZOU1.missouri.edu ([128.206.2.2]) by life.ai.mit.edu (4.1/AI-4.10) id AA17944; Tue, 23 Jun 92 13:31:40 EDT
Message-Id: <9206231731.AA17944@life.ai.mit.edu>
Received: from MIZZOU1 by MIZZOU1.missouri.edu (IBM VM SMTP V2R1)
with BSMTP id 4884; Tue, 23 Jun 92 12:31:20 CDT
Received: by MIZZOU1 (Mailer R2.08) id 2332; Tue, 23 Jun 92 12:31:19 CDT
Date: Tue, 23 Jun 92 12:21:27 CDT
From: MONET01@mizzou1.missouri.edu
To: cube-lovers@life.ai.mit.edu
Subject: Ultimate cube
The recent posting about cubes with photos has prompted me to post about my
favorite cube. I picked this one up around the end of the BIG cube craze and
have kept it in my desk every since. The cube looks like someone took a knife
to a normal solved cube and cut a diagonal 'x' through each face and folded the
flaps back down the sides. This leads to a cube where opposing centers have an
'x' that has four colors in a mirror image. (It is hard to describe, sorry.)
This cube has to be solved and then the centers oriented properly. The slick
thing about the cube is that part way through the solution (fairly early on),
you may have to swap top for bottom and start over. I like to fiddle with it
because at first glance it looks impossible to determine which cubelet is which
to a novice and to a semi-experienced cubist it is not as easy as it looks.
The cube was made by ULTRACO and is called ULTIMATE CUBE (copyrighted 1982).
Unfortunately, when I went back a few weeks later to buy a couple more cubes,
they were all gone and the sales people had no idea what I was talking about.
I think I got this cube at a Mall toy store. If anyone knows where I can get
a replacement, I would be interested as the printing on a few squares has faded
just like my youth.
From hoey@aic.nrl.navy.mil Wed Jun 24 15:45:56 1992
Return-Path: <hoey@aic.nrl.navy.mil>
Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA23541; Wed, 24 Jun 92 15:45:56 EDT
Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0)
id AA09989; Wed, 24 Jun 92 15:45:51 EDT
Date: Wed, 24 Jun 92 15:45:51 EDT
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Message-Id: <9206241945.AA09989@Sun0.AIC.NRL.Navy.Mil>
To: MONET01@mizzou1.missouri.edu, cube-lovers@life.ai.mit.edu
Subject: Re: Ultimate cube
MONET01@mizzou1.missouri.edu writes of a cube that
``looks like someone took a knife to a normal solved cube and cut
a diagonal 'x' through each face and folded the flaps back down
the sides. This leads to a cube where opposing centers have an
'x' that has four colors in a mirror image. (It is hard to
describe, sorry.)''
I would appreciate a few more details. I think the color scheme of
each face you describe is something like
+-----+-----+-----+
|.1111|11111|1111.|
|44.11|11111|11.22|
|4444.|11111|.2222|
+-----+-----+-----+
|44444|.111.|22222|
|44444|44.22|22222|
|44444|.333.|22222|
+-----+-----+-----+
|4444.|33333|.2222|
|44.33|33333|33.22|
|.3333|33333|3333.|
+-----+-----+-----+
where 1,2,3, and 4 are distinct colors, but there are still several
ways to make the colors on different faces match up. Look at a
corner, where the colors are
+-------+
/a.bbbbb/c\
/aaa.bbb/ccc\
/aaaaa.b/ccccc\
+-------+.......+
\fffff.e\ddddd/
\fff.eee\ddd/
\f.eeeee\d/
+-------+
That is, one corner is colored a/b, another c/d, and the third e/f,
where I expect some of a,b,c,d,e,f will be the same color.
One possibility was pictured in Hofstatder's Scientific American
article of February, 1981. It had b=c,d=e,f=a and used twelve colors.
Jim Saxe and I were impressed by its wasteful use of color and its
failure to exhibit edge orientation. From your remarks about turning
it over, I suspect this isn't what you mean.
You may be talking about the cube in which a=d,b=e,c=f which uses six
colors. I would say it is as if you cut an 'x' on a cube and
exchanged each triangle with the other triangle on the same edge of
the cube. That is a reasonably good coloring. It isn't really
necessary to solve it twice, though. To find out whether a given
corner goes on the top or bottom, look at the two colors that the
corner shares with the top face center. Either the corner will have
the two colors in the same order as the top, or they will be reversed,
and that determines whether that corner goes on the top or bottom.
That tells you where the third color on that corner goes, and the
last color is determined by elimination.
There is an even more interesting coloring that uses only four colors.
In this coloring a=c=e and the other three colors are distinct. Jim
Saxe and I came up with this coloring in our discussions of
Hofstatder's article. It isn't quite symmetric enough, since its
reflection is a coloring in which b=d=f, a slightly different pattern.
Our discussions then led to the Tartan coloring we talked about in our
article of 16 February 1981.
The only cube in the archives called the Ultimate Cube is the one that
has ``over 43 quintillion solutions.'' It has all six sides colored
the same.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil
From pbeck@pica.army.mil Fri Jun 26 13:39:29 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA04500; Fri, 26 Jun 92 13:39:29 EDT
Date: Fri, 26 Jun 92 13:36:42 EDT
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: cube-lovers@ai.mit.edu
Subject: ultimate cube
Message-Id: <9206261336.aa13556@COR4.PICA.ARMY.MIL>
name "ultimate cube" is a brand name or TM
my description of its coloring is:
still has six solid colors
if we visualize a standard cube as
a box with 6 stickers, 1 to each flat face
the the ultimate cube has these 6 stickers
translated and rotated so that
4 of these squares come together on a center
cubie with the diagonal of each square
laying along the edge of the cube.
From mouse@lightning.mcrcim.mcgill.edu Fri Jun 26 20:06:15 1992
Return-Path: <mouse@lightning.mcrcim.mcgill.edu>
Received: from Lightning.McRCIM.McGill.EDU by life.ai.mit.edu (4.1/AI-4.10) id AA17340; Fri, 26 Jun 92 20:06:15 EDT
Received: by Lightning.McRCIM.McGill.EDU (5.65)
id <9206270006.AA06976@Lightning.McRCIM.McGill.EDU>; Fri, 26 Jun 92 20:06:06 -0400
Date: Fri, 26 Jun 92 20:06:06 -0400
From: der Mouse <mouse@lightning.mcrcim.mcgill.edu>
Message-Id: <9206270006.AA06976@Lightning.McRCIM.McGill.EDU>
To: cube-lovers@ai.mit.edu
Subject: Re: ultimate cube
> my description of ["ultimate" cube's] coloring is:
> still has six solid colors
> if we visualize a standard cube as a box with 6 stickers, 1 to each
> flat face the the ultimate cube has these 6 stickers translated and
> rotated so that 4 of these squares come together on a center cubie
> with the diagonal of each square laying along the edge of the cube.
Hm, there are 12 edges on a cube. That leaves half of them unaccounted
for. What do they get? (Note that you also have to shrink the 6
stickers, beacuse the face diagonal is longer than an edge.)
der Mouse
mouse@larry.mcrcim.mcgill.edu
From pbeck@pica.army.mil Tue Jun 30 10:05:29 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA10327; Tue, 30 Jun 92 10:05:29 EDT
Received: by COR4.PICA.ARMY.MIL id aa22053; 30 Jun 92 7:56 EDT
Date: Tue, 30 Jun 92 7:46:56 EDT
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: cube-lovers@ai.mit.edu
Subject: ultimate cube, correction
Message-Id: <9206300746.aa20380@COR4.PICA.ARMY.MIL>
ULTIMATE CUBE - sorry for my past misinformation this time I will try
and describe it by observation.
The cube is covered with 24 (4 to a face) 45, 45, 90 deg triangles.
These triangles have there hypotenuse along the edge of the cube and
their 90 deg apex at the center of the center cubie.
Each opposite face has the same coloring except that the rotation of
the colors is opposite. For example if the front face has a green,
orange, yellow and red triangle in clockwise order then in order for
the rear face to correspond it has a color rotation that is counter
clockwise.
The coloring scheme is that the top and bottom triangles on the side
faces (ie, front, right, left, back) are the same. Six colors (
green, orange, yellow , red for the front and rear, and green,
white,yellow,blue for the sides and white, orange, blue, red for the
top and bottom) are used and there are 4 triangles of each color.
I think this is an accurate description, if there are questions please
ask I have my cube at my desk.
From tjj@rolf.helsinki.fi Tue Jun 30 15:36:42 1992
Return-Path: <tjj@rolf.helsinki.fi>
Received: from funet.fi by life.ai.mit.edu (4.1/AI-4.10) id AA22260; Tue, 30 Jun 92 15:36:42 EDT
Received: from rolf.Helsinki.FI by funet.fi with SMTP (PP)
id <03302-0@funet.fi>; Tue, 30 Jun 1992 22:35:13 +0300
Received: by rolf.helsinki.fi (5.57/Ultrix3.0-C) id AA10177;
Tue, 30 Jun 92 22:34:53 +0300
Date: Tue, 30 Jun 92 22:34:53 +0300
From: tjj@rolf.helsinki.fi (Timo Jokitalo)
Message-Id: <9206301934.AA10177@rolf.helsinki.fi>
To: cube-lovers@ai.mit.edu
Subject: Please, quick, I need the address of the new puzzle shop in Amsterdam
I believe it was with the Dutch Cubists' Club newletter that I got an
advertisement of a new puzzle shop in Amsterdam. I tried to find this
advertisement, not, but could not. I'm leaving Finland on Thursday
afternoon, and will be passing through Amsterdam, so I would be forever
grateful to any kind soul who would send me the address!!!
Thanks,
Timo (tjj@rolf.helsinki.fi)
From tjj@rolf.helsinki.fi Tue Jun 30 21:15:05 1992
Return-Path: <tjj@rolf.helsinki.fi>
Received: from funet.fi by life.ai.mit.edu (4.1/AI-4.10) id AA02935; Tue, 30 Jun 92 21:15:05 EDT
Received: from rolf.Helsinki.FI by funet.fi with SMTP (PP)
id <03311-0@funet.fi>; Tue, 30 Jun 1992 22:36:55 +0300
Received: by rolf.helsinki.fi (5.57/Ultrix3.0-C) id AA10181;
Tue, 30 Jun 92 22:36:37 +0300
Date: Tue, 30 Jun 92 22:36:37 +0300
From: tjj@rolf.helsinki.fi (Timo Jokitalo)
Message-Id: <9206301936.AA10181@rolf.helsinki.fi>
To: cube-lovers@ai.mit.edu
In the mail I just sent, there were a couple of serious typing errors, but
I think the gist of the message should be clear... sorry!
Timo
From news@cco.caltech.edu Thu Jul 9 19:28:24 1992
Return-Path: <news@cco.caltech.edu>
Received: from gap.cco.caltech.edu by life.ai.mit.edu (4.1/AI-4.10) id AA03141; Thu, 9 Jul 92 19:28:24 EDT
Received: by gap.cco.caltech.edu (4.1/1.34.1)
id AA17055; Thu, 9 Jul 92 13:17:25 PDT
Newsgroups: mlist.cube-lovers
Path: nntp-server.caltech.edu!ph
From: ph@vortex.ama.caltech.edu (Paul Hardy)
Subject: Re: Name query.
In-Reply-To: ACW@riverside.scrc.symbolics.com's message of Thu, 1 Jan 1970 00: 00:00 GMT
Message-Id: <PH.92Jul9131157@vortex.ama.caltech.edu>
Sender: news@cco.caltech.edu
Nntp-Posting-Host: ama.caltech.edu
Organization: California Institute of Technology
References: <9206112052.AA18593@strident.think.com>
<19920611213942.5.ACW@PALLANDO.SCRC.Symbolics.COM>
Distribution: mlist
Date: Thu, 9 Jul 1992 21:11:57 GMT
Apparently-To: mlist-cube-lovers@nntp-server.caltech.edu
In article <19920611213942.5.ACW@PALLANDO.SCRC.Symbolics.COM> ACW@riverside.scrc.symbolics.com (Allan C. Wechsler) writes:
> While I'm reminiscing, I should confess that my standard corner operator
> is still the same as it was then: (FUR)^5, which exchanges two corners,
> leaves the rest of the corners alone, and fucks the edges completely.
> (Prudes, do not hassle me. This has been a technical term in cubing
> around MIT since The Beginning.) Because of this property of "furry
> five", I have to home and orient all the corners first, before I touch
> the edges. It's the kind of quirky algorithm you don't see among
> younger cubers, because everybody these days learns how to solve the
> thing from a book. In the Beginning, there were no books, and I proudly
> state that I solved the cube from scratch, by brainpower. Later I
> discovered that there were easier ways to do things than (FR)^105! I
> had pages and pages covered with little cube diagrams with arrows
> showing how the stickers were permuted by a particular sequence.
>
> I'm interested in hearing other reminiscences from people who actually
> solved the cube -- you're disqualified if you learned how to solve it
> from somebody else, or from a book.
I also solved the cube alone at first. I solved the top and middle first,
then spent some time pondering the final face. I realized that manipulating
the corners was trickier than the edges because there were three faces rather
than two, so I solved the bottom corners and then got the bottom edges in
place.
I eventually got Singmaster's book, and found that my method of solving two
layers was faster than his. I don't quite remember now, but I think it was
because I had found a quick method for flipping a piece on the middle edge
around if necessary (i.e., if it was in the correct position but flipped the
wrong way) without disturbing anything else on the top or middle of the cube.
Still, Singmaster's book had many patterns that were fun to go through and
see evolve.
I've long since lost my copy of Singmaster's book (one move too many); is it
still available?
--Paul
--
This is my address: ph@ama.caltech.edu
This is UUCP: ...!{decwrl,uunet}!
This is my address on UUCP: ...!{decwrl,uunet}!caltech.edu!ama!ph
Any questions?
"Does Emacs have the Buddha nature?" --Paul Hardy "Yow!" --Zippy
From alan@ai.mit.edu Sun Jul 19 10:51:55 1992
Return-Path: <alan@ai.mit.edu>
Received: from august (august.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA08560; Sun, 19 Jul 92 10:51:55 EDT
Received: by august (4.1/AI-4.10) id AA14998; Sun, 19 Jul 92 10:52:45 EDT
Date: Sun, 19 Jul 92 10:52:45 EDT
Message-Id: <9207191452.AA14998@august>
From: Alan Bawden <Alan@lcs.mit.edu>
Sender: Alan@lcs.mit.edu
To: Cube-Lovers
Subject: [nick@lcs.mit.edu: In-sol-u-bil-i-tyyyy!]
Date: Thu, 16 Jul 92 13:11 EDT
From: nick@lcs.mit.edu
Reply-To: nick@lcs.mit.edu
Subject: In-sol-u-bil-i-tyyyy!
To: qotd@ghoti.lcs.mit.edu
In a wonderful article about Claude Shannon in the April 92 IEEE
Spectrum, a few lines from his poem called "a Rubric on Rubik's Cubics"
(to the tune of Ta-ra-ra-boom-de-ay):
Respect your cube and keep it clean,
Lube your cube with Vaseline.
Beware the dreaded cubist's thumb,
the calloused hands and fingers numb.
No borrower nor lender be,
Rude folk might switch two tabs on thee.
The most unkindest switch of all,
Into Insolubility.
[Chorus]
In-sol-u-bil-i-ty!
The strangest place to be
However you persist
Solutions don't exist!
From @mail.uunet.ca:mark.longridge@canrem.com Mon Aug 3 02:57:03 1992
Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com>
Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA13101; Mon, 3 Aug 92 02:57:03 EDT
Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <9679>; Mon, 3 Aug 1992 02:46:15 -0400
Received: from canrem.com by unixbox.canrem.COM
id aa15064; Mon, 3 Aug 92 2:39:20 EDT
Received: by canrem.com (PCB/Usenet Gateway)
Path-id <19923.104.88888@dosgate>; 3 Aug 92 (02:30)
Message-Id: <19923.104.88888@dosgate>
From: Mark Longridge <mark.longridge@canrem.com>
Date: Sun, 2 Aug 1992 20:00:00 -0400
To: cube-lovers@ai.mit.edu
Subject: cube theory
I've been doing some research to try and figure out some things
about the cube. I've also tried (unsuccessfully) to develop a sort
of CRC or checksum for the cube. With this cube "signature" I could
then find out which depth each pattern requires. I'm puzzled how
some computer types managed to find God's Algorithm for the 3x3x3
squares group. How do you keep track of all the patterns without
repeating yourself? If it is by holding all patterns in an array
the array must become huge.
Maximum Depth (using q and h moves)
-------------
2x2x2 sq group 4 (24 total states)
Pyraminx 11 (or 14 including the 3 tips)
2x2x2 11 (14 using q turns only)
3x3x3 corners only 11
3x3x3 sq group 15 (half turns only, don't know if using q improves
this)
3x3x2 domino 18 (for 1 solution)
A local maxima is a state where any possible move will bring you closer
to a solution. This can occur on the 2x2x2 at depth 4 and on the 3x3x3
at
depth 3. Note that all possible patterns at maximum depth are local
maxima,
however it is surprising that local maxima may occur in patterns much
closer to the surface.
To date, no work has been done to determine the depth of the
dodecahedron
(megaminx) or square 1.
Some questions:
What pattern is an example of local maxima? e.g. 3x3x3 at depth 3
-> 12-flip, 12-flip 8-twist
q+h Depth Patterns
2x2x2 1 9
3x3x3 1 18
Dodecahedron 1 48
Analysis of the full cube group
-------------------------------
Moves Deep arrangements (q+h) arrangements (q only) *
0 1 1
1 18 12
2 243 114
3 3,240 1,068
4 > 48,600 10,011
* Work by Zoltan Kaufmann
Notes: At 1 move deep each of the 6 sides can turn 3 ways (+ - 2)
giving 18 distinct patterns
At 2 moves deep it is redundant to turn the same side again
so 5 sides can turn 3 ways so 18x15=270
However, with opposite turns order is not significant,
e.g. T,D = D,T F,B = B,F L,R = R,L
since each of these can occur in 9 different ways there are 27
redundancies so 270 - 27 = 243
At 3 moves deep with the first 2 moves on opposite faces don't
turn the face used in move one since:
T,D,T = T2,D F,B,F = F2,B L,R,L = L2,R
This can occur in 3x3x3=27 ways for each case so 81 are dropped
(Remember the first 2 moves have already been weeded of
redundancy!)
Also when the 2nd and 3rd moves are of opposite faces
e.g. T,R,L = T,L,R B,R,L = B,L,R F,R,L = F,L,R D,R,L = D,L,R
T,B,F = T,F,B D,B,F = D,F,B L,B,F = L,F,B R,B,F = R,F,B
F,T,D = F,D,T B,T,D = B,D,T L,T,D = L,D,T R,T,D = R,D,T
since each of these can occur 27 different ways in each of the
cases this gives 27x12 = 324 redundancies
Thus 243x15 = 3645, removing the redundancies gives 3645-81-324=3240
At 4 moves deep.... still working on this one! Zoltan Kaufmann
has done 4 moves deep using quarter turns, but has anyone
calculated farther using q and h turns? I'd be interested in
the source code of any programs people have written on finding
path-lengths. Also what is an example of a local maxima close
to the surface, e.g. 4 moves. I believe Jim Saxe and Dan Hoey
have done some work in this regard.
One more question: What is the maximum number of moves required
if you do the 3x3x3 one face last? The best results I've seen
are 19 q and h moves.
-> Mark Longridge
--
Canada Remote Systems - Toronto, Ontario/Detroit, MI
World's Largest PCBOARD System - 416-629-7000/629-7044
From hoey@aic.nrl.navy.mil Mon Aug 3 11:10:35 1992
Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA00431; Mon, 3 Aug 92 11:10:35 EDT
Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0)
id AA22147; Mon, 3 Aug 92 11:10:30 EDT
Return-Path: <hoey@aic.nrl.navy.mil>
Received: by sun13.aic.nrl.navy.mil; Mon, 3 Aug 92 11:10:29 EDT
Date: Mon, 3 Aug 92 11:10:29 EDT
From: hoey@aic.nrl.navy.mil
Message-Id: <9208031510.AA03178@sun13.aic.nrl.navy.mil>
O: Mark Longridge <mark.longridge@canrem.com>
Cc: Cube-lovers@life.ai.mit.edu
Subject: Re: cube theory
Dear Mark,
I sent you email on 8 June. Did you receive it? Are you interested
in acquiring a Tartan cube? Did you ever get Minh Thai's book on
Rubik's Revenge?
With respect to the 3x3x3 squares group,
> How do you keep track of all the patterns without repeating yourself?
there are several ways. The most general is to use the Sims table
(aka the FHL table) for the subgroup, which gives a mixed-base
enumeration of the positions. See my message of 1 February 1981.
> A local maxima is a state where any possible move will bring you
> closer to a solution.
That's ``local maximum''. ``Maxima'' is the plural of ``maximum''.
> Note that all possible patterns at maximum depth are local
> maxima, ....
We call such positions ``global maxima'' because they are at the
overall maximum depth. The statement is then that every global
maximum is a local maximum.
> To date, no work has been done to determine the depth of the
> dodecahedron (megaminx)....
Well, I've done some looking at it. Since my initial remarks on 23
September 1982, I've figured out a way to generate a recurrence for
it, but it seems I haven't put it down anywhere. Are you interested?
(Do I have to tell anyone to answer that question only to
Hoey@AIC.NRL.Navy.Mil, not the list?)
> What pattern is an example of local maxima? e.g. 3x3x3 at depth 3
> -> 12-flip, 12-flip 8-twist
Jim Saxe and I listed the 25 symmetric local maxima in our message on
Symmetry and Local Maxima, dated 14 December 1980. We verified Jim's
conjecture that the four-spot is a local maximum, but not on the
grounds of symmetry, and reported that on 22 March 1981. Do you have
access to the cube-lovers archives?
> Moves Deep arrangements (q+h) arrangements (q only) *
> 0 1 1
> 1 18 12
> 2 243 114
> 3 3,240 1,068
> 4 > 48,600 10,011
This is in the archives, too
5 93,840 (22 March 1981)
6 878,880 (14 August 1981)
7 8,221,632 (7 December 1981)
David C. Plummer and I had hoped to use his program (which counted the 7
QT positions) to extend this to 8 QT, but we got busy. I still have
hopes....
> At 2 moves deep it is redundant to turn the same side again....
> However, with opposite turns order is not significant, e.g. T,D = D,T....
This approach appeared on 9 January 1981. It showed how to follow the
argument to 25 QT, and to get what are still the best known lower bounds
for the ordinary cube and for the supergroup.
> Also what is an example of a local maxima close to the surface, e.g.
> 4 moves. I believe Jim Saxe and Dan Hoey have done some work in this
> regard.
It's known there are no local maxima at 7 QT or less. The shortest
known local maxima are Pons Asinorum and the four-spot, both at 12 QT.
I know of no results between 8 and 11 QT.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil
From pbeck@pica.army.mil Mon Aug 10 14:13:31 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA14995; Mon, 10 Aug 92 14:13:31 EDT
Date: Mon, 10 Aug 92 14:11:51 EDT
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: CUBE-LOVERS@life.ai.mit.edu
Subject: smart alex - new puzzle
Message-Id: <9208101411.aa10693@COR4.PICA.ARMY.MIL>
SMART ALEX about $14 retail
from 2 MCH FUN
777-108th Avenue N.E. #2340
Bellevue, WA 98004
206-453-5659
purchase source
PUZZLETTS
MIKE GREEN
24843 144th Place S.E.
KENT, WA 98042
206-630-1432
or
from myself
DESCRIPTION: This puzzle is similar to the Hungarian puzzle UFO but a
little more complex. The puzzle has a cube at its center which is
cut in half and these two pieces can rotate with respect to each
other. In the plane of this cut there are four hubs that can rotate,
2 on the x-axis and 2 on the y-axis. These hubs have a hexagonal
cross section and are divided into six equilateral pie shaped wedges.
By rotating the hubs and then by rotating the center cube 3 hub
pieces on each axle are moved at a time.
The hub pieces have 2 colors each one perpendicular to the axis and
the other on its edge. There are 2 wedges with the same coloring.
The center cube is also colored. The object is to arrange the hub
pieces so that the edge pieces align with the center cube and that the
perpendicular sides are solid colored and aligned.
From pbeck@pica.army.mil Mon Aug 10 16:22:55 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA19839; Mon, 10 Aug 92 16:22:55 EDT
Date: Mon, 10 Aug 92 14:12:42 EDT
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: CUBE-LOVERS@life.ai.mit.edu
Subject: mazeland
Message-Id: <9208101412.aa10952@COR4.PICA.ARMY.MIL>
MAZELAND GARDENS and DISCOVERY CENTER
POB 573
Alexandria Bay, NY 13607
315-482-LOST
800-585-FUNN
Alexandria Bay is in the Thousand Islands region near where I-81
crosses into Canada.
Mazeland Gardens is an entertainment center (the property was formerly
a miniature golf course) of mazes. It has 2 mazes that are
constructed with arborvitae hedges, one a 1/2 acre in size and the
other a full acre. In addition it has 2 mazes that are constructed
with stakes and colored tape to mark the walls.
I went without kids and had fun. The families with kids looked like
they were having a good time. If you are in the area you might want
to give it a try.
PS The fellow working the desk said that there is are 2 other mazes
on the east coast that he is aware of:
one is in North Carolina somewhere and the other is in Daytona beach
Florida/
Pete Beck
From hoey@aic.nrl.navy.mil Thu Aug 20 13:51:38 1992
Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA16858; Thu, 20 Aug 92 13:51:38 EDT
Received: from sun30.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0)
id AA19935; Thu, 20 Aug 92 13:51:25 EDT
Return-Path: <hoey@aic.nrl.navy.mil>
Received: by sun30.aic.nrl.navy.mil; Thu, 20 Aug 92 13:51:25 EDT
Date: Thu, 20 Aug 92 13:51:25 EDT
From: hoey@aic.nrl.navy.mil
Message-Id: <9208201751.AA14111@sun30.aic.nrl.navy.mil>
To: Allan C Wechsler <ACW@riverside.scrc.symbolics.com>,
wft@math.canterbury.ac.nz (Bill Taylor)
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: subgroups
On 14 Jan 1992, Allan C. Wechsler posted
>Regarding the meta-approach of descending through a series of subgroups,
>how much leverage does properly selecting the chain give you? It seems
>like the jump from <F2,B2,...> to <F2,B2,L2,R2,...> is pretty large.
>There are probably other paths through the subgroup lattice.
>Does anyone have a table of subgroups?
As far as selecting the chain goes, I have been meaning to look into
that a bit. Of course, since Bill posted, the results of Hans
Kloosterman, Michael Reid, and Dik Winter have shown that you indeed
get a lot of leverage. I would like to get some idea of the possible
group towers, for a more general approach to selecting which towers
give you leverage. But what I haven't been able to figure out is how
to figure out which coset of G1 wrt G2 you're in. I've been able to
figure it out for specific groups, but if we wanted to do this for a
lot of chains, we would need to do coset identification given G1 and
G2 as a table of strong generators. We could in fact ensure that the
strong generators of G1 form a subset of those of G2. Is that a hard
thing to do?
More to the point, I've heard that the FHL algorithm should more
properly be called Sims's algorithm and that Furst, Hopcroft, and Luks
mostly analyzed the performance. I haven't read anything by Sims on
it, though. Is there a good reference that treats this sort of
algorithm in a more general setting? I have toyed with implementing
the Jerrum improvements to FHL, but it is a mighty complicated beast.
Also, a talk announced in the archives mentioned 1987 work by Akos
Seress that was supposed to be an improvement, but I don't know
whether it got published. Anyway, if not, do you know if there is a
good general way of finding out which coset a given position is in.
On 29 Jan 1992, wft@math.canterbury.ac.nz (Bill Taylor) posted
> There hasn't been any response to this, seemingly, which is a pity.
For some reason, I never saw Bill's message. I just noticed it when
comparing my files against the archives. [ Archives seekers note: the
archives have moved to FTP.LCS.MIT.EDU (18.26.0.36), still in
directory /pub/cube-lovers. ]
> In any event, I would like to know of any other well-known subgroups.
> There are the slice group; double-slice group; U group; square group;
> anti-slice group. How many others are there not mentioned here, that
> people know of ?
There were some tables in Singmaster with more examples, and there are
the stuck-faces groups that I wrote about on 21 July 1981. I seem to
recall there was some non-obvious equivalence between two groups,
perhaps the slice group and the antislice group. But a general list
of popular subgroups would be interesting. Of course a list of *all*
the subgroups would have, um, over three beelion of them. I suspect
it has more than 4.3x10^19. Does anyone know a good way of counting
how many subgroups there are? Or even a way of estimating the number?
By comparison, the symmetries of the cube form a 48-element group with
98 subgroups.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil
From ACW@riverside.scrc.symbolics.com Thu Aug 20 16:24:45 1992
Return-Path: <ACW@riverside.scrc.symbolics.com>
Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA22502; Thu, 20 Aug 92 16:24:45 EDT
Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 888810; 20 Aug 1992 16:25:53-0400
Date: Thu, 20 Aug 1992 16:25-0400
From: Allan C. Wechsler <ACW@riverside.scrc.symbolics.com>
Subject: Re: subgroups
To: hoey@aic.nrl.navy.mil, ACW@riverside.scrc.symbolics.com,
wft@math.canterbury.ac.nz
Cc: Cube-Lovers@ai.mit.edu
In-Reply-To: <9208201751.AA14111@sun30.aic.nrl.navy.mil>
Message-Id: <19920820202540.7.ACW@PALLANDO.SCRC.Symbolics.COM>
Date: Thu, 20 Aug 1992 13:51 EDT
From: hoey@aic.nrl.navy.mil
[...]
Of course a list of *all*
the subgroups would have, um, over three beelion of them. I suspect
it has more than 4.3x10^19. Does anyone know a good way of counting
how many subgroups there are? Or even a way of estimating the number?
By comparison, the symmetries of the cube form a 48-element group with
98 subgroups.
All we should really be interested in are conjugate classes of
subgroups. I think.
From alan@ai.mit.edu Thu Aug 20 20:03:23 1992
Return-Path: <alan@ai.mit.edu>
Received: from transit (transit.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA29289; Thu, 20 Aug 92 20:03:23 EDT
Received: by transit (4.1/AI-4.10) id AA15643; Thu, 20 Aug 92 20:06:49 EDT
Date: Thu, 20 Aug 92 20:06:49 EDT
Message-Id: <9208210006.AA15643@transit>
From: Alan Bawden <Cube-Lovers-Request@ai.mit.edu>
Sender: Cube-Lovers-Request@ai.mit.edu
To: Cube-Lovers
Subject: Archives
Date: Thu, 20 Aug 92 13:51:25 EDT
From: hoey@aic.nrl.navy.mil
... [ Archives seekers note: the archives have moved to FTP.LCS.MIT.EDU
(18.26.0.36), still in directory /pub/cube-lovers. ] ...
No, that isn't right. The correct address is FTP.AI.MIT.EDU (which is at
128.52.32.11 -- at least this week). Here is the text I currently send to
people who are new subscribers or who express interest in the archives:
Using FTP, connect to FTP.AI.MIT.EDU, login as "anonymous" (any password),
and in the directory "pub/cube-lovers" you will find the nine (compressed)
files "cube-mail-0.Z" through "cube-mail-8.Z".
Archive vital statistics (when uncompressed):
File From To Size (bytes)
---- ---- -- ------------
cube-mail-0 12 Jul 80 23 Oct 80 185037
cube-mail-1 3 Nov 80 9 Jan 81 135719
cube-mail-2 10 Jan 81 3 Aug 81 138566
cube-mail-3 3 Aug 81 3 May 82 137753
cube-mail-4 4 May 81 11 Dec 82 139660
cube-mail-5 11 Dec 82 6 Jan 87 173364
cube-mail-6 10 Jan 87 13 Apr 90 216733
cube-mail-7 12 Oct 90 9 Sep 91 137508
cube-mail-8 1 Nov 91 25 May 92 171205
In addition, the file "recent-mail" contains a copy of the currently active
section of the archive. (Unfortunately, due to the way mail works here at
the AI Lab, it is not possible to have new mail accumulate directly into
this file, so there may be some delay before a new message arrives here.)
- Alan
From reid@math.berkeley.edu Thu Aug 20 20:10:13 1992
Return-Path: <reid@math.berkeley.edu>
Received: from math.berkeley.edu by life.ai.mit.edu (4.1/AI-4.10) id AA29497; Thu, 20 Aug 92 20:10:13 EDT
Received: from phnom-penh.berkeley.edu.berkeley.edu by math.berkeley.edu (4.1/1.33(math))
id AA19361; Thu, 20 Aug 92 17:03:31 PDT
Date: Thu, 20 Aug 92 17:03:31 PDT
From: reid@math.berkeley.edu (michael reid)
Message-Id: <9208210003.AA19361@math.berkeley.edu>
To: ACW@riverside.scrc.symbolics.com, hoey@aic.nrl.navy.mil,
wft@math.canterbury.ac.nz
Subject: Re: subgroups
Cc: Cube-Lovers@ai.mit.edu
dan hoey writes:
> On 14 Jan 1992, Allan C. Wechsler posted
> >Regarding the meta-approach of descending through a series of subgroups,
> >how much leverage does properly selecting the chain give you? It seems
> >like the jump from <F2,B2,...> to <F2,B2,L2,R2,...> is pretty large.
> >There are probably other paths through the subgroup lattice.
> >Does anyone have a table of subgroups?
well, i don't know ALL the subgroups, but i did some investigation
before devising my three stage algorithm. one of the great advantages
of thistlethwaite's four stage method is that since each subgroup
restricts the motion of various faces, it is routine to exhaustively
search the cosets spaces at each stage, since we only make twists that
leave us in the given space. so i looked at all possible ways of
restricting various faces, up to symmetry. there are three possible
restrictions for a face: no restriction, half turns only, no turns.
our problem is then coloring the faces of the cube with 3 colors, up
to symmetry (rigid and non-rigid). the polya polynomial for the faces
of the cube under this group of symmetries is:
( x^6 + 3 x^5 + 9 x^4 + 13 x^3 + 14 x^2 + 8 x ) / 48
so there are 56 different ways to three-color the faces. i spent the
better part of an evening and most of the night calculating (by hand)
the orders of these subgroups. shortly thereafter, i saw an announcement
for the group theory package GAP, which specifically mentions calculating
the order of the rubik's cube group. so i used the package to verify my
answers. here's the list (i don't see a canonical way of ordering them):
1. |<>| = 1
2. |<U2>| = 2 = 2
3. |<U>| = 2^2 = 4
4. |<U2, D2>| = 2^2 = 4
5. |<U, D2>| = 2^3 = 8
6. |<U, D>| = 2^4 = 16
7. |<U2, R2>| = 2 3 = 12
8. |<U, R2>| = 2^6 3^2 5^2 = 14400
9. |<U, R>| = 2^6 3^8 5^2 7 = 73483200
10. |<U2, R2, L2>| = 2^5 3 = 96
11. |<U, R2, L2>| = 2^12 3^4 5^2 7 = 58060800
12. |<U2, R, L2>| = 2^12 3^4 5^2 7 = 58060800
13. |<U2, R, L>| = 2^14 3^4 5^2 7^2 = 1625702400
14. |<U, R, L2>| = 2^14 3^11 5^2 7^2 = 3555411148800
15. |<U, R, L>| = 2^14 3^13 5^3 7^2 = 159993501696000
16. |<U2, R2, F2>| = 2^5 3^4 = 2592
17. |<U, R2, F2>| = 2^8 3^5 5^2 7 = 10886400
18. |<U, R, F2>| = 2^10 3^12 5^2 7^2 = 666639590400
19. |<U, R, F>| = 2^18 3^12 5^2 7^2 = 170659735142400
20. |<U2, R2, L2, D2>| = 2^6 3 = 192
21. |<U, R2, L2, D2>| = 2^13 3^4 5^2 7 = 116121600
22. |<U, R2, L2, D>| = 2^15 3^4 5^2 7^2 = 3251404800
23. |<U, R, L2, D2>| = 2^15 3^11 5^2 7^2 = 7110822297600
24. |<U, R, L, D2>| = 2^15 3^13 5^3 7^2 = 319987003392000
25. |<U, R, L, D>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
26. |<U2, R2, L2, F2>| = 2^11 3^4 = 165888
27. |<U, R2, L2, F2>| = 2^13 3^5 5^2 7^2 = 2438553600
28. |<U2, R, L2, F2>| = 2^14 3^5 5^2 7^2 = 4877107200
29. |<U2, R, L, F2>| = 2^14 3^5 5^2 7^2 = 4877107200
30. |<U, R2, L2, F>| = 2^14 3^13 5^3 7^2 11 = 1759928518656000
31. |<U2, R2, L, F>| = 2^14 3^13 5^3 7^2 11 = 1759928518656000
32. |<U2, R, L, F>| = 2^14 3^13 5^3 7^2 11 = 1759928518656000
33. |<U, R, L2, F>| = 2^24 3^13 5^3 7^2 11 = 1802166803103744000
34. |<U, R, L, F>| = 2^24 3^13 5^3 7^2 11 = 1802166803103744000
35. |<U2, F2, R2, L2, B2>| = 2^13 3^4 = 663552
36. |<U, F2, R2, L2, B2>| = 2^16 3^5 5^2 7^2 = 19508428800
37. |<U2, F, R2, L2, B2>| = 2^16 3^5 5^2 7^2 = 19508428800
38. |<U2, F2, R, L, B2>| = 2^16 3^5 5^2 7^2 = 19508428800
39. |<U2, F, R, L2, B2>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
40. |<U, F, R2, L2, B2>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
41. |<U2, F, R, L, B2>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
42. |<U, F2, R, L, B2>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
43. |<U, F, R, L2, B2>| = 2^27 3^14 5^3 7^2 11 = 43252003274489856000
44. |<U2, F, R, L, B>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
45. |<U, F2, R, L, B>| = 2^27 3^14 5^3 7^2 11 = 43252003274489856000
46. |<U, F, R, L, B>| = 2^27 3^14 5^3 7^2 11 = 43252003274489856000
47. |<U2, F2, R2, L2, B2, D2>| = 2^13 3^4 = 663552
48. |<U, F2, R2, L2, B2, D2>| = 2^16 3^5 5^2 7^2 = 19508428800
49. |<U, F2, R2, L2, B2, D>| = 2^16 3^5 5^2 7^2 = 19508428800
50. |<U, F, R2, L2, B2, D2>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
51. |<U, F, R2, L2, B, D2>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
52. |<U, F, R, L2, B2, D2>| = 2^27 3^14 5^3 7^2 11 = 43252003274489856000
53. |<U, F, R2, L2, B, D>| = 2^16 3^14 5^3 7^2 11 = 21119142223872000
54. |<U, F, R, L, B2, D2>| = 2^27 3^14 5^3 7^2 11 = 43252003274489856000
55. |<U, F, R, L, B, D2>| = 2^27 3^14 5^3 7^2 11 = 43252003274489856000
56. |<U, F, R, L, B, D>| = 2^27 3^14 5^3 7^2 11 = 43252003274489856000
subgroups with the same order are equal (possibly after necessary rotation
of the cube) with the following exceptions: (3, 4), (11, 12), (30, 31) and
(30, 32). equality of various pairs of subgroups can be obtained from the
three maneuvers:
R L F2 R2 F B L F2 B2 R2 F2 B2 L F B3 R3 L3 ~ U2 ,
so that <L, F, R, B> = <L, F, R, B, U2>,
F2 U2 L2 F2 R2 U2 F2 R F2 U2 R2 F2 L2 U2 F2 ~ L ,
so that <U2, F2, R, L2> = <U2, F2, R, L> and
R2 F2 B2 L2 U2 L2 F2 B2 R2 ~ D2 ,
so that <U2, F2, R2, L2, B2> = <U2, F2, R2, L2, B2, D2>.
thistlethwaite's filtration is 56 --> 53 --> 49 --> 47 --> 1.
kloosterman replaced 47 by a subgroup not on this list (one not obtained
by restricting face turns). call this 56 --> 53 --> 49 --> kl --> 1.
(in his final stage, kloosterman allows all twists available in the
subgroup 49.) my filtration is 56 --> 19 --> 17 --> 1 , which was
chosen precisely because it had the smallest size of the largest coset
space amongst all three stage filtrations with subgroups from the above.
winter's filtration is 56 --> 49 --> kl --> 1. it may be the case that
this can be improved by replacing kl with 17 , and allowing all face
turns available in the subgroup 49. i haven't had the time to look into
this yet.
using subgroups on the list above, we see that the only reasonable two
stage filtrations are:
56 --> 29 --> 1 with coset sizes 8868372480 and 4877107200
56 --> 22 --> 1 with coset sizes 13302558720 and 3251404800
56 --> 27 --> 1 with coset sizes 17736744960 and 2438553600
56 --> 49 --> 1 with coset sizes 2217093120 and 19508428800
56 --> 13 --> 1 with coset sizes 26605117440 and 1625702400
of these, the best seems to be 56 --> 49 --> 1 , since it has the most
symmetries (16). the number of symmetries the others have is 4 (for 29),
8 (for 22), 2 (for 27) and 2 (for 13). furthermore, aside from subgroup
49, the other intermediate groups seem to have too much restriction to
be efficient. also, of course, dik winter has already calculated that
the stage 56 --> 49 can always be accomplished in 12 face turns.
> On 29 Jan 1992, wft@math.canterbury.ac.nz (Bill Taylor) posted
> > There hasn't been any response to this, seemingly, which is a pity.
> For some reason, I never saw Bill's message. I just noticed it when
> comparing my files against the archives. [ Archives seekers note: the
> archives have moved to FTP.LCS.MIT.EDU (18.26.0.36), still in
> directory /pub/cube-lovers. ]
i also seem to have missed both allen's post as well as bill's reply.
perhaps 'twas the twilight zone between the start of my subscription to
cube-lovers and the time it takes recent messages to reach the archives.
however, i don't find the archives on ftp.lcs , but rather on
ftp.ai.mit.edu. also i see we've spawned cube-mail-8.Z.
> > In any event, I would like to know of any other well-known subgroups.
> > There are the slice group; double-slice group; U group; square group;
> > anti-slice group. How many others are there not mentioned here, that
> > people know of ?
in addition to those listed above there are subgroups generated by
combinations of face turns and slice turns, e.g. <U, M_R>, <U2, M_R>,
<U, R, M_R>, etc. i haven't looked at these at all. there's a lot of
work to be done here.
mike
From ronnie@cisco.com Thu Aug 20 23:32:46 1992
Return-Path: <ronnie@cisco.com>
Received: from ale.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA04659; Thu, 20 Aug 92 23:32:46 EDT
Received: by ale.cisco.com; Thu, 20 Aug 92 17:52:40 -0700
Date: Thu, 20 Aug 92 17:52:40 -0700
From: Ronnie B. Kon <ronnie@cisco.com>
Message-Id: <9208210052.AA00306@ale.cisco.com>
To: Cube-Lovers@life.ai.mit.edu
Subject: Back to the 16th century
In the Cube Lovers' archives we have files:
From To Size Time Bytes/Month
---- -- ---- ------ -----------
12 Jul 80 23 Oct 80 185037 3 months 61679
3 Nov 80 9 Jan 81 135719 2 months 67860
10 Jan 81 3 Aug 81 138566 6 months 23094
3 Aug 81 3 May 82 137753 9 months 15306
4 May 81 11 Dec 82 139660 19 months 7351
11 Dec 82 6 Jan 87 173364 48 months 3612
10 Jan 87 13 Apr 90 216733 39 months 5557
12 Oct 90 9 Sep 91 137508 12 months 11459
1 Nov 91 25 May 92 171205 7 months 24458
Ladies and Gentlemen, I believe we are witnessing a Renaissance of Cubing!
Ronnie
From @mail.uunet.ca:mark.longridge@canrem.com Tue Sep 1 22:09:24 1992
Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com>
Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA14825; Tue, 1 Sep 92 22:09:24 EDT
Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10871>; Tue, 1 Sep 1992 22:09:02 -0400
Received: from canrem.com by unixbox.canrem.COM
id aa23354; Tue, 1 Sep 92 20:43:35 EDT
Received: by canrem.com (PCB/Usenet Gateway)
Path-id <1992Sep1.104.99639@dosgate>; 1 Sep 92 (19:39)
Message-Id: <1992Sep1.104.99639@dosgate>
From: Mark Longridge <mark.longridge@canrem.com>
Date: Mon, 31 Aug 1992 20:00:00 -0400
To: cube-lovers@ai.mit.edu
Subject: rare variants
Hello fellow cube lovers...
Although this is not a strict Rubik's cube type post I think it should
be of some interest to the other subscribers.
I collect cube variants, and although the variations in cube colours are
interesting my great interest is variants in mechanisms (which require
different solving techniques).
I've been corresponding with other cube buffs around the world in an
effort to record ALL the significant cube variants and I use the
following classification system:
M = Manufactured in quantity, readily available
S = Produced in small quantities only
R = Rare, a few prototypes exist, difficult to get
P = Prototype, the inventor has the only one!
C = Exists only as a computer simulation and/or cardboard mockup
I = Intellectual idea only (perhaps on pencil and paper)
In my opinion, Square 1 is the most interesting cube variant in recent
years, and it gave me the most trouble! Here are some of the tough ones
to get, and if anyone knows of any others email me and I'll maintain a
list of them...
Trajber's Octahedron (R) Evidently Greg Stevens owns one
Octahedral puzzle with rotating faces
Extended Missing Link (S)
Missing Link with 6 tiers
Master Pyraminx (P??)
Looks like a normal pyraminx BUT it's edges can turn (just the
strip) 180 degrees and 2 vertices can be swapped
Space Grenade (P??)
Other weird one from Uwe Meffert. Mike Green of Puzzletts showed
me a picture of this, still not sure how it moves
Pyraminx Disc Chess (S)
Planar puzzle with 6 rotating discs, similar to Raba's Rotoscope
Masterball (S)
This seems to be a recent one, it's like a VIP sphere but it has
8 vertical cuts instead of one (like a tangerine) and 4
hortizontal sections
Halpern's Tetrahedron (P) Also called Pyraminx Tetrahedron
Like a pyraminx BUT it has face centres which are small
triangles
and it's faces rotate. Very rare.
Pyraminx Hexagon (C) Jerry Slocum says he got a cardboard mockup
of this from Meffert. I wrote a computer simulation of it.
Imagine a Rubik's cube with an N-prism shape, thus the top and bottom
are hexagons, and there are 6 (rather than 4) adjacent sides.
The top and bottom can rotate 60 degrees and the adjacent sides
can only rotate 180 degrees.
Twist Torus (I) My own concept. Imagine a torus segmented 4 ways
length-wise so it can slide around. Additionally there are
12 rings around the circumference which can rotate at right
angles to the segments. Still thinking of a good colour
arrangement for this one.
Super Skewb (I) Another idea of mine. It's a skewb and a 2x2x2
cube! A compound of two mechanisms.
Honourable mention: Pyraminx Star, Puck, Ufo, Trick Haus
(Anyone got multiples of these)
Imagine my disappointment when I found out the Mach Ball, Skewb and
Moody Ball all have the same basic mechanism! Anyway if anyone has
a rare variant or puzzle idea please post here or email me...
Mark Longridge
259 Thornton Rd N
Oshawa Ontario Canada L1J 6T2
Email: mark.longridge@canrem.com
--
Canada Remote Systems - Toronto, Ontario/Detroit, MI
World's Largest PCBOARD System - 416-629-7000/629-7044
From diamond@jit081.enet.dec.com Tue Sep 1 23:58:01 1992
Return-Path: <diamond@jit081.enet.dec.com>
Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA19112; Tue, 1 Sep 92 23:58:01 EDT
Received: by enet-gw.pa.dec.com; id AA21950; Tue, 1 Sep 92 20:57:51 -0700
Message-Id: <9209020357.AA21950@enet-gw.pa.dec.com>
Received: from jit081.enet; by decwrl.enet; Tue, 1 Sep 92 20:58:00 PDT
Date: Tue, 1 Sep 92 20:58:00 PDT
From: 02-Sep-1992 1249 <diamond@jit081.enet.dec.com>
To: cube-lovers@ai.mit.edu
Apparently-To: cube-lovers@ai.mit.edu
Subject: Re: rare variants
Mr. Longridge! Please post the addresses of where to buy that stuff!
Oh, I'm drooling all over my keyboard.
Meanwhile, Jean-Claude Constantin (Pirkach 14, D-8535 ah I forgot the name
of the city but it's near Nurnberg, Germany) makes lots of variations on
cubes and skewbs etc., but the underlying mechanisms all seem to be standard
ones.
> Honourable mention: Pyraminx Star, Puck, Ufo, Trick Haus
> (Anyone got multiples of these)
Hmmmm. I might confess to owning two Trick Hauses in exchange for something
of sufficient persuasion, such as some of the others listed in Mr. Longridge's
post. In hopes of being able to exchange for some neat stuff, I'd better not
mention that Mr. Constantin can sell Trick Hauses for something around DM 20.
>Imagine my disappointment when I found out the Mach Ball, Skewb and
>Moody Ball all have the same basic mechanism!
Yeah, but with Mach Ball you have to orient the square-like pieces.
Haven't seen Moody Ball.
-- Norman Diamond diamond@jit081.enet.dec.com
[Digital did not write this.]
From ronnie@cisco.com Wed Sep 2 00:50:42 1992
Return-Path: <ronnie@cisco.com>
Received: from wolf.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA20803; Wed, 2 Sep 92 00:50:42 EDT
Received: from lager.cisco.com by wolf.cisco.com with TCP; Tue, 1 Sep 92 21:50:36 -0700
Message-Id: <9209020450.AA01685@wolf.cisco.com>
To: 02-Sep-1992 1249 <diamond@jit081.enet.dec.com>
Cc: cube-lovers@ai.mit.edu
Subject: Re: rare variants
In-Reply-To: Your message of "Tue, 01 Sep 92 20:58:00 PDT."
<9209020357.AA21950@enet-gw.pa.dec.com>
Date: Tue, 01 Sep 92 21:50:35 PDT
From: "Ronnie B. Kon" <ronnie@cisco.com>
> Mr. Longridge! Please post the addresses of where to buy that stuff!
> Oh, I'm drooling all over my keyboard.
>
> Meanwhile, Jean-Claude Constantin (Pirkach 14, D-8535 ah I forgot the name
> of the city but it's near Nurnberg, Germany) makes lots of variations on
> cubes and skewbs etc., but the underlying mechanisms all seem to be standard
> ones.
>
> > Honourable mention: Pyraminx Star, Puck, Ufo, Trick Haus
> > (Anyone got multiples of these)
>
> Hmmmm. I might confess to owning two Trick Hauses in exchange for something
> of sufficient persuasion, such as some of the others listed in Mr. Longridge'
s
> post. In hopes of being able to exchange for some neat stuff, I'd better not
> mention that Mr. Constantin can sell Trick Hauses for something around DM 20.
Unless the the Bundespost will deliver mail to "a city near Nurnberg" I think
you're pretty safe. Unless of course you'll post the full address?
Ronnie
From diamond@jit081.enet.dec.com Wed Sep 2 00:59:23 1992
Return-Path: <diamond@jit081.enet.dec.com>
Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA20846; Wed, 2 Sep 92 00:59:23 EDT
Received: by enet-gw.pa.dec.com; id AA24605; Tue, 1 Sep 92 21:59:05 -0700
Message-Id: <9209020459.AA24605@enet-gw.pa.dec.com>
Received: from jit081.enet; by decwrl.enet; Tue, 1 Sep 92 21:59:19 PDT
Date: Tue, 1 Sep 92 21:59:19 PDT
From: 02-Sep-1992 1356 <diamond@jit081.enet.dec.com>
To: cube-lovers@ai.mit.edu
Cc: ronnie@cisco.com
Apparently-To: ronnie@cisco.com, cube-lovers@ai.mit.edu
Subject: Re: rare variants
>> Meanwhile, Jean-Claude Constantin (Pirkach 14, D-8535 ah I forgot the name
>> of the city but it's near Nurnberg, Germany) makes lots of variations on
ronnie@cisco.com writes:
>Unless the the Bundespost will deliver mail to "a city near Nurnberg" I think
>you're pretty safe. Unless of course you'll post the full address?
I was telling the truth -- I forgot the name of the city while in the middle
of typing. Meanwhile, I had posted the full address on rec.puzzles a few
months ago.
Just now, I have recalled the name of the city, and here is the full address
(if I don't forget again while typing :-)
Constantin Geduldspiele
Pirkach 14
D-8535 Emskirchen
Germany
-- Norman Diamond diamond@jit081.enet.dec.com
[Digital did not write this.]
From Don.Woods@eng.sun.com Sun Sep 6 14:40:52 1992
Return-Path: <Don.Woods@eng.sun.com>
Received: from Sun.COM by life.ai.mit.edu (4.1/AI-4.10) id AA03843; Sun, 6 Sep 92 14:40:52 EDT
Received: from Eng.Sun.COM (zigzag-bb.Corp.Sun.COM) by Sun.COM (4.1/SMI-4.1)
id AA15176; Sun, 6 Sep 92 11:40:50 PDT
Received: from colossal.Eng.Sun.COM by Eng.Sun.COM (4.1/SMI-4.1)
id AA20580; Sun, 6 Sep 92 11:40:53 PDT
Received: by colossal.Eng.Sun.COM (4.1/SMI-4.1)
id AA28174; Sun, 6 Sep 92 11:42:34 PDT
Date: Sun, 6 Sep 92 11:42:34 PDT
From: Don.Woods@eng.sun.com (Don Woods)
Message-Id: <9209061842.AA28174@colossal.Eng.Sun.COM>
To: cube-lovers@ai.mit.edu, mark.longridge@canrem.com
Subject: Re: rare variants
I have a puzzle not in your list; it's rare enough that I've never seen
it in a store or catalog. I got mine from a friend. He calls it "The
Barrel".
Imagine a transparent cylinder divided into 6 circular slices. Slices 2-5
each have 5 pockets equally spaced around the circumference, just below the
surface (see left figure below). Slices 1 and 6 have three pockets in
positions corresponding to 3 of the 5 (see right figure).
******* *******
**** **** **** ****
***** ***** ***** *****
****** ****** ****** ******
***************** *****************
******************* *******************
******************* *******************
* ************* * *********************
* ************* * *********************
* ************* * *********************
********************* *********************
********************* *********************
********************* *********************
********************* *********************
******************* *******************
*** ******* *** *** ******* ***
* ******* * * ******* *
* ********* * * ********* *
************* *************
*********** ***********
******* *******
Through the center of the cylinder runs a piece with a cap on each end.
The caps each have 3 prongs poking into the cylinder, lined up on the
3 openings in slices 1 and 6. However, the central piece is long enough
that if the prongs are pushed into slice 6, the prongs at the other end
are lifted out of slice 1, and vice versa. So, at any given time, three
of the end pockets are filled by one of the endcaps. The other 23 pockets
contain colored balls.
Originally, the 3 balls in the unpronged endcap are black, and the balls
in the other slices are lined up by color; i.e. 4 blue balls lined up
above one another in slices 2-5, 4 green balls, likewise lined up, etc.
The possible moves are:
1) Slide the endcaps up and down. E.g., from the starting position, this
would push three balls of different colors into slice 6, and push the 3
black balls from slice 1 into slice 2 (and also push various balls of the
same color down, but that has no visible effect).
2) Turn slices 2 and 3 together; they do not move separately.
3) Turn slices 4 and 5 together; ditto.
Slices 1 and 6 are fixed, so they always line up with the end cap prongs
and with each other.
That's all there is to it. It certainly has the "cubish" feel to me, in
that it's impossible to make single moves that affect only a small portion
of the puzzle.
-- Don.
From diamond@jit081.enet.dec.com Sun Sep 6 20:11:45 1992
Return-Path: <diamond@jit081.enet.dec.com>
Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA08950; Sun, 6 Sep 92 20:11:45 EDT
Received: by enet-gw.pa.dec.com; id AA29557; Sun, 6 Sep 92 17:11:40 -0700
Message-Id: <9209070011.AA29557@enet-gw.pa.dec.com>
Received: from jit081.enet; by decwrl.enet; Sun, 6 Sep 92 17:11:44 PDT
Date: Sun, 6 Sep 92 17:11:44 PDT
From: 07-Sep-1992 0906 <diamond@jit081.enet.dec.com>
To: cube-lovers@ai.mit.edu
Apparently-To: cube-lovers@ai.mit.edu
Subject: Re: rare variants
Don.Woods@eng.sun.com writes about the barrel puzzle. If I recall correctly,
this was made by Nintendo before they switched to computer games. I also
believe the name was "Ten Billion Barrel." (Mine is buried somewhere and
I couldn't solve it, but I didn't want to buy the book that was published
at one time, sigh... I didn't need a book for the cube, so why should I
cheat for a piddly little barrel... sigh.)
Anyway, there are still a few available. But I have to warn, if anyone wants
one, it will cost more for postage and for my train fare going to the store,
than to buy the thing. If anyone wants one, we can arrange it by e-mail.
But I'd really prefer to trade for some of those wonderful things that
Mr. Longridge described. (I'm drooling all over my keyboard again, just
remembering them.)
-- Norman Diamond diamond@jit081.enet.dec.com
[Digital did not write this.]
From @mail.uunet.ca:mark.longridge@canrem.com Tue Sep 15 17:14:06 1992
Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com>
Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA11012; Tue, 15 Sep 92 17:14:06 EDT
Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10252>; Tue, 15 Sep 1992 17:13:51 -0400
Received: from canrem.com by unixbox.canrem.COM
id aa27586; Tue, 15 Sep 92 17:06:45 EDT
Received: by canrem.com (PCB/Usenet Gateway)
Path-id <199215.104.106005@dosgate>; 15 Sep 92 (16:56)
Message-Id: <199215.104.106005@dosgate>
From: Mark Longridge <mark.longridge@canrem.com>
Date: Mon, 14 Sep 1992 20:00:00 -0400
To: cube-lovers@ai.mit.edu
Subject: pyraminx revisited
Notes on the Pyraminx
---------------------
According to Dr. Ronald Turner-Smith there are 5 different Pyraminx
puzzles, all of different complexity. The following are listed in
order from easiest to hardest (to the best of my knowledge):
Pyraminx Star: Easiest of all pyraminx??
A simplification of the popular pyraminx because of the
little uni-coloured (usually grey or silver) tetrahedrons
on the 3 middle pieces of each face. Effectively all
middle pieces on this pyraminx are the same colour!
Snub Pyraminx: Same as standard pyraminx with tips removed
Popular Pyraminx: The standard pyraminx of which appeared in vast
quanities after the cube caught on.
Senior Pyraminx: This is a mystery puzzle. No one seems to know anything
about it, yet Turner-Smith's book refers to it and
gives the maximum number of moves for it! It is between
the Popular Pyraminx and Master Pyraminx in difficulty.
Master Pyraminx: All the moves of the standard pyraminx plus 180 degree
turns of the edges (just the strip, not the whole face)
446,965,972,992,000 combinations. Interestingly in
--
Canada Remote Systems - Toronto, Ontario, Canadas
World's Largest PCBOARD System - 416-629-7000/629-7044
From @mail.uunet.ca:mark.longridge@canrem.com Tue Sep 15 18:37:59 1992
Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com>
Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA14355; Tue, 15 Sep 92 18:37:59 EDT
Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10307>; Tue, 15 Sep 1992 18:37:53 -0400
Received: from canrem.com by unixbox.canrem.COM
id aa00746; Tue, 15 Sep 92 18:30:29 EDT
Received: by canrem.com (PCB/Usenet Gateway)
Path-id <199215.104.106150@dosgate>; 15 Sep 92 (18:27)
Message-Id: <199215.104.106150@dosgate>
From: Mark Longridge <mark.longridge@canrem.com>
Date: Mon, 14 Sep 1992 20:00:00 -0400
To: cube-lovers@ai.mit.edu
Subject: pyraminx revisited
Notes on the Pyraminx
---------------------
According to Dr. Ronald Turner-Smith there are 5 different Pyraminx
puzzles, all of different complexity. The following are listed in
order from easiest to hardest (to the best of my knowledge):
Pyraminx Star: Easiest of all pyraminx??
A simplification of the popular pyraminx because of the
little uni-coloured (usually grey or silver) tetrahedrons
on the 3 middle pieces of each face. Effectively all
middle pieces on this pyraminx are the same colour!
Snub Pyraminx: Same as standard pyraminx with tips removed
Popular Pyraminx: The standard pyraminx of which appeared in vast
quanities after the cube caught on.
Senior Pyraminx: This is a mystery puzzle. No one seems to know anything
about it, yet Turner-Smith's book refers to it and
gives the maximum number of moves for it! It is between
the Popular Pyraminx and Master Pyraminx in difficulty.
Master Pyraminx: All the moves of the standard pyraminx plus 180 degree
turns of the edges (just the strip, not the whole face)
446,965,972,992,000 combinations. Interestingly in the
ads for Dr. Ronald Turner-Smith's book "The Amazing
Pyraminx" the Master Pyraminx is cited as a puzzle
superior to Rubik's Cube because there are no centre
pieces and it's harder! (Both points debatable IMHO)
Also Turner-Smith gives the following maximum number of moves for
each type of Pyraminx puzzle: (The popular pyraminx is now known
to be 11 moves at most or 14 moves if the tips are included)
Type 1 24 moves ??
Type 2 28 moves ??
Type 3 38 moves ??
Type 4 215 moves (Senior Pyraminx)
Type 5 255 moves (Master Pyraminx)
Also it is known that transparent pyraminx puzzles were made. This
would be a good idea for the cube as well. Meffert also considered
a textured pyraminx for the blind, and ones with leather and wood
finishes.
All the post-cube puzzles compare themselves to the cube, such as
the Master Pyraminx, and more recently Smart Alex. It seems that
Rubik's Cube is the benchmark for all others to compare with.
Alas, Uwe Meffert's puzzle club was a bust. Barring unique prototypes
(perhaps Singmaster or Hofstadter have a Master Pyraminx, I'll check)
none of the following were produced: (most of these are documented in
the extremely rare "Pyraminx The Exciting new 1982 range" or the even
more obscure 1983 edition booklet. Both of these have full colour pages
with photos of cardboard mockups of all the variants.)
1982:
Pyraminx Star (exists in small quanities, in Constantin's catalog)
Pyraminx Pentagon, Pyraminx Hexagon (Computer Simulation/Mockup only)
One can also imagine Septagons, Octagons etc...
Pyraminx Barrel Junior, Pyraminx Barrel Senior (Mockup only)
Pyraminx Disc Chess (Prototypes exist)
Pyraminx Ultimate (as shown in July 82 Scientific American, Mockup)
Pyraminx Crystal, Pyraminx Ball (Mockups, July 82 S.A)
I'd really like to see the mechanism for a working crystal!
Pyraminx Assembly Puzzles, 4 types (They exist)
Pyraminx Octahedron (An octahedral skewb, I believe Braun & Bandelow
made
some) Gerd Braun is the inventor of the Moody Ball (rare but exists)
Pyraminx Tetrahedron (Ben Halpern made a prototype)
1983:
Space Grenade (???)
Crystal Ball (Looks like an orb, definitely not the same though)
However....
Just a few days ago I got Constantin's catalog. Surprisingly there is
a picture of Josef Trajber's Octahedron inside. Also there is a picture
of what appears to be a Pyraminx Ball. Other ideas he includes are a
2x2x2 siamese cube, new variants on Fisher's cube, e.g. Fisher's Domino,
and a Pyraminx Ultimate for 180 DM!
...and so the search for new cube variants continues. Please send me
your
comments (Does anyone actually own a working Master Pyraminx??) I'm
also interested in exchanging full cube lists with other collectors.
Mark Longridge Email: mark.longridge@canrem.com
259 Thornton Rd N
Oshawa Ontario Canada L1J 6T2
--
Canada Remote Systems - Toronto, Ontario, Canadas
World's Largest PCBOARD System - 416-629-7000/629-7044
From tom@goat.clipper.ingr.com Tue Sep 15 21:28:48 1992
Return-Path: <tom@goat.clipper.ingr.com>
Received: from ingr.ingr.com by life.ai.mit.edu (4.1/AI-4.10) id AA19739; Tue, 15 Sep 92 21:28:48 EDT
Received: from clipper.clipper.ingr.com by ingr.ingr.com (5.65c/1.920611)
id AA01312; Tue, 15 Sep 1992 20:34:24 -0500
Received: from goat by clipper.clipper.ingr.com (5.61/1.910401)
id AA28058; Tue, 15 Sep 92 16:21:41 -0700
Received: by goat.clipper.ingr.com (5.61/1.910201)
id AA00258; Tue, 15 Sep 92 15:56:26 -0700
Subject: Re: pyraminx revisited
To: cube-lovers@ai.mit.edu
Date: Tue, 15 Sep 92 15:56:22 PDT
In-Reply-To: <199215.104.106150@dosgate>; from "Mark Longridge" at Sep 14, 92 8:00 pm
X-Mailer: ELM [version 05.00.01.20]
Message-Id: <9209151556.AA00256@goat.UUCP>
From: tom@goat.clipper.ingr.com (Tom Granvold)
> Also it is known that transparent pyraminx puzzles were made. This
> would be a good idea for the cube as well. Meffert also considered
> a textured pyraminx for the blind, and ones with leather and wood
> finishes.
I have one of the textured pyraminx. I had to mail order it from
Hong Kong in the early '80s.
> Just a few days ago I got Constantin's catalog.
Where can one get a copy of Constantin's catalog?
> ...and so the search for new cube variants continues. Please send me
> your
> comments (Does anyone actually own a working Master Pyraminx??) I'm
> also interested in exchanging full cube lists with other collectors.
I wish I did :-)
Tom Granvold
------------------------------------------------------
Mail: 2400 Geng Rd., Palo Alto, Calif., 94303
Email: tom@clipper.ingr.com
------------------------------------------------------
From @mail.uunet.ca:mark.longridge@CANREM.COM Mon Sep 21 00:05:34 1992
Return-Path: <@mail.uunet.ca:mark.longridge@CANREM.COM>
Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA04313; Mon, 21 Sep 92 00:05:34 EDT
Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10241>; Mon, 21 Sep 1992 00:05:23 -0400
Received: from canrem.com by unixbox.canrem.COM
id aa02771; Sun, 20 Sep 92 23:58:15 EDT
Received: by canrem.com (PCB/Usenet Gateway)
Path-id <199220.104.108416@dosgate>; 20 Sep 92 (23:51)
Message-Id: <199220.104.108416@dosgate>
From: Mark Longridge <mark.longridge@canrem.com>
Date: Sat, 19 Sep 1992 20:00:00 -0400
To: cube-lovers@ai.mit.edu
Subject: pyraminx revisted
Notes on the Pyraminx
---------------------
According to Dr. Ronald Turner-Smith there are 5 different Pyraminx
puzzles, all of different complexity. The following are listed in
order from easiest to hardest (to the best of my knowledge):
Pyraminx Star: Easiest of all pyraminx??
A simplification of the popular pyraminx because of the
little uni-coloured (usually grey or silver) tetrahedrons
on the 3 middle pieces of each face. Effectively all
middle pieces on this pyraminx are the same colour!
Snub Pyraminx: Same as standard pyraminx with tips removed
Popular Pyraminx: The standard pyraminx of which appeared in vast
quanities after the cube caught on.
Senior Pyraminx: This is a mystery puzzle. No one seems to know anything
about it, yet Turner-Smith's book refers to it and
gives the maximum number of moves for it! It is between
the Popular Pyraminx and Master Pyraminx in difficulty.
Master Pyraminx: All the moves of the standard pyraminx plus 180 degree
turns of the edges (just the strip, not the whole face)
446,965,972,992,000 combinations. Interestingly in the
ads for Dr. Ronald Turner-Smith's book "The Amazing
Pyraminx" the Master Pyraminx is cited as a puzzle
superior to Rubik's Cube because there are no centre
pieces and it's harder! (Both points debatable IMHO)
Also Turner-Smith gives the following maximum number of moves for
each type of Pyraminx puzzle: (The popular pyraminx is now known
to be 11 moves at most or 14 moves if the tips are included)
Type 1 24 moves ??
Type 2 28 moves ??
Type 3 38 moves ??
Type 4 215 moves (Senior Pyraminx)
Type 5 255 moves (Master Pyraminx)
Also it is known that transparent pyraminx puzzles were made. This
would be a good idea for the cube as well. Meffert also considered
a textured pyraminx for the blind, and ones with leather and wood
finishes.
All the post-cube puzzles compare themselves to the cube, such as
the Master Pyraminx, and more recently Smart Alex. It seems that
Rubik's Cube is the benchmark for all others to compare with.
Alas, Uwe Meffert's puzzle club was a bust. Barring unique prototypes
(perhaps Singmaster or Hofstadter have a Master Pyraminx, I'll check)
none of the following were produced: (most of these are documented in
the extremely rare "Pyraminx The Exciting new 1982 range" or the even
more obscure 1983 edition booklet. Both of these have full colour pages
with photos of cardboard mockups of all the variants.)
1982:
Pyraminx Star (exists in small quanities, in Constantin's catalog)
Pyraminx Pentagon, Pyraminx Hexagon (Computer Simulation/Mockup only)
One can also imagine Septagons, Octagons etc...
Pyraminx Barrel Junior, Pyraminx Barrel Senior (Mockup only)
Pyraminx Disc Chess (Prototypes exist)
Pyraminx Ultimate (as shown in July 82 Scientific American, Mockup)
Pyraminx Crystal, Pyraminx Ball (Mockups, July 82 S.A)
I'd really like to see the mechanism for a working crystal!
Pyraminx Assembly Puzzles, 4 types (They exist)
Pyraminx Octahedron (An octahedral skewb, I believe Braun & Bandelow
made
some) Gerd Braun is the inventor of the Moody Ball (rare but exists)
Pyraminx Tetrahedron (Ben Halpern made a prototype)
1983:
Space Grenade (???)
Crystal Ball (Looks like an orb, definitely not the same though)
However....
Just a few days ago I got Constantin's catalog. Surprisingly there is
a picture of Josef Trajber's Octahedron inside. Also there is a picture
of what appears to be a Pyraminx Ball. Other ideas he includes are a
2x2x2 siamese cube, new variants on Fisher's cube, e.g. Fisher's Domino,
and a Pyraminx Ultimate for 180 DM!
...and so the search for new cube variants continues. Please send me
your
comments (Does anyone actually own a working Master Pyraminx??) I'm
also interested in exchanging full cube lists with other collectors.
Mark Longridge Email: mark.longridge@canrem.com
259 Thornton Rd N
Oshawa Ontario Canada L1J 6T2
--
Canada Remote Systems - Toronto, Ontario, Canadas
World's Largest PCBOARD System - 416-629-7000/629-7044
From mb8d+@andrew.cmu.edu Mon Sep 21 21:31:25 1992
Return-Path: <mb8d+@andrew.cmu.edu>
Received: from po3.andrew.cmu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA08728; Mon, 21 Sep 92 21:31:25 EDT
Received: by po3.andrew.cmu.edu (5.54/3.15) id <AA01154> for cube-lovers@ai.mit.edu; Mon, 21 Sep 92 21:31:22 EDT
Received: via switchmail; Mon, 21 Sep 1992 21:31:21 -0400 (EDT)
Received: from pcs6.andrew.cmu.edu via qmail
ID </afs/andrew.cmu.edu/service/mailqs/q003/QF.EejZvxe00WBKA0jcIg>;
Mon, 21 Sep 1992 19:47:17 -0400 (EDT)
Received: from pcs6.andrew.cmu.edu via qmail
ID </afs/andrew.cmu.edu/usr25/mb8d/.Outgoing/QF.QejZvYK00WBKA8jY9B>;
Mon, 21 Sep 1992 19:46:44 -0400 (EDT)
Received: from mms.0.1.873.MacMail.0.9.CUILIB.3.45.SNAP.NOT.LINKED.pcs6.andrew.cmu.edu.pmax.ul4
via MS.5.6.pcs6.andrew.cmu.edu.pmax_ul4;
Mon, 21 Sep 1992 19:46:44 -0400 (EDT)
Message-Id: <0ejZvYC00WBK48jY0m@andrew.cmu.edu>
Date: Mon, 21 Sep 1992 19:46:44 -0400 (EDT)
From: Matthew John Bushey <mb8d+@andrew.cmu.edu>
To: cube-lovers@ai.mit.edu
Subject: cubes are great
Cc:
Does anyone out there know what is the cubed root of 81?
Just wondering....
From yekta@huey.jpl.nasa.gov Mon Sep 21 22:38:35 1992
Return-Path: <yekta@huey.jpl.nasa.gov>
Received: from huey.Jpl.Nasa.Gov by life.ai.mit.edu (4.1/AI-4.10) id AA10534; Mon, 21 Sep 92 22:38:35 EDT
Received: from hercules.JPL.NASA.GOV ([128.149.68.28]) by huey.Jpl.Nasa.Gov (4.1/SMI-4.1+DXRm2.2)
id AA00679; Mon, 21 Sep 92 19:34:07 PDT
Date: Mon, 21 Sep 92 19:34:07 PDT
From: yekta@huey.jpl.nasa.gov (Yekta Gursel)
Message-Id: <9209220234.AA00679@huey.Jpl.Nasa.Gov>
Received: by hercules.JPL.NASA.GOV (4.1/SMI-4.1)
id AA09483; Mon, 21 Sep 92 19:38:22 PDT
To: mb8d+@andrew.cmu.edu
Cc: cube-lovers@ai.mit.edu
In-Reply-To: Matthew John Bushey's message of Mon, 21 Sep 1992 19:46:44 -0400 (EDT) <0ejZvYC00WBK48jY0m@andrew.cmu.edu>
Subject: cubes are great
Little smaller than its square root. Are you having fun yet?
--Yekta
From gls@think.com Tue Sep 22 11:50:24 1992
Received: from mail.think.com by life.ai.mit.edu (4.1/AI-4.10) id AA26659; Tue, 22 Sep 92 11:50:24 EDT
Return-Path: <gls@Think.COM>
Received: from Strident.Think.COM by mail.think.com; Tue, 22 Sep 92 11:50:22 -0400
From: Guy Steele <gls@think.com>
Received: by strident.think.com (4.1/Think-1.2)
id AA24984; Tue, 22 Sep 92 11:50:22 EDT
Date: Tue, 22 Sep 92 11:50:22 EDT
Message-Id: <9209221550.AA24984@strident.think.com>
To: mb8d+@andrew.cmu.edu
Cc: cube-lovers@ai.mit.edu
In-Reply-To: Matthew John Bushey's message of Mon, 21 Sep 1992 19:46:44 -0400 (EDT) <0ejZvYC00WBK48jY0m@andrew.cmu.edu>
Subject: cubes are great
Date: Mon, 21 Sep 1992 19:46:44 -0400 (EDT)
From: Matthew John Bushey <mb8d+@andrew.cmu.edu>
Does anyone out there know what is the cubed root of 81?
Just wondering....
Well, the "root of 81" is 9 (recall that when you don't say
what kind of root you want, the default is "square"), and
9 cubed is 729.
... Eh? Oh, you meant the "cube root", not the "cubed root"?
Well, that's another kettle of fish entirely. The n'th root
of x is equal to x raised to the power 1/n. I fed this to
my friendly Common Lisp system:
> (expt 81 1/3)
4.3267487109222245
If I were you, I wouldn't trust the last few digits of this
approximation, but fifteen decimal places ought to hold you
for now.
Here's how you could estimate it in your head.
Note that 81 = 3 to the fourth power, so
1/3 4 1/3 4/3 1/3
81 = ( 3 ) = 3 = 3 ( 3 )
Now, the cube root of 3 is surely between 1 and 2,
because 1 cubed is 1 and 2 cubed is 8. So the cube
root of 3 is 1 plus some smaller fractional amount x.
3 2 3
So 3 = (1 + x) = 1 + 3 x + 3 x + x (binomial expansion).
3
Let's ignore the x term, which is probably small because
x is sort of small. Then
2 2
1 + 3 x + 3 x = 3 so x + x = 2/3 .
2
Hm... if x = 1/2, then x + x = 3/4, which is a bit
2
too big. So figure x is about 0.4; then x + x = .4 + .16 = .56
which is too small. So probably x is about 0,45 or so.
So the cube root of 3 is about 1.45, and the cube root of
81 is 3 times that, or about 4.35 -- not a bad approximation.
--Guy STeele
From bosch@smiteo.esd.sgi.com Tue Sep 22 12:10:48 1992
Return-Path: <bosch@smiteo.esd.sgi.com>
Received: from sgi.sgi.com (SGI.COM) by life.ai.mit.edu (4.1/AI-4.10) id AA27072; Tue, 22 Sep 92 12:10:48 EDT
Received: from [192.48.193.1] by sgi.sgi.com via SMTP (920330.SGI/910110.SGI)
for cube-lovers@ai.mit.edu id AA11832; Tue, 22 Sep 92 09:10:45 -0700
Received: by smiteo.esd.sgi.com (911016.SGI/920502.SGI.AUTO)
for @sgi.sgi.com:cube-lovers@ai.mit.edu id AA02225; Tue, 22 Sep 92 09:10:44 -0700
From: bosch@smiteo.esd.sgi.com (Derek Bosch)
Message-Id: <9209221610.AA02225@smiteo.esd.sgi.com>
Subject: Gaby Games address needed
To: cube-lovers@ai.mit.edu
Date: Tue, 22 Sep 92 9:10:44 PDT
X-Mailer: ELM [version 2.3 PL4]
Does anyone out there in cube-land know the address for Gaby Games?
They are an Israeli manufacturer of interesting 3-d interlocking wooden
puzzles. I have posted this to rec.puzzles, with no help so far.
Derek Bosch
bosch@sgi.com
From ronnie@cisco.com Tue Sep 22 13:39:12 1992
Return-Path: <ronnie@cisco.com>
Received: from wolf.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA29713; Tue, 22 Sep 92 13:39:12 EDT
Received: from lager.cisco.com by wolf.cisco.com with TCP; Tue, 22 Sep 92 10:37:51 -0700
Message-Id: <9209221737.AA25287@wolf.cisco.com>
To: Matthew John Bushey <mb8d+@andrew.cmu.edu>
Cc: cube-lovers@ai.mit.edu
Subject: Re: cubes are great
In-Reply-To: Your message of "Mon, 21 Sep 92 19:46:44 EDT."
<0ejZvYC00WBK48jY0m@andrew.cmu.edu>
Date: Tue, 22 Sep 92 10:37:50 PDT
From: "Ronnie B. Kon" <ronnie@cisco.com>
>
> Does anyone out there know what is the cubed root of 81?
>
> Just wondering....
>
Let's see: the root of 81 is 9. 9 cubed is 729.
Ronnie
From azimmerm@rnd.stern.nyu.edu Mon Oct 5 16:53:08 1992
Return-Path: <azimmerm@rnd.stern.nyu.edu>
Received: from rnd.stern.nyu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA17208; Mon, 5 Oct 92 16:53:08 EDT
Received: by rnd.stern.nyu.edu (4.1/1.34)
id AA28496; Mon, 5 Oct 92 16:51:57 EDT
Date: Mon, 5 Oct 92 16:51:57 EDT
From: Al Zimmermann <azimmerm@rnd.stern.nyu.edu>
To: Cube-Lovers@ai.mit.edu
Subject: Reminiscences
Message-Id: <CMM.0.90.2.718318317.azimmerm@rnd.gba.nyu.edu>
Is everybody ready for more reminiscences?
I got my first cube at Harrad's in London in October of 1980 while I was on
vacation there with my girl friend. I spent every non-touristy moment
working out and recording moves until, on day 13, I got the final face.
When we got back to the States, my girl friend and I broke up. Do you think
there's a moral here?
Al Zimmermann
From diamond@jit081.enet.dec.com Mon Oct 5 20:54:41 1992
Return-Path: <diamond@jit081.enet.dec.com>
Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA22961; Mon, 5 Oct 92 20:54:41 EDT
Received: by enet-gw.pa.dec.com; id AA18960; Mon, 5 Oct 92 17:54:31 -0700
Message-Id: <9210060054.AA18960@enet-gw.pa.dec.com>
Received: from jit081.enet; by decwrl.enet; Mon, 5 Oct 92 17:54:38 PDT
Date: Mon, 5 Oct 92 17:54:38 PDT
From: 06-Oct-1992 0949 <diamond@jit081.enet.dec.com>
To: cube-lovers@ai.mit.edu
Apparently-To: cube-lovers@ai.mit.edu
Subject: Re: miniscences
Al Zimmermann <azimmerm@rnd.stern.nyu.edu> writes:
>I spent every non-touristy moment
>working out and recording moves until, on day 13, I got the final face.
>When we got back to the States, my girl friend and I broke up. Do you think
>there's a moral here?
Yes, at least three:
(1) Every non-touristy moment that you weren't recording moves, you should
have spent with your girl friend instead of working out :-)
(2) You should have given equal attention to the final face and to your
girl friend's face :-)
(3) You should have chosen a girl friend who could solve the cube, like
I did.
But I thank you for warning about the dangers of getting back to the States.
Maybe I shouldn't go back after all :-)
-- Norman Diamond diamond@jit081.enet.dec.com
[Digital did not write this.]
From imp@kolvir.solbourne.com Fri Oct 9 17:59:44 1992
Return-Path: <imp@kolvir.solbourne.com>
Received: from Solbourne.COM by life.ai.mit.edu (4.1/AI-4.10) id AA25966; Fri, 9 Oct 92 17:59:44 EDT
Received: from kolvir.Solbourne.COM by Solbourne.COM (4.1/Solbourne-4.1)
id AA16499; Fri, 9 Oct 92 15:59:44 MDT
Received: from localhost by kolvir.Solbourne.COM (4.1/SMI-4.1)
id AA06857; Fri, 9 Oct 92 15:59:40 MDT
Message-Id: <9210092159.AA06857@kolvir.Solbourne.COM>
To: cube-lovers@ai.mit.edu
Subject: Quick question....
Date: Fri, 09 Oct 1992 15:59:40 MDT
From: Warner Losh <imp@kolvir.solbourne.com>
I was wondering if there were any X programs out there that allowed
one to play with a rubic's cube (3x3x3 ... nxnxn) on a workstation?
Archie didn't seem to know of any.
Warner
From pbeck@pica.army.mil Fri Oct 23 07:49:03 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA12683; Fri, 23 Oct 92 07:49:03 EDT
Date: Fri, 23 Oct 92 7:47:24 EDT
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: cube-lovers@life.ai.mit.edu
Cc: pbeck@pica.army.mil
Subject: 13th IPP (1993)
Message-Id: <9210230747.aa12659@COR4.PICA.ARMY.MIL>
...............................................................
<--> 13th International puzzle collector's party and fair "
and the 1993 Dutch Cube Day party
transcribed by pbeck, 10/23/92
...............................................................
WHEN ---- 8/20 - 8/22/93
WHERE ---- Amsterdam vicinity
LODGING ---- about $90 per night at
MOTEL BREUKELEN
STATIONSWEG 91
3621 LK BREUKELEN
NETHERLANDS
TEL: 03462 - 65888
FAX: 03462 - 62894
*** INVITATIONS *** Admission by invitation only!!!
Contact: Mr. W.G.H. STRIJBOS
BREDEROSTRAAT 18
5921 BM VENLO
NETHERLANDS
TEL: +31 (0) 77 -826213
FAX: +31 (0) 4704 - 4656
AGENDA:
8/20 13:00 - 16:30 PUZZLE EXCHANGE
17:30 - 22:00 DINNER AND MAGIC SHOW
8/21 10:00 - 17:00 PUZZLE PARTY AND FAIR (SALES)
COST FOR ABOVE 150 DUTCH GUILDERS & IT WILL BE HELD AT MOTEL
BREUKELEN - 100 EXTRA FOR SAT SALES TABLE
8/22 10:00 - 17:00 CUBE DAY _ THIS WILL BE HELD AT CHESS & GO
CENTER IN AMSTELVEEN AND PROBABLY HAS AN EXTRA
COST.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
*** 4 NIGHT RESERVATION (8/19 - 8/23) AT " MOTEL BREUKELEN" IS 625
DUTCH GUILDERS AND INCLUDES BREAKFAST - ONE DOUBLE BED IN EACH ROOM.
---> RESERVATION REQUEST FOR THIS PACKAGE MUST BE MADE BY
*** JAN 10 1993 *** TO STRIJBOS
>>>>>>>>>>>>>>>>>> If I was unclear or if you have other questions
ask them to the list since several members of the Dutch Cube Club
(party hosts) are subscribers.
From hirsh@cs.rutgers.edu Wed Nov 4 15:18:42 1992
Return-Path: <hirsh@cs.rutgers.edu>
Received: from pei.rutgers.edu by life.ai.mit.edu (4.1/AI-4.10) id AA20102; Wed, 4 Nov 92 15:18:42 EST
Received: by pei.rutgers.edu (5.59/SMI4.0/RU1.5/3.08)
id AA09079; Wed, 4 Nov 92 15:18:38 EST
Sender: Haym Hirsh <hirsh@cs.rutgers.edu>
Date: Wed, 4 Nov 92 15:18:37 EST
From: Haym Hirsh <hirsh@cs.rutgers.edu>
Reply-To: Haym Hirsh <hirsh@cs.rutgers.edu>
To: cube-lovers@ai.mit.edu
Subject: masterball
Cc: Haym Hirsh <hirsh@cs.rutgers.edu>
Message-Id: <CMM-RU.1.3.720908317.hirsh@pei.rutgers.edu>
A friend just sent me email about a new (to him and to me) puzzle
called "masterball". Anyone know anything about it? Is it worth
getting?
Haym
> I saw a Rubik's cube variant today called "Masterball." Have you
> seen it? It is a sphere with 32 faces. If you consider the sphere
> to be a world globe, there are 8 longitudinal slices each going
> through the axis of the globe, dividing the sphere into 8 segments
> like a sliced orange (sorry for starting to mix my metaphors [actually,
> I guess I was mixing similes, but I know *you* wouldn't bring up
> such a trivial point]).
>
> Oooops I guess there are only 4 longitudinal slices, each through
> the axis, to divide the globe into 8 segments.
>
> There are also 3 slices of latitude, one through the equator one
> each in the northern and the southern hemisphere parallel to
> the equator.
>
> Resultant 32 faces. Mechanism has some similarities to Square One.
>
> Two different versions of Masterball are available. One has eight
> different colors, corresponding to 8 segments. The other has only
> black and white. I don't remember the home pattern of the black
> and white sphere, I presume it is a degenerate case of the 8 color
> sphere with black and white alternating slices.
>
> Cost: $24.95 each. My source is the same store in San Francisco
> (Stonestown mall) that provided the Rubiks Tangle, Rubiks Dice, etc.
From azimmerm@rnd.stern.nyu.edu Wed Nov 4 17:30:12 1992
Return-Path: <azimmerm@rnd.stern.nyu.edu>
Received: from rnd.stern.nyu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA29389; Wed, 4 Nov 92 17:30:12 EST
Received: by rnd.stern.nyu.edu (4.1/1.34)
id AA20786; Wed, 4 Nov 92 17:12:04 EST
Date: Wed, 4 Nov 92 17:12:04 EST
From: Al Zimmermann <azimmerm@rnd.stern.nyu.edu>
To: Haym Hirsh <hirsh@cs.rutgers.edu>
Cc: cube-lovers@ai.mit.edu, Haym Hirsh <hirsh@cs.rutgers.edu>
Subject: Re: masterball
In-Reply-To: Your message of Wed, 4 Nov 92 15:18:37 EST
Message-Id: <CMM.0.90.2.720915124.azimmerm@rnd.stern.nyu.edu>
> A friend just sent me email about a new (to him and to me) puzzle
> called "masterball". Anyone know anything about it? Is it worth
> getting?
>
> Haym
>
> > I saw a Rubik's cube variant today called "Masterball." Have you
> > seen it? It is a sphere with 32 faces. If you consider the sphere
> > to be a world globe, there are 8 longitudinal slices each going
> > through the axis of the globe, dividing the sphere into 8 segments
> > like a sliced orange (sorry for starting to mix my metaphors [actually,
> > I guess I was mixing similes, but I know *you* wouldn't bring up
> > such a trivial point]).
> >
> > Oooops I guess there are only 4 longitudinal slices, each through
> > the axis, to divide the globe into 8 segments.
> >
> > There are also 3 slices of latitude, one through the equator one
> > each in the northern and the southern hemisphere parallel to
> > the equator.
> >
> > Resultant 32 faces. Mechanism has some similarities to Square One.
> >
> > Two different versions of Masterball are available. One has eight
> > different colors, corresponding to 8 segments. The other has only
> > black and white. I don't remember the home pattern of the black
> > and white sphere, I presume it is a degenerate case of the 8 color
> > sphere with black and white alternating slices.
> >
> > Cost: $24.95 each. My source is the same store in San Francisco
> > (Stonestown mall) that provided the Rubiks Tangle, Rubiks Dice, etc.
>
>
Games magazine seems to like the puzzle. They included Mastermind Rainbow
(the polychromatic version) in this year's "Games 100" listing. Their
write-up isn't very informative, but there's a picture. It appears on page
59 of the Dec. '92 issue.
By the way, they indicate that the puzzle is available from Baekgaard at
1-800-323-5413.
From pbeck@pica.army.mil Thu Nov 5 21:56:13 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA15053; Thu, 5 Nov 92 21:56:13 EST
Date: Thu, 5 Nov 92 11:11:51 EST
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: cube-lovers@life.ai.mit.edu
Subject: RE: MASTERBALL
Message-Id: <9211051111.aa20544@COR4.PICA.ARMY.MIL>
Masterball is similar to the VIP Sphere and Orb puzzles.
It is nicely made and has been available in europe for
a couple of years.
I believe ISHI is also selling it.
If you don't have the ISHI x-mas flyer call them
and get it. This flyer has several unique items
for slae, eg, 5x5x5, skewb
From yekta@huey.jpl.nasa.gov Fri Nov 6 11:46:52 1992
Return-Path: <yekta@huey.jpl.nasa.gov>
Received: from huey.Jpl.Nasa.Gov by life.ai.mit.edu (4.1/AI-4.10) id AA16339; Fri, 6 Nov 92 11:46:52 EST
Received: from hercules.JPL.NASA.GOV ([128.149.68.28]) by huey.Jpl.Nasa.Gov (4.1/SMI-4.1+DXRm2.2)
id AA25552; Fri, 6 Nov 92 08:41:53 PST
Date: Fri, 6 Nov 92 08:41:53 PST
From: yekta@huey.jpl.nasa.gov (Yekta Gursel)
Message-Id: <9211061641.AA25552@huey.Jpl.Nasa.Gov>
Received: by hercules.JPL.NASA.GOV (4.1/SMI-4.1)
id AA07093; Fri, 6 Nov 92 08:40:56 PST
To: cube-lovers@life.ai.mit.edu
In-Reply-To: Peter Beck (BATDD)'s message of Thu, 5 Nov 92 11:11:51 EST <9211051111.aa20544@COR4.PICA.ARMY.MIL>
Subject: MASTERBALL
Could you post ISHI's phone number?
--Yekta
From dik@cwi.nl Sun Nov 8 16:16:16 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA02772; Sun, 8 Nov 92 16:16:16 EST
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA20202 (5.65b/3.2/CWI-Amsterdam); Sun, 8 Nov 1992 21:15:15 GMT
Received: by boring.cwi.nl
id AA21833 (5.65b/2.10/CWI-Amsterdam); Sun, 8 Nov 1992 22:15:14 +0100
Date: Sun, 8 Nov 1992 22:15:14 +0100
From: Dik.Winter@cwi.nl
Message-Id: <9211082115.AA21833.dik@boring.cwi.nl>
To: cube-lovers@life.ai.mit.edu
Subject: Where are the archives?
Can somebody tell me where the cube-lovers archives are maintained at this
moment?
Thanks, dik
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl
From dik@cwi.nl Sun Nov 8 16:15:20 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA02765; Sun, 8 Nov 92 16:15:20 EST
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA20202 (5.65b/3.2/CWI-Amsterdam); Sun, 8 Nov 1992 21:15:15 GMT
Received: by boring.cwi.nl
id AA21833 (5.65b/2.10/CWI-Amsterdam); Sun, 8 Nov 1992 22:15:14 +0100
Date: Sun, 8 Nov 1992 22:15:14 +0100
From: Dik.Winter@cwi.nl
Message-Id: <9211082115.AA21833.dik@boring.cwi.nl>
To: cube-lovers@life.ai.mit.edu
Subject: Where are the archives?
Can somebody tell me where the cube-lovers archives are maintained at this
moment?
Thanks, dik
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl
From alan@ai.mit.edu Sun Nov 8 17:00:39 1992
Return-Path: <alan@ai.mit.edu>
Received: from corn-pops (corn-pops.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA03727; Sun, 8 Nov 92 17:00:39 EST
Received: by corn-pops (4.1/AI-4.10) id AA03768; Sun, 8 Nov 92 17:00:08 EST
Date: Sun, 8 Nov 92 17:00:08 EST
Message-Id: <9211082200.AA03768@corn-pops>
From: Alan Bawden <Alan@lcs.mit.edu>
Sender: Alan@lcs.mit.edu
To: Dik.Winter@cwi.nl
Cc: Cube-Lovers@ai.mit.edu
In-Reply-To: Dik.Winter@cwi.nl's message of Sun, 8 Nov 1992 22:15:14 +0100 <9211082115.AA21833.dik@boring.cwi.nl>
Subject: Where are the archives?
Date: Sun, 8 Nov 1992 22:15:14 +0100
From: Dik.Winter@cwi.nl
Can somebody tell me where the cube-lovers archives are maintained at this
moment?
Such questions should be addressed to Cube-Lovers-Request@AI.MIT.EDU. But
since you asked publicly, I might as well answer publicly as well.
If you are interested in the archives of the Cube-Lovers mailing list:
Using FTP, connect to FTP.AI.MIT.EDU, login as "anonymous" (any password),
and in the directory "pub/cube-lovers" you will find the nine (compressed)
files "cube-mail-0.Z" through "cube-mail-8.Z".
Archive vital statistics (when uncompressed):
File From To Size (bytes)
---- ---- -- ------------
cube-mail-0 12 Jul 80 23 Oct 80 185037
cube-mail-1 3 Nov 80 9 Jan 81 135719
cube-mail-2 10 Jan 81 3 Aug 81 138566
cube-mail-3 3 Aug 81 3 May 82 137753
cube-mail-4 4 May 81 11 Dec 82 139660
cube-mail-5 11 Dec 82 6 Jan 87 173364
cube-mail-6 10 Jan 87 13 Apr 90 216733
cube-mail-7 12 Oct 90 9 Sep 91 137508
cube-mail-8 1 Nov 91 25 May 92 171205
In addition, the file "recent-mail" contains a copy of the currently active
section of the archive. (Unfortunately, due to the way mail works here at
the AI Lab, it is not possible to have new mail accumulate directly into
this file, so there may be some delay before a new message arrives here.)
Finally, the file "README" contains the information you are currently reading.
- Alan
From dik@cwi.nl Sun Nov 8 17:35:25 1992
Return-Path: <dik@cwi.nl>
Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA04853; Sun, 8 Nov 92 17:35:25 EST
Received: from boring.cwi.nl by charon.cwi.nl with SMTP
id AA20522 (5.65b/3.2/CWI-Amsterdam); Sun, 8 Nov 1992 22:35:18 GMT
Received: by boring.cwi.nl
id AA21966 (5.65b/2.10/CWI-Amsterdam); Sun, 8 Nov 1992 23:35:17 +0100
Date: Sun, 8 Nov 1992 23:35:17 +0100
From: Dik.Winter@cwi.nl
Message-Id: <9211082235.AA21966.dik@boring.cwi.nl>
To: cube-lovers@ai.mit.edu
Subject: Reply to old message
Going through the archives I found a message from Mike Reid, dated
August 20, where he investigates the filtration through different
subgroups. I missed the original, but he says:
> winter's filtration is 56 --> 49 --> kl --> 1. it may be the case that
> this can be improved by replacing kl with 17 , and allowing all face
> turns available in the subgroup 49. i haven't had the time to look into
> this yet.
For the record, the filtration is from Herbert Kociemba, and it is:
56 --> 49 --> 1.
So there is no intermediate stage between 49 and 1.
dik
From pbeck@pica.army.mil Mon Nov 9 11:22:43 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA12080; Mon, 9 Nov 92 11:22:43 EST
Date: Mon, 9 Nov 92 9:04:24 EST
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: yekta@huey.jpl.nasa.gov
Cc: cube-lovers@life.ai.mit.edu
Subject: ishi's phone number
Message-Id: <9211090904.aa25819@COR4.PICA.ARMY.MIL>
ISHI PRESS
408-944-9110
From gk1k+@andrew.cmu.edu Wed Nov 18 22:59:34 1992
Return-Path: <gk1k+@andrew.cmu.edu>
Received: from po2.andrew.cmu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA10302; Wed, 18 Nov 92 22:59:34 EST
Received: by po2.andrew.cmu.edu (5.54/3.15) id <AA01154@X> for cube-lovers@ai.mit.edu; Wed, 18 Nov 92 22:59:30 EST
Received: via switchmail; Wed, 18 Nov 1992 22:59:27 -0500 (EST)
Received: from pcs16.andrew.cmu.edu via qmail
ID </afs/andrew.cmu.edu/service/mailqs/q000/QF.kf2l2sm00iV200T9J7>;
Wed, 18 Nov 1992 22:57:46 -0500 (EST)
Received: from pcs16.andrew.cmu.edu via qmail
ID </afs/andrew.cmu.edu/usr14/gk1k/.Outgoing/QF.Uf2l2rS00iV289Tuo7>;
Wed, 18 Nov 1992 22:57:43 -0500 (EST)
Received: from mms.0.1.873.MacMail.0.9.CUILIB.3.45.SNAP.NOT.LINKED.pcs16.andrew.cmu.edu.sun4c.411
via MS.5.6.pcs16.andrew.cmu.edu.sun4c_411;
Wed, 18 Nov 1992 22:57:43 -0500 (EST)
Message-Id: <Yf2l2rO00iV249Tug6@andrew.cmu.edu>
Date: Wed, 18 Nov 1992 22:57:43 -0500 (EST)
From: George Cornelius Kuhl <gk1k+@andrew.cmu.edu>
To: cube-lovers@ai.mit.edu
Subject: cube question
Cc:
what is the cube root of 81?
George
From ACW@riverside.scrc.symbolics.com Thu Nov 19 09:31:59 1992
Return-Path: <ACW@riverside.scrc.symbolics.com>
Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA01606; Thu, 19 Nov 92 09:31:59 EST
Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 954415; 19 Nov 1992 09:33:00-0500
Date: Thu, 19 Nov 1992 09:32-0500
From: Allan C. Wechsler <ACW@riverside.scrc.symbolics.com>
Subject: cube question
To: gk1k+@andrew.cmu.edu, cube-lovers@ai.mit.edu
In-Reply-To: <Yf2l2rO00iV249Tug6@andrew.cmu.edu>
Message-Id: <19921119143257.8.ACW@PALLANDO.SCRC.Symbolics.COM>
Date: Wed, 18 Nov 1992 22:57 EST
From: George Cornelius Kuhl <gk1k+@andrew.cmu.edu>
what is the cube root of 81?
George
[4 /3 /16 /1 /1 /5 /1 /1 /2 /16 /1 /44 /1 /2 /1 /1 /1 /1 /1 /3 /12 ...]
From pbeck@pica.army.mil Mon Nov 23 09:35:38 1992
Return-Path: <pbeck@pica.army.mil>
Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA26950; Mon, 23 Nov 92 09:35:38 EST
Date: Mon, 23 Nov 92 9:32:46 EST
From: Peter Beck (BATDD) <pbeck@pica.army.mil>
To: cube-lovers@life.ai.mit.edu
Subject: ATLANTA Puzzle Party & exhibition
Message-Id: <9211230932.aa13919@COR4.PICA.ARMY.MIL>
1/14 - 4/10/93 there will be a puzzle exhibition
at
THE ATALANTA INTERNATIONAL MUSEUM OF ART AND DESIGN
SPECIAL EVENTS FOR PUZZLERS:
1/14 opening and reception at 6PM
1/15 eve reception
1/16 open to puzzlers only 9am - 13:00
1/17 puzzlers only
there will be puzzle trades and sales (for puzzlers)
on either sat or sun
POCs
TOM RODGERS 404-351-7744
TYLER BARREET 404-998-7432
Please distribute to all interested parties.
From @mail.uunet.ca:mark.longridge@canrem.com Fri Jan 8 00:57:44 1993
Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com>
Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA01786; Fri, 8 Jan 93 00:57:44 EST
Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10049>; Fri, 8 Jan 1993 00:57:22 -0500
Received: from canrem.com by unixbox.canrem.COM
id aa04957; Fri, 8 Jan 93 0:55:40 EST
Received: by canrem.com (PCB/Usenet Gateway)
Path-id <19938.104.159061@dosgate>; 8 Jan 93 (00:49)
Message-Id: <19938.104.159061@dosgate>
From: Mark Longridge <mark.longridge@canrem.com>
Date: Thu, 7 Jan 1993 19:00:00 -0500
To: cube-lovers@ai.mit.edu
Subject: computer cubing
With thanks to Dan Hoey for getting me on the right track, I
have finally got most of the squares group evaluted. The big
breakthru was developing a checksum for a squares position.
I know it's been done before, but I wanted to prove to myself
I could do it on a mere 386 with 4 megs of memory. My
latest program (rubik5.exe) took 24 hours to number the squares
group up to 8 moves deep. The point of all this was to create
a squares group database to aid in developing an optimal
solver for the cube. Ultimately the database will have an entry
for every squares group position, along with it's optimal
solution. I would be interested in hearing from any others who
have created such a database, and what type of compression or
checksum was used for the arrangement.
Also I've received a call from Richard Schneider. He is publishing
a comprehensive book on square 1, plus a follow-up book on
pretty patterns and shapes. This will be available in the States
shortly.
I haven't been seeing anything from cube-lovers in a while,
I hope it's still up and running.
To: Mike Reid --- Hope you see this! Any further progress been
made on God's Algorithm? I'm still trying to
catch up. I'm still interested in that code of yours to find
improvements on some pretty patterns I've discovered.
Anyways here is what my program has found so far:
Squares group (u2, d2, l2, r2, f2, b2)
Moves Deep Number of patterns
---------- ------------------
0 1
1 6
2 27
3 120
4 519
5 1932
6 6484
7 20310
8 53000 (and counting) :->
Got to improve it's speed....
-> Mark <-
--
Canada Remote Systems - Toronto, Ontario
World's Largest PCBOARD System - 416-629-7000/629-7044