Date: 10 Dec 1981 1903-PST
From: ISAACS at SRI-KL
Subject: Ten Billion Puzzle (the Barrel)
To: CUBE-LOVERS at MIT-MC

  It just happens that the day before I got the Nourse solution to the
Ten Billion (Barrel, Magic Barrel, etc), I got another solution by Naoaki
Takashima, that he presented to the ARMJ (??) meeting in Japan.  His 
notation:  
Columns and stages are defined             5-   x   x x
as shown on the right.                 s   4- x x x x x
                                       t   3- x x x x x
                                       a   2- x x x x x
                                       g   1- x x x x x
                                       e
                                       s      | | | | |
                                              1 2 3 4 5

                                             columns
He moves the drums, rather than the plunger.
   U--move drums up.       D--move drums down.
   Ru (R sub u) Rotate upper drum right one column; superscripts for number
   Rl (R sub l)   "    lower  "     "    "    "     of columns
   R for upper and lower together to the right
   L for left similarly.

 At any rate, a cumbersome notation for computer writing.  What's more, unit
move people (who don't like slice moves or half turns on the cube) won't like
combining the two drums (rings).  BUT since I hold my barrel by the bottom
plunger, I do find it easier to think of moving the rings up and down, rather
than pushing the plunger.  I like rings better than drums, but columns better
than notches.  And I would rather take standard numbering to start from 
what is above defined as column 2, ie, the one under the neighborless plunger.

  So, all in all, I would like to propose accepting Don Lynns' notation
with the following changes:  U and D would refer to the rings moving up
and down, instead of the plunger.  "Notches" would become "columns".
The picture above, moved to reflect the new columns, could be used when it
is needed for clarity.  

* * * * Small spoiler * *
  The Takashima solution is really based on one move, which cycles 3 balls
more-or-less vertically:
  let X = U T D -T / U B D -B / (TB) U -(TB) D
  then  X2 moves 3F TO 4F TO 2L TO 3F;  ie, the 3-cycle (3F,4F,2L).
* * * * End of small spoiler * * *

 I already propose apostrophe instead of -, so the above becomes:
   UTDT'-UBDB'-(TB)U(TB)'D    which I think is a little clearer.

Notice that the above can be simplified to:
     UTDT'UBDTUB'DT'UBDTUTBD  (I think.  I find it hard to think about in
                               such small pieces; I like the less efficient
                               chunks at first.)
* * * * * * * * SPOILER WARNING * * * * * * * * * * *SPOILER WARNING * * * *

Anyway, with this move alone (plus some playing at the beginning), I can
solve the puzzle as follows:  First get the three black balls at the top,
under the plungers, any old way.  Then put 5 different colors in the first
layer, moving only the bottom ring (to leave the black balls alone).  Then
use the move above to move in the matching second layer from the top (layer
4).  When necessary, use the same move to get from 3 to 4 first, or even from
2 to 4.  Using the inverse move can speed things up, but just doing it twice
is easier on the memory.  Move the space in layer 2 you want to fill, to column
5 (new numbering; col. 1 in above diagram, or "2L" in notation), by moving
the lower ring; move the column containing the ball you want in that space
to col. 1 (F), apply the move once or twice, and move the rings back.  Rest
position has the barrel down (and the plunger up).  Once the second layer
is complete (or, for efficiency, complete except for 1), do similar things
for layers 3 and 4, using the second layer for temporary storage.

* * * * * * * END OF SPOILER * * * * * * * * * * * * * * * * * * * * * * * * *

  Don Lynn:  I would be very interested if you would summarize Nourse's
approach, and either send it to me or to cube-lovers.  I have not had the
time to break it down into chunks to see what basic moves he really uses.
I find that to be the main trouble with most of the solution books to 
the cube as well as this - that they offer long sequences to memorize, with
little thought to learning what is going on, and what the logic of the moves
are.  In Nourse's solution, what are the basic pieces of his moves. Can they
be expressed (as above) as a "simple to comprehend" set as well as a "shortest
number of unit moves" way.  By use of parenthesis (or maybe slashes), U and D
could probably be eliminated; at each slash, switch from up to down or vice
versa.  Would this be useful notationally.  What are some basic algorithms
to exchange 2 balls on a single layer or in a single column?  To cycle a
layer or column?  
  Enough questions for tonight.  When's the first Barrel contest?
   --- Stan
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