Message #1340

From: "Galla, Matthew" <mgalla@trinity.edu>
Subject: Re: [MC4D] noteworthy new hypersolutions
Date: Sun, 23 Jan 2011 02:15:49 -0600

Hi everyone,

Thanks for the mention and congrats on completing the 120Cell. In my email
to Roice I mentioned several things about my solve that he suggested I
forward to the group:

"By 4 separate samples I took throughout my solve, I estimate the actual
time of completion to be right around 150 hours. To be honest I am quite
frustrated the solve took as long as it did, but as a college student, I
have very little free time to begin with and it is difficult to dedicate so
much time to something that remains so repetitive for large amounts of time.
As you may remember, I solved the puzzle by piece type, 2C, 3C, and then 4C.
Partway through solving the 3C pieces I lost a significant portion of my
solve and I had to just about restart solving the 3C pieces. During this
time I actually interacted with the log file, opening it up and seeing how
the information was stored. After doing this I decided to write a simple
Java program that would scan through the current state of the puzzle and
give an estimate of how many of each type of piece I had solved, along with
a percentage completed of the entire puzzle. After writing this program I
discovered that 2 2C pieces were actually flipped in place, one of which was
two slightly different shades of green, the other two slightly different
shades of off-white/light gray. I decided to leave these until the very end
of my solve. They were corrected in the final 40 moves.

You might find it interesting to know that I solved the 2C by layer, the 3C
by ring, and the 4C by hypercube cell, thereby incorporating every given
symmetry into my solve except the tori. After our conversation about
building up solves in local groups to minimize the distance between the
remaining unsolved pieces I solved all of the 4C pieces in odd numbered cube
cells first and then finished up with the evens (so my solving order was 1 3
5 7 2 4 6 8, but by the time I got to 6 just the stragglers remained
["hypercube cell" 9 was solved automatically as no 4C piece lies exclusively
in those 8 cells]) I can actually confirm that your hypothesis holds true
and makes a very noticeable difference while solving. Once I got to the even
numbered hypercube cells, my solving speed nearly doubled with the improved
proximity of the remaining pieces. Had I realized this early, I would not
have solved the 2C pieces by ring as this doesn’t consolidate unsolved
pieces well at all (but then I couldn’t say I solved by rings either so
maybe it was worth it).

As an experienced enthusiast not only in playing with and solving Rubik-like
puzzles but also in the group theory and related mathematics behind the
puzzles, I believe the most "naturally correct" way of counting moves in any
Rubik puzzle is face-turn metric and for the 120Cell this means any number
of clicks made on a single cell should count as a single move. From
experience solving the 120Cell any desired rotation of a cell can be
accomplished in exactly one click EXCEPT one that requires 2/5 of a rotation
about a pentagonal face piece. Interestingly enough, from experience it
seems to be possible to never require such a turn in 2 moves to get any
given piece to a specific destination 1 cell away. There are typically 3
ways to pass a piece from one cell to the next and if one of these three
ways requires a 2/5 rotation about a pentagonal face of the new cell to get
the piece to its final destination, then it seems guaranteed that another
way will also require a 2/5 turn but the third way will not. So starting
from the point in my solve where I lost so much of my log, I successively
avoiding using 2/5 rotations during any setups or algorithms. This meant
there were MANY times that I attempted to do a set up only to find I would
need a 2/5 rotation so I would undo and try one of the other 2 possibilities
of manipulating a piece along the same "path" of cells. Half of these
resulted in another 2/5 rotation so once again I would undo to use the final
choice instead. The completed log does not show the hundreds of setups that
were rejected due to a 2/5 rotation ;) However I was occasionally forced to
use a 2/5 rotation whenever a setup required only one "movement" of a cell.
Although this felt like one true move, it of course counted as two since a
2/5 rotation requires you to click twice. If any number of clicks on a cell
counted as only one move, this could have saved me A LOT of time as I could
have always accepted the first setup I came up with and skipped the checking
process. It would be interesting to see how Noel’s solve compared with mine
in move count if moves were counted only when a new cell was clicked on. I
don’t know about his solve, but since I specifically set out to avoid them
in my solve, the decrease in move count may be much less for me and may even
make his solution shorter than mine. Then again maybe I shouldn’t tell you
about this and just silently let the numbers stand as is ;) After all, a
good 30 hours of my solve was probably dedicated purely to undoing and
finding new setups for the purpose of saving 2 moves by avoiding a 2/5 turn.
"

-UPDATE-

After this conversation, Roice wrote a program to calculate the number of
moves required by each of myself and Noel using the above metric (where any
number of consecutive clicks on a single face only counts as 1 move). Sadly,
I must report that under this metric, my solution is actually LONGER than
Noel’s! (looking through parts of Noel’s solve, it seems he was reluctant to
use any rotation except those around the pentagonal faces of each cell,
while in my solve, I explicitly did everything I could to avoid using more
than one consecutive click on a single cell).

The exact move count under each metric is given below:
Original Metric (analogue of quarter turn metric)
Noel-33,546
Matt-23,185

My Suggested Metric (analogue of face turn metric)
Noel-22,576
Matt-22,856

Despite its size and the time it took me to solve it, the 120Cell really
isn’t that hard, as I’m sure some of you have already concluded. Yes it is
4-Dimensional, and certainly that makes it inherently confusing. Yes there
are a large number of pieces, and certainly that makes it a very, VERY LONG
puzzle. But when it comes down to it, the moves are SOOOO shallow compared
to the whole puzzle that you have an extraordinary amount of room to
manipulate the pieces however you want. Solving the puzzle by piece type
basically amounted to repeating the same process up to 600 times, and then
finding a little bit of excitement when the last few pieces of the current
type required an extra little bit of trickery to get into place and/or
reorient. The main reason this solve took so long was honestly, in the midst
of solving the 1200 3C pieces, it got quite boring and I had to constantly
force myself to reserve an hour or so each night to make progress. I would
keep this up for maybe a week or two before eventually dropping it, picking
it up again a few months down the road (often times, the only motivation I
had during this time was a desire to not see my previous work on the puzzle
be wasted). But I am glad I stuck it out because I can now say I am one of
two people to have "walked the walk" and solved the entire puzzle by hand
(although I suspect many others have already found a minimal set of
algorithms that legitimately prove they could solve the puzzle).
Based on this observation, I have been experimenting with other 4D shapes
and trying to visualize the 4D analogues of other twisty puzzles like the
Skewb and Helicopter Cube. I am about to share some of my results on these
in another topic :)

I want to thank Roice for writing the Magic120Cell program and giving me the
opportunity to complete a journey so rare, it has only been completed twice
:)

-Matt Galla

PS If anyone is curious to know, I am a 20yo undergraduate pursuing a double
degree in Mathematics and Engineering at Trinity University in San Antonio