Message #1462

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Re: slicing up MagicTile puzzles without triangle vertex figures
Date: Mon, 28 Feb 2011 14:38:07 -0800

On 2/28/2011 1:05 PM, Brandon Enright wrote:
> "Matthew"<damienturtle@hotmail.co.uk> wrote:
>
> […]
>> Another is to use the "fudging" concept (see
>> http://www.shapeways.com/model/204721/futtminx.html?gid=ug13603 for
>> one example, though the idea is a little different here) started by
>> Oscar van Deventer to tweak the geometry and remove small pieces if
>> so desired, if any ideas from this are implemented.
> Speaking for my own aesthetics, I don’t think fudging has much of a
> place in the world of computer puzzling. Cuts that require pieces to
> be fudged to eliminate the infinite cascade of pieces bug me.
>
> On the topic of fudging though, there was a long rambling discussion
> about fudging, what it is, etc here:
>
> http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=20105
>
> Most of the interesting discussions revolve around the posts by user
> ‘wwwmwww’ (Carl). Woven in with the fudging discussion is a debate
> about what it means for a puzzle to be "deep-cut" which I don’t think
> was resolved.
>
> The interesting animations don’t start until about mid-way through page
> 2.
>
> In short though, fudging is a messy, seemingly non-mathematical trick
> to make cuts that shouldn’t work actually work. Considering how many
> great puzzles there are that do work, I think we shouldn’t focus too
> much energy on fudging puzzles that wouldn’t work right otherwise.

I’ve been thinking about this for a while and agree that this is a
question of aesthetics, but I can see how different viewpoints can be
equally valid mathematically. Matthew’s link shows a twisty Buckyball
which makes for a nice-looking puzzle. It took me a while to see what
was fudged about it. The pentagonal faces give no problems but the
hexagonal faces are only completely kosher for 120 degree turns. 60
degree turns of the hexagonal faces move some stickers from hexagonal
faces to and from pentagonal ones. Given my aesthetic preferences, I
would simply not allow those 60 degree twists and everything would be fine.

I guess I would call the two valid definitions "geometric" versus
"permutation" orientations. If we think only in terms of generating
puzzles by slicing uniform polytopes, then we’re taking a geometric
orientation and find ourselves struggling over tiny inelegant pieces
that are sometimes generated. However if you are more interested in the
group theoretic aspects of twisty puzzles, then you don’t need to worry
about the geometry at all. From that point of view, "fudged" puzzles are
just nice visual aids that embody the essence of a particular puzzle
group, and any fudging needed to build one is completely unimportant.

One thing that I think we can all appreciate are those puzzles which
happen to satisfy both points of view, but beyond that we just have to
pick one or the other perfectly reasonable sides. This exercise has
clarified some of my previous feelings of ambivalence that I wasn’t
quite able to settle. My particular problem was with the subject of
piece intersections during twisting. Some people on this list are
bothered by twisting geometry that doesn’t smoothly slide in either the
model space or the viewing space, but that issue never bothered me. On
that point I was only interested in the geometric result of twisting.
Now I see that in those cases I was interested in the resulting
permutations and therefore didn’t care what happened during the twist,
however my geometric aesthetic governed my interest in particular puzzles.

On a related note, I recently received the face-turning octahedron
puzzle that I ordered from the link
<http://www.hknowstore.com/item.aspx?corpname=nowstore&itemid=058fbd01-4a44-4e6f-99ec-71ae3bd9eb23>
that Nan posted, along with a floppy cube. The connection to this
discussion is that the plastic of the floppy cube bends strangely as
parts of the central cube gets pried open while twisting. It’s a very
odd mechanism that may not stand up to a lot of use, but then the puzzle
is not at all floppy, possibly because of that very design choice. I
don’t want it to break, but the bending doesn’t bother me at all if that
is what allowed them to build a non-floppy 3x3x1 puzzle.

Just for completeness sake, I can report that the octahedron puzzle is
quite nice though it does take some care to keep some twists from
locking up. I really wish that the twists would snap into place like my
Chinese megaminx, but that’s not a big problem. The most surprising
thing I found about this puzzle was just how quickly it becomes
scrambled. Even a single twist from the pristine position is not
immediately clear that it is only one twist, and two intersecting twists
are not nearly as obvious how to untwist as similar twists on the
original Rubik’s cube. It is a very handsome puzzle and makes me
appreciate the humble octahedron all the more.

-Melinda