Apparently correct figures.

Inspired from a mail I received, my thoughts wandered along the trail.
"How do we know if a solved SOMA is really the solution, or is it just an Imagination.?"

It started back on 3 january 2015 with a mail - as things often do.

3. januar 2012 from JaggerG
Hello Mr. Thorleif. I like puzzles. Puzzles are fun.

I have recently revisited my childhood through a Christmas gift my brother gave me: Soma cubes. He got them from MonkeyPodGames.com, and the box contained a sheet of various figures to build, including the memorial.
http://www.fam-bundgaard.dk/SOMA/SOMA1/SOMAFIG1.HTM#LINKA024
I came to the conclusion, as you did, that it is impossible
(object 7 only has 2 possible positions, forcing object 4 in the position across it, forcing object 3 on top/underneath, forcing object 5 or 6 to form a corner, any formation of the remaining 3 pieces results in a hole in an edge).

But then I started thinking: This list includes 2-dimensional diagrams of 3-dimensional objects, and is angled in such a manner that, actually, you can't be certain if the shape is disymmetrical. You can't actually prove from the image given that the front and back are mirror images. It's only implied that it's symmetrical from the y axis.
From there, I started thinking of other means of solving the particular puzzle, and ended up finding a symmetrical figure that looks just like the diagram if observed from the same angle.

Some might be considered cheating, but if that's true, then the creator of the list is cheating by providing an impossible puzzle.
Still haven't figured out 29 and 33, but I thought I recall having solved 29 in third grade, and I remember 33 being proclaimed impossible, though maybe not proven.
I've also "solved" that through alternative interpretation of the diagram, but given the name, it's probably not passable.

Thanks for hosting that site!

Of course I answered right away.

Hi JaggerG

Thank you so much for your mail
Basically you are right, there is no real way to determine the figure shape unless it is viewed from several angles.
Though there is a general consensus that the figures are mirror like if they look as if they are.
Unless they carry a note stating otherwise (ie. The figure has a missing cube inside)
When drawing the figures the viewing angle is always selected to show any irregularities.

On the other hand – as you state – there might be some great fun in making a figure that “look” identical
So, given that – your solution to the memorial is very interesting.
(If you send me the solution, I will add it to the website)

Regarding the “W” figure 33. YES it has been proven to be impossible (Of course assuming that the shape is smooth)
http://www.fam-bundgaard.dk/SOMA/NEWS/N990226.HTM
Again, If you make a “W” that looks right from the normal viewing point, then please send me the solution.

Hmm. Maybe that could be a SOMA-Newsletter about “Impossible figures that seem to be right” ?

22 days later I tried to regain contact - with no luck

Subject: Soma: Do you have solutions?

Hi JaggerG

In Januar 2012 you mailed me about solutions that are “apparently” correct.
You mentioned that you had found such semi-solutions for: The memorial and the “W” figure 33.

Is it possible that you could write down these solutions (or draw them, or photo) ??

And so - the "Trail grew cold".
Now the question of course is - can you find 'this' solution.?
If so, then send me a mail - and I will post it here. :-)

The solutions.

We are now 2017-04-18 and I received a good answer from my friend Bob Nungester.
Bob has found 3 figures that may be build so that they "look right" without being right.
He wrote:

I was reading through the newsletters and saw the one on Apparently correct figures.
That looked like an interesting problem, so I tried it.
Actually, I cheated and used the Windows SOMA solver to do this.
I analyzed the Memorial, W Wall and Skyscraper figures and found that:

Semi-solution for: The memorial.
The Memorial has 15 visible cubes and 12 hidden.
In the standard view there are only two spots where any hidden cube can be moved and not be visible, or disconnected from the main figure.
Of the 12 hidden cubes, seven of them can be relocated to these possible move sites and the figure is solvable.

/SOMAMemorial1?         /SOMAMemorial2?         /SOMAMemorial3?         /SOMAMemorial4?
/...../..2../..3..      /...../..6../..6..      /...../..7../.1...      /...../..6../..6..
/...../.426./1.33.      /...../.667./4477.      /...../.177./.173.      /...../.662./7222.
/..2../14266/14376      /..2../22255/.4475      /..6../56644/55233      /..5../74551/77.11
/...../.557./.477.      /...../.115./.333.      /...../.644./.523.      /...../.445./.333.
/...../..5../..5..      /...../..1../..3..      /...../..2../..2..      /...../..4../..3..

/SOMAMemorial5?         /SOMAMemorial6?         /SOMAMemorial7?
/...../..6../..6..      /...../..7../.77..      /...../..2../..2..
/...../.662./7222.      /...../.661./.37..      /...../..71./5521.
/..3../73355/77345      /..6../44611/33255      /..7../57716/54266
/...../.115./..44.      /...../.445./.325.      /...../.333./.446.
/...../..1../..4..      /...../..2../..2..      /...../..3../..4..

Semi-solution for: The “W” figure 33.
The W Wall has 25 visible and only 2 hidden cubes,
but there are 12 spots available to move a hidden cube.
Only one of the hidden cubes can be moved and have a possible solution (the bottom center cube).

/SOMAWWall?
/....3/....3/....3
/....5/....3/....7
/..255/..457/...77
/..2../..4../.64..
/112../612../664..

Semi-solution for: The Skyscraper.
The Skyscraper has 19 visible and 8 hidden, with 8 hiding places.
Five of the eight hidden can be solved.

/SOMASkyscraper1?
/.../.../.../.../.../.../4..
/2../2../22./66./67./45./455
/.../.../11./16./77./47./.35
/.../.../.../.../.../.../333

/SOMASkyscraper2?                        /SOMASkyscraper3?
/.2../.2../.21./.51./.57./.44./44.3      /.2../.2../.24./.44./.47./..5./6655
/..../..../.21./.55./.77./.67./.633      /..../..../.21./.11./.77./.67./.635
/..../..../..../..../..../..../.663      /..../..../..../..../..../..../.333

/SOMASkyscraper4?                        /SOMASkyscraper5?
/.2../.2../.21./.11./..6./.75./7755      /.2../.2../.26./..4./.14./.75./7755
/..../..../.24./.44./.46./.66./.735      /..../..../.26./.66./.14./.14./.735
/..../..../..../..../..../..../.333      /..../..../..../..../..../..../.333

Some thoughts about Apparently correct figures.

The date is: 2017-08-07 and Bob Nungester notes.

Regarding the "Apparently correct figures" I noticed that this came up in Vol. 2, No. 1 of the SOMA Addict. Newsletter 1999.01.18
The first article is titled "W-Wall Is Possible!".
It mentions Jean Paul Francillon of Montreal Canada who submitted a solution that looks correct from the front,
but not from the top or back. This is "Apparently correct figures".

They also note two others who submitted correct solutions by using a double set of SOMA pieces.
So I ran the W-Wall through the Double SOMA program (See the Newsletter 2000.01.12 Solver )
and it found a solution using two each of pieces 2 and 7, and neither of pieces 4 or 5.

       /..773/..733/..223
       /..7../..1../..2..
       /LLL../P11../PP2..
       /L..../T..../P....
       /T..../T..../T....

Regarding the two correct solutions, the article states "They sit before us, as we write. We have only two SOMA sets
to work with, we assure you." This implies that both solutions can be built at the same time using a double SOMA set.
However, I ran a figure containing two separate W-Walls through the "Double SOMA program" and it took 20 seconds
to analyze and say it's impossible.

      /SOMADoubleW-Wall
      /..***..../..***..../..***....
      /..*....../..*....../..*......
      /***....../***....../***......
      /*......../*......../*........
      /*.....***/*.....***/*.....***
      /......*../......*../......*..
      /....***../....***../....***..
      /....*..../....*..../....*....
      /....*..../....*..../....*....

So it looks like there are at least two ways to make a W-Wall with two sets of SOMA pieces, but you can't make two
at once with two sets.
Using a program doesn't actually prove it's impossible, but can anyone prove it? (or find a solution)?

Starting 1971 - now 2012/2017, it's interesting to see how a problem may persist for many years,
being attacked several times, and then when time is ripe, the solution (or falsification) show itself.

More searching with the program found five solutions to the W-Wall that involve swapping of only a single piece
from another set of SOMA pieces. These use:

Extra piece  No piece
    2         4
    2         5
    2         7
    3         4
    3         5

The SOMA Addict article mentions that Mr. Naylor had the simpler solution of the two submitted,
so it must have been one of these. I think these are the only possibilities swapping only one piece,
other than the obvious reflected solutions using no piece 6 instead of no piece 5.

The other solutions found by the program involve swapping two pieces. These are:

Extra pieces  No piece
  2 and 5      6 and 7
  2 and 7      4 and 5
  2 and 7      5 and 6
  3 and 4      5 and 6
  3 and 5      4 and 6

Any of the solutions can be reflected, which means changing 5 to 6 and 6 to 5 in the above descriptions
will technically give a new solution.
It's difficult to force the program to use duplicates of specific pieces and none of others so there may
be a few other double-swap possibilities, but none that allow both sets to make a W-Wall after the swap.
I didn't check any swapping of three pieces at a time.

I then decided to investigate the mathematics involved in swapping pieces and got into some interesting work
with combinations. The number of combinations noted below applies to any SOMA figure formed using two sets of
pieces, not just the W-Wall.

The number of combinations of N elements from a set of R elements, when order doesn't matter is R!/(N! * (R-N)!).
I did have to look that up. Looking only at SOMA pieces #2-#7 there are six elements to choose from.
Using the formula, when swapping one piece there are 6 choices to give up (choose 1 of 6) and 5 to get back,
so the total possibilities are 6*5=30.
When exchanging 2 pieces there are 15 combinations of 2 pieces to give up (choose any 2 of 6) and 6 to get
back (choose 2 of 4), so the total is 15*6=90.
For 3 there are 20 choices to give up (choose 3 of 6) and only 1 choice to get back (choose 3 of 3),
so the total is 20*1=20.

I ran through all 30 of the 1-piece exchanges and all 20 of the 3-piece exchanges.
Excluding ones where piece 6 was exchanged (unless 5 and 6 were exchanged together) due to reflective
symmetry considerations.
I believe there are now 21 1-piece exchanges and 14 3-piece exchanges to check.
This analysis resulted in the same list of 5 solutions for 1-piece exchanges given above.
I didn't run through all the 2-piece exchanges since there are so many of them. I believe there are 36 that can
be eliminated due to reflective symmetry of pieces #5 and #6, but that still leaves 54 possibilities to check.
I found two more 2-piece exchanges that work, but finally lost interest.

The key point found when running through the 3-piece exchanges is that only ONE gives a successful solution.
This creates an interesting problem if anyone wants to solve this without looking at the solution.

Given two sets of SOMA pieces, give up any three of pieces #2-#7 and replace them with the three
other pieces from the other set.
In other words, you end up with two each of three of #2-#7 SOMA pieces, plus the #1 piece.
There are 20 ways to do this selection, but only one allows building the W-Wall.
The problem is to find the one combination of pieces that allow construction of the W-Wall.
For a hint, the existence of only one solution means certain combinations using piece #5 and/or #6
are impossible since they would give either zero or two valid combinations.
This reduces the number of possible combinations from 20 to 8.

I did save this one solution

       /SOMAW-Wall3Substitutions
       /..773.../..733.../..223...
       /..7...../..1...../..2.....
       /LLL...../P11...../PP2.....
       /L......./T......./P......V
       /T......./T......./T.....VV
       /......../......../........
       /......../.......6/..44..66
       /......../......../.44...6.
       /......../......../........
       /......../......../........
       /......../.......B/..ZZ..BB
       /......../......../.ZZ...B.
       /......../......../........
       /......../......../........
       /......../......../.55...AA
       /......../.5....A./.5....A.

The figure shows the W-Wall figure and separately all the unused individual pieces off to the side.
This way the program is forced to use only the remaining pieces to solve the W-Wall.

Inspired by JaggerG. <jaggerg@gmail.com>
Augmented by Bob Nungester <bnungester@comcast.net>
Adjusted by Thorleif Bundgaard <thorleif@fam-bundgaard.dk>

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