SOMA News 14 Aug 2000 + 5 April 2021 E-Mail.

# Is there a SOMA+plus ?

When you have played the SOMA puzzle for a while, many players come up with the next logical question.
"Is there anything beyond SOMA" ? And indeed there is.
But the alternatives are not always to our liking.

• Pentominoes: using 12 flat pieces.
• Pentacubes: using 29 pieces.
Now these puzzles seem like expansions of SOMA, but they all lack the inherent simplicity we so love in SOMA. And most of them will take almost forever to solve just the simplest constellations.

Well - Not that a puzzle should be easy. BUT if a puzzle takes too long to solve, it clutters your mind, you cannot let it go. And inevitably it will lead to a lesser selfesteem.
On the other hand, our beloved SOMA, is both complex, but also simple enough to enable most of us "Normal people" to have SUCCESS in solving many of its puzzle questions. And having fun with all the pretty figures we can build.

Could we think of something between SOMA and Pentominoes, that still leaves both the beauty and the logic of SOMA.?

Until last month, I would have answered NO.
But then a German SOMA player pointed me in a totally new (and extremely logic) direction. !!

Volker Pöschel, teacher in comprehensiv-school. Email: PoeGot@aol.com showed me the "Herzberger Quader" discovered by Mr. Gerhard Schulze (Germany)

Lets first go back to the original idea of soma. As it is described in Newsletter 99-03-08 .

Polyominoes are a special set of figures, that may be made, by joining squares at their edges. If these squares are cubes, we can make 3 dimensional shapes, by joining their faces.

Let us now examine all the possible combinations of 1,2,3 and 4 squares. Allowing them to bend into 3 dimensions, we get:

THIS is where SOMA is defined.!!!!
If we now select all the irregular figures, (ie. not straight connections) we get the 7 SOMA pieces.

Let us INSTEAD select ALL figures (Except the singular cube) Then we get the 'Herzberger set' which I choose to call;

SOMA+plus.

With these 11 puzzle pieces we still retain the original 7 SOMA pieces V,L,T,Z,A,B,P and additionally gain 4 new pieces D,I,Q,S. In total:

 (V) Shape 'V' (L) Shape 'L' (T) Shape 'T' (Z) Shape 'Z' (A) Left Hand Turn (B) Right Hand Turn (P) Shape 'Pyramid' (D) Duo (I) shape 'I' (Q) Quad stick (S) Square 'S'
( It is 'I' because the 'T' for Trio was already in use. )

I suggest these piece colors.

This SOMA+plus set give us a whole range of new puzzling tasks.

(More SOMA+plus figures are found in the figure collection. SP Figures .)

Note: That we have a SOMA+ solver program Available here

 ``` SOMAplus 001 The 2,4,5 Base /BSSTP/BZZPP /DSSTT/BBZZP /DVVTL/AALLL /AVIII/AQQQQ``` ``` ``` ``` SOMAplus 002 The dual towers. /P...../PP..../BBZZVV /....../PB..../QBTZZV /....../....../QTTTSS /....../AA..../QIIISS /A...../AL..../QLLLDD``` ``` ```

Now - if YOU make any nice figures with SOMA+plus, then please send their description to me.
Initially, I suggest that the figures should use ALL 11 pieces.

## The PRE history:

As I acquired my Granddad's SOMA i found that this was just enough. It is only 7 pieces, It is manageable.
Then after some years I got inspired by a German puzzler who pointed me to a puzzle made in a very limited set.
A teacher in Germany had discovered a puzzle in 1984, intended for the 800 year anniversary of the town Herzberg.
At that time, the city was part of East Germany. They called the puzzle the "Herzberger Quader".

I was inspired, and used a more general name when I started to experiment with these shapes, the SOMA+Plus (Or just SOMAplus+)
Take ALL the possible combinations of multible cubes up to 4.
I found that it would make for 11 pieces, and at total of 40 cubes.
The single cube was left out for three reasons
- 1: It is not a polycube.
- 2: It has no connection. And
- 3: If included, the cube count would be 41, a prime number, indivisible.
So we would not be able to make boxes and such.

This puzzle has many additional interesting aspects.
BUT at the same time it remain on the simple side - It is Manageable :-)
SOMA and the SOMA+ are something to enjoy without taking too much time for each figure.

By the way. Bob Nungester has also made a PC solver program for the SOMAplus The 2023.07.14 Newsletter The newest advanced SOMA & SOMAplus programs.

# The story of the Herzberger Quader.

2021-01-06 6:18 I received a mail from my friend Bob Nungester. Continuing some talks we had while Bob was programming the new SOMA+ solver program.
Bob had also found that there existed some magazines (Called alpha) describing both the history as well as figures.
Fortunately, I also speak German, so I checked these magazines and found that here was a lot of interesting information.

• A teacher's guide with links to various tasks in spatial geometry using the Herzberger Quader (SOMA+)   [Very fine material - in German]
Herzberger Quader papers for a teacher

2021-04-02 During some months now, I have learnt more about the Herzberger Quader. ( The puzzle I have named SOMA+plus, or just SOMA+ )
It seems that there are some more information to this inspiring version of our SOMA puzzle than first met my eyes.

Much of this information is available in German, so along with the English translations and explanations. I will also give you the option to read the German versions if you like.
Though I will not translate it all. In some cases the pictures speak for themselves.
Do also note that you can either use Google-Translate. Or maybe just look at the German text with an open mind.
Of course the word sequence is often differing from English, but still some meaning should be immediately recognizable. Look at this.

```English: "This is  my   house and I   live here, seing my    friends and hear music"
German:  "Dies ist mein Haus  und ich lebe hier, sehe  meine Freunde und höre Musik"
```
Now, that wasn't difficult was it ?.

 Actually the gray at the top is partly Denmark, I live approximately HERE      * Germany [Deutschen Reich] was a very large country untill the end of the World War II.   The time of the "Nationalsozialismus" (1933 through 1945) ended, and the 4 winning powers   met at Berlin. They were USA, England, France and The Soviet Union.   They divided Germany i 4 sections, but where the three western sections was combined   into one country "Germany". The fourth section remained controlled by the Soviet Union   This part became the DDR Deutsche Demokratische Republik [German Democratic Republic].     The Republic ended on the day 1989-11-09,   and was later reattached with the western Germany, to form what we know today with   the name Germany.     Note the position of the town Herzberg.     Obs: There are more than one Herzberg (Hearts Mountain) in Germany.           But this is Herzberg am Elster (Herzberg by the river Elster)

## But now the Herzberger Quader:

The creator of the Herzberger Quader [cuboids] is Oberstudienrat [Senior teacher] Gerhard Schulze (1919 - 1995),
he worked intensively with mathematical games as an extracurricular activity from 1982 - 1994.
 Gerhard Schulze, worked at the   Philipp Melanchthon Gymnasium in Herzberg.

At the 800th anniversary of his hometown Herzberg he developed the Herzberger Quader
And it was thus first produced in 1984. When it was also presented to the general public.
In the mathematical school magazine "Mathematische Schülerzeitschrift alpha" Gerhard Schulze also published several articles about the Herzberger Quader.
• alpha 1984/6 See page 128.       That is page 10 in the 'N000814-A84_6.pdf'
• alpha 1986/3 See page 66+67.   That is page 20 + 21 in the 'N000814-A86_3.pdf'
• alpha 1990/6 See page 137.       That is page 19 in the 'N000814-A90_6.pdf'
• alpha 1991/5 See page 22+23.   That is page 18 + 19 in the 'N000814-A91_5.pdf'

Since 2008 the Herzberger Quader has been used in special school work, by interested Mathematics teachers.
These documents were mentioned above, but here is a copy of the figure sheet. "Arbeitsblätter".
• Baupläne   ' N000814-Bau_was_2.pdf '
 The Herzberger Quader, as it was on the occasion of the anniversary 50 years high school graduation in Herzberg presented by the Friends of the "Philipp-Melanchton-Gymnasium" The Herzberger Quader is a game to develop the spatial imagination.   It has the size of 40 unit cubes (5 x 4 x 2) and consists of eleven different poly cubes.   All poly-cubes from the bi-cube to the Tetra cubes.   The game is similar to the Soma cube, if you use only 7 of its pieces.

On 2009-02-24 The history of the Herzberger Quader was told in an article, when the newspaper 'Lausitzer Rundschau' re- presented the Herzberger Quader.
[The highlighted words in the middle are on my account, because this folder can be seen in this Newsletter.]

NOW in january 2021-01-06 my friend Bob Nungester, tried a hunch and contacted the US sister City to Herzberg, which ment contacting "Sister Cities Association in Dixon, Illinois" to see if they have a copy of the Herzberg Quader instruction book ... and they did!
 Open in a separate Tab, to Zoom. The contact was made to "Ellen Mumford" , (President of Dixon Sister Cities Association) who has one of the original Herzberger Quader, including its booklets. Meaning both the German version and the small leaflet written in English.

AND ... Ellen even made photos of both the puzzle and of some of the booklet pages,
 The Jubilee box of the Herzberger Quader The content The 11 pieces of the puzzle

Initially we received 4 document photos.
However Bob Nungester had an accord with Ellen Mumford, that would give us access to all pages of these documents.

NOW: 2021-05-20 Bob received the original herzberger Quader puzzle, and made a set of really nice scans of the booklets.
So it is now possible to download the complete original booklets HERE. ↓

## The English supplement

 First page of the English version Page 2 and 3 of the English version Page 4 of the English version The complete English supplement folder.

## More details of the Herzberger Quader

As Bob Nungester got his hands on the Quader, he also made carefull notes of the puzzle, as well as a long number of nice photos.

These pictures can be enlarged:
Right_click the image and select "Open image in a new tab" and thereafter 1 left_click to enlarge.

Dimensions:
I often hear stories from people who make their own puzzles. So maybe the dimensions of this puzzle will be interesting.
Bob found a centimeter ruler (Europeans use the Metric system, whereas Americans 'mostly' use Inches [1" = 2,54 cm]).

Bob writes.
The Herzberger Box has the millimeter reading 106.5 mm wide x 86.6 mm deep x 55.5 mm high.
The sides are 5 mm thick with a 2.5 mm deep groove for the top lid.
The bevel at the front top of the sides cuts a 45 degree line with the ends 8.5 mm from the corner.
The top is 3 mm thick and is 100.5mm wide x 84.0 mm deep.
The front side that the top slips over measures 47.1 mm high.
All the cubes have 17.5 mm edge length.
I knew all that engineering training would be useful someday :-)

I was surprised that the dimensions of the Quader aren't some integer of centimeters.
Maybe lumber in Germany has properties similar to our American lumber where the finished piece is smaller than the stated starting dimensions of the rough wood.
For example a 2x4 here is actually 1.5" x 3.5" and thinner boards listed as 1" thick are actually 3/4".
Maybe 2 cm x 2 cm lumber is actually 1.7 x 1.7 as a finished piece.
Anyway the puzzle is fairly small and easily fits in one hand.
The manuals are equally small as well. Each page measures about 7 x 9 cm so the printing on the page is quite small.
In the PDF file you can see a standard sized staple in the German manual between pages 8 and 9.
Figures:
How about the figures then, if you are like I am, then you would certainly want to see some real SOMA+Plus figures.
So here is what Bob Nungester has made for us:
 Figure SP025 Figure SP001 Figure SP010 Figure SP002 Figure SP006 Figure SP022

During all this, Bob made some interesting observations, he wrote ...
The puzzle is actually smaller than we thought. Each small cube is only 1.7 cm wide,
and the box containing all the pieces has exterior dimensions of appr. 5.6 x 8.7 x 10.6 cm. (All dimensions are noted in the previous paragraph)

There are two rules that they seem to have followed when assembling the 11 pieces:

1. The wood grain will never align when the small cubes are glued together.
2. After following rule #1, maximize the number of faces with end grain exposed.

These had to be rules for the initial construction because the odds of #1 happening randomly are less than 1 in about 192,000. The odds of #2 happening randomly are less than 1 in at least a million.

This drawing show the location of end grain on each of the pieces.

The one hidden cube on piece 7 has end grain pointing to the right,
and piece \$ has it pointing to the left (that's why this image is named SomaPlusGrain7R\$L).
The maximum number of faces that can have end grain exposed is 76.
4 end-grain faces have to be at glue joints (pieces 4,7,\$).
The image shows 75 so I imagine there was a slight error in the assembly.
The fourth cube on piece 3 has an end-grain face glued to the central cube. If it had been rotated 90 degrees both end-grain faces could be exposed in an up/down orientation while still satisfying rule #1.
However I doubt most people would notice these things.
Thorleif think:
"I could imagine that the command was. Glue the cubes together, But never have two end grains besides each other."

## 2021-01-10 Mails regarding Herzberger Quader

In the process of acquiring all the pages of the booklets, we had the following positive communication:

Email from Bob to Ellen
We looked at the manual in more detail and have a couple of questions.
The pictures show three single pages and one double page.
Is that a 4-page manual (one folded sheet of paper) and a separate single page, or are there other pages?
In other words, the manual must have an even number of pages, but 5 pages are shown.

The last page of the manual says "One particular subset of polycubes (number 11 in the booklet) gives the well-known SOMA Cube".
Is there a booklet with the set too?
It seems to indicate there's a separate booklet with drawings of various figures that can be built.

This is probably the only Herzberger Quader we'll ever find,
so I'd like to get pictures of everything that's available, if possible.

Email from Ellen to Bob
It's 4 pages. The 5th is the first page of the German instruction which is a separate booklet.

Email from Ellen to Bob
I am in Florida at our winter home and won't return until May 1. The drawings may be in the German instruction.
I did not look at that. I can send it all to you in May if you wish. That is, if you will return it.

Email from Bob to Ellen
I would be happy to see anything you'd send, and yes I would definitely return it.
However, I think just good pictures of all the pages is all I'm really looking for.
Whatever's easiest for you would be fine for me. My friend Thorleif in Denmark speaks German so he can easily translate the booklet.

That's a long trip! Have a good time. It's been decades since the Quader was first created, so there's no rush at this point.
If you could send me pictures or the booklet in May I'd appreciate it.
Since it's a while in the future I'll make a note to send a friendly reminder if you get too busy and forget in May.

Email from Ellen to Bob
I'll send you the entire puzzle and the contents.

Email from Bob to Ellen
Thanks so much. I'll just take good photos of everything and then send it back.
Thorleif will probably post your photos soon, and then we can update with new pictures when you send the puzzle in May.

Oh, I'll send my address again when we get in contact in May,

Email from Bob to Thorleif
That's great that she's going to send the whole puzzle and manuals!
I'll let you know when I get it in May. I'll send pictures of everything and then do
different views or details of anything you'd like. Before returning the Quader.
Pictures are easy.

At the end of May 2021 the package containing the Herzberger Quader, arrived at Bob's address.
Bob made several high resolution photos of both the puzzle and the documents.
And mailed it to Thorleif.

Email from Thorleif to Bob
All the pictures and the Newsletter text has now been updated.
Your photos and the help from Ellen has made it all possible.
Will you forward my many thank's to Ellen.
Friendliest Thorleif

Having read the booklet, Bob Nungester (our specialist in SOMA mathematics) made a number of calculations,
relating to the task of building the SOMA cube from SOMA+ / Herzberger pieces. So, in Bob's words:

2021-01-10 BOB NUNGESTER
Yes, forming the 3x3x3 cube with SOMA+Plus is indeed interesting. Using the SOMA+Plus lettering

The manual notes that piece Q can't fit and piece D can't be used because that would leave 25 cubes to fill and that's
impossible with only two 3-cube pieces and the rest 4-cubes each.
It can only use one 3-cube piece and six 4-cube pieces.
With two 3-cube and seven 4-cube pieces remaining, that means one of each is eliminated.
That makes 2 x 7 = 14 different combinations of pieces to try.
They note that one of these combinations results in no solutions.
So I ran them all through the program [SOMA+Plus solver] and got these results:
```Pieces left out,   Number of solutions,

1, 2  | V, L        27
1, 3  | V, T         0
1, 4  | V, Z        99
1, 5  | V, A       245
1, 6  | V, B       245
1, 7  | V, P        32
1, \$  | V, S       138
8, 2  | I, L        39
8, 3  | I, T        47
8, 4  | I, Z       221
8, 5  | I, A       337
8, 6  | I, B       337
8, 7  | I, P       261
8, \$  | I, S       240

Total                2,268 solutions```
Running it with all pieces available gives 1,686 unique solutions. That's because the solutions above, with piece 5 or 6
removed are reflections of each other, so removing either piece creates the same number of unique solutions for that piece.
Subtracting these reflected solutions (245 + 337) gives 1,686.
That's another good test of the Solver and Find Unique programs. to have both methods agree.

I suspect the fact there's no solution with pieces 1 and 3 removed has to do with parity and other considerations since
Schulz was able to prove that one combination has no solutions. It's not just parity because pieces 1 and 8 can have the
same parity (and removing 3 and 8 works), so there is another reason.

It is interesting to see pieces in orientations that are not permitted with the normal SOMA set.
ie. I saw piece 1 with the middle cube in a corner, and piece 7 with its middle cube located on an edge.

2021-01-14 BOB NUNGESTER
I did analyze creating a 3x3x3 cube with pieces 1 and 3 left out.
There's a short proof it's impossible, using the same analysis techniques used in Winning Ways. Newsletter 99-02-01

They analyzed the number of vertices that can be filled by the pieces and then the results of coloring the individual cubes
with two alternating colors (which we call parity).

The vertex analysis shows only one piece can be deficient by only one vertex, similar to the SOMA pieces.
Piece 8 can only have a deficiency of 0 or 2, so it can't be deficient. It must occupy two vertices and an edge cell.
Looking at colors, all pieces have two FV cells and two EC cells except pieces 7 and 8. We know piece 8 has to have two FV
and one EC cell, so piece 7 must have two FV and two EC cells to total 14 FV and 13 EC cells.
Piece 7 can only be placed to have three of one color and one of the other, so this is impossible.

I am sure that with more analysis on the other configurations, especially the ones with a small number of solutions.
We will find that they'll have very restrictive rules limiting possible positions for various pieces, but they all have solutions.

2023-01-25 BOB NUNGESTER "Cuboids with SOMA and SOMA+Plus"
I found some interesting information regarding the Herzberger Quader (SOMA+Plus) and how many Quaders (Cuboids in English)
can be made with the pieces.
It turns out that Soma+ pieces can make 47 different cuboids ranging from 1x1x2 up to 3x3x4 sizes but the standard Soma pieces can
only make 6 different size cuboids.
I worked out the number of unique solutions to each cuboid and the results are even more surprising.
Soma+ can make a total of 10,166,227unique cuboid solutions whereas Soma can only make 315.
It's no wonder they named the puzzle the Herzberger Quader. Soma and Soma+Plus possible box sizes

```    Box      Soma+Plus          Soma
Size     Unique Solutions   Unique Solutions
------------------------------------------------
1x1x2            1
1x1x3            1
1x1x4            1
1x1x5            1
1x1x6            1
1x1x7            1
1x1x9            3
1x2x2            1
1x2x3            2
1x2x4            1
1x2x5            5
1x2x6           11
1x2x7           11
1x2x8           54
1x2x9           16
1x2x10         132
1x2x11          14
1x2x12         160
1x2x13           4
1x2x14          84
1x3x3            6
1x3x4           24
1x3x5           17            4
1x3x6           60
1x3x7          243
1x3x8        1.046
1x4x4           99
1x4x5          754
1x4x6        1.688
1x4x7        1.522
1x5x5          163
2x2x3           12            2
2x2x4          151            5
2x2x5        1.267            3
2x2x6        7.775
2x2x7       35.685
2x2x8      117.150
2x2x9      244.434
2x2x10     243.564
2x3x3          356
2x3x4       40.017           61
2x3x5       54.793
2x3x6    2.008.419
2x4x4      597.066
2x4x5    4.441.090
3x3x3        1.686          240
3x3x4    2.366.636

Count               47            6
Total       10.166.227          315
```

Historic research by Bob Nungester.