14 Aug 2000 + 5 April 2021
|History is expanding.|
Note 2021-04-05 This Newsletter have been expanded with further knowledge about the Herzberger Quader.
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.
Could we think of something between SOMA and Pentominoes, that still leaves both the beauty and the logic of SOMA.?
Volker Pöschel, teacher in comprehensiv-school.
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;
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'|
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.
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 SOMA+)
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 SOMA+ The 2020.12.05 Newsletter The newest advanced SOMA & SOMA+ programs.
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.
He mentioned that He had read an article about the Herzberger Quader, and remembered that this was the same as the SOMA+plus.
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.
Bob found a series of links relating to the Herzberger Quader:
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.
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)
Gerhard Schulze, worked at the
Philipp Melanchthon Gymnasium in Herzberg.
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.
Herzberger Quader erlebt Neuauflage zu den Jubiläen
Auf eine weitere Initiative zum Herzberger Stadtjubiläum macht der Kultur- und Heimatverein aufmerksam. Der Herzberger Mathematiklehrer Oberstudienrat Gerhard Schulze (bei vielen Schülergenerationen unter "Mathe-Schulze" bekannt) hatte einst ein räumliches Puzzle erdacht.
Es war sein persönlicher Beitrag zu Herzbergs 800-Jahr-Feier. [in 1984]
Als man das Jubiläum "50 Jahre Abitur in Herzberg" im Jahre 1999 vorbereitete, hat der Förderverein des Philipp-Melanchthon-Gymnasiums den Quader, vermehrt um ein Anleitungsheft, neu herausgebracht.
Der Zeitpunkt war auch der 80. Geburtstag seines inzwischen verstorbenen Erfinders.
Mehrere Sätze gingen zunächst an einige Herzberger Schulen, doch die größere Zahl der durch die Elsterwerkstätten produzierten 200 Stück wurde beim Schultreffen der ehemaligen Abiturienten frei verkauft. Viele Teilnehmer der Delegation, die der zu Herzbergs Partnerstadt Dixon (USA) einen besuch abstatteten, hatten den Quader als besonderes Gastgeschenk in ihre Koffer gepackt, und dafür war das Anleitungsheft extra ins Englische übertragen worden.
Selbst ein Gymnasium in Achern bei Baden-Baden hatte das geistreiche Beschäftigungsmaterial 1999 im Internet entdeckt und einen Satz für den Mathematik-Leistungskurs geordert.
Die verbliebenen Restexemplare wurden auf dem Weihnachtsmarkt verkauft, und seitdem gibt es immer wieder Nachfragen bei den Initiatoren. Warum soll zur Herzberger 825' Jahr-Feier der Quader nicht neu aufgelegt werden, zumal in diesem Jahr auch zum Jubiläum "60 Jahre Abitur in Herzberg" eingeladen wird? Diese Frage stellte man sich kürzlich im Kultur- und Heimatverein, und das Ergebnis dieser Initiative ist die Neuauflage mit einer Stückzahl von mindestens 100 Exemplaren. Das soll ein weiterer Beitrag des Vereins zum Jubiläumsjahr sein, und das geschmackvoll aufgemachte Holzkästchen wird aus diesem Anlass auch das Stadtfest-Logo tragen.
Wird es in wiederum zehn Jahren eine weitere Auflage geben, wenn der 100. Geburtstag von Gerhard Schulze ins Haus steht?
Herzberger Quader appear in a new edition for the anniversaries
On a continued initiative for the city anniversary of Herzberg, the 'Culture and Homestead club' reminded that. The Herzberg Mathematics teacher, senior teacher Gerhard Schulze (known as "Mathe-Schulze" among many generations of students) once had devised a spatial puzzle.
This was his personal contribution to Herzberg's 800'th anniversary celebration. As they in 1999 prepared the anniversary of "the 50 year Abitur in Herzberg", The sponsoring association of the Philipp-Melanchthon-Gymnasiums suggested that the Quader was re-introduced with an instruction booklet. This period was also the 80th birthday of the (late) inventor. Several puzzle set were initially sent to some Herzberg schools, but the larger amount of the 200 puzzles produced by the Elster workshops was sold free at school meetings, by the former high school graduates. Many participants in the Delegation who paid a visit to Herzberg's twin town Dixon (USA), had brought the Quader in their suitcase as a special gift, and for that, the instruction folder had been specially translated into English.
Even a high school in Achern near Baden-Baden had discovered the ingenious activity material in 1999 on the Internet and ordered a set for the advanced mathematics course.
The remaining copies were sold at the Christmas market, and since then there are always inquiries by the initiators. Why shouldn't the Quader not be reissued at the Herzberger 825-Anniversary celebration, especially since this year is also the Anniversary of "60 years Abitur in Herzberg"? That was the question asked recently in the culture and hometown association, and the result of this initiative is a new reissue with a quantity of at least 100 copies. That's supposed to be another contribution by the association to the anniversary year. And this neatly made, wooden box will for this occation also bear the logo of the city festival.
Will there also be yet another reissue after ten years more - when the 100th birthday of Gerhard Schulze arise?
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.
The Jubilee box of the Herzberger Quader
The 11 pieces of the puzzle
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.
The Herzberger Box has the millimeter reading 106.5 mm wide x 86.6 mm deep x 55.5 mm high.Figures:
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.
The puzzle is actually smaller than we thought. Each small cube is only 1.7 cm wide,Thorleif think:
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.
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.
The Quader arrived at Bob
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.
Having read the booklet, Bob Nungester (our specialist in SOMA mathematics) made a number of
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
Running it with all pieces available gives 1,686 unique solutions. That's because the solutions above, with piece 5 or 6Pieces 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
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