The Magic Rhombicube


By Giuseppe Gori – Aug. 19, 2019. Last updated March 19, 2020.


[PART 1] Last week, I discovered the amazing properties of a planar representation of a solid cube which I call a rhombicube. It is a flat, 2-dimensional figure made of rhombi oriented in three different directions to form a hexagonal shape (This is not to be confused with a hemicube, or half-cube, which is an abstract solid polyhedron with three square faces).
Patterns of rhombicubes were used since the time of the ancient Romans for tiling arrangements and they also occurs in basket making.
The rhombi’s 3-directional orientation makes them appear as 3D cubes with three visible faces and three hidden faces.



Mosaic floor, House of the Faun, Pompei Basket making example
The simplest rhombicube is composed of only three rhombi as it appears on the figures above. The next simplest, the order-2 rhombicube, is shown below on the right.





The order-1 rhombicube 12 rhombi form the order-2 rhombicube
The order-1 rhombicube has no “magic” solution. We will focus instead on the order-2 rhombicube, composed by 12 equal rhombi, or cells, oriented in groups of four (its faces).
Notice that an order-1 rhombicube appears at the center of our order-2 rhombicube.
Our rhombicube will be filled with the numbers from 1 to 12, one number for each cell.
We divide our cells into six groups of four cells, marked below by a colored ribbon.
Because of their 3D appearance, we can also describe these groups of cells as the faces of a slice of a Rubik’s cube. We can imagine two slices for each of the three apparent dimensions:


In the figure above the numbers 1 to 4 are placed in the cells of slice g1. Here the color (green) identifies one of the three directions we could slice the imaginary cube in 3D.
Each one of the six colored ribbons b1, b2, p1, p2, g1 and g2 identifies a group of four cells.
Can our rhombicube become “magic”? That is, can we arrange its numbers so that all six groups of four cells have numbers that add up to the same magic constant?
The answer is yes! Our magic constant is 26.
As we looked for solutions, not only we found magic solutions, but we discovered a remarkable feature of a Magic Rhombicube:
Surprise number 1:

 When all six groups of four cells have numbers that add up to 26, then necessarily the sums of the four numbers on each face - white, gray and orange, also add up to 26!
For example:




It looks like the face numbers want to rearrange themselves in an orderly fashion without us even asking! Just this discovery is worth sharing.
But wait, that’s not all. Another surprise awaits us.
Surprise number 2:
We can rearrange the numbers on our rhombicube so that the numbers in the three cells at its center, themselves an order-1 rhombicube, also add up to 26.
Now that’s truly magic!



This implies that the numbers in black in the above figure, in the external cells at the perimeter of our hexagon, always add up to 52 - the sum of the numbers from 1 to 12 (78) minus 26.
Oh, did I miss something?
Did you notice that only three cells of our rhombicube are not adjacent to the central order-1 rhombicube?
These cells completely own three of the six corners of the external hexagon, which is the perimeter of the rhombicube. In the above figure, these cells have numbers 3, 11 and 12.
Surprise number 3: (Well, you may have guessed)
The numbers in these three corner-owning cells also add up to 26!
OK, now brace yourself.
What if I told you that we can find one more group of cells in a symmetrical arrangement, that have numbers adding up to our magic constant?
Look at the diagram above. Can you see it? It should not be difficult.
The remaining three corners of the external hexagon of our rhombicube are bisected by an edge line shared by two cells. We call these corner-sharing cells.
That’s six corner-sharing cells in total.
Surprise number 4:
The numbers in these corner-sharing cells (5+6+1+2+8+4) also add up to 26.
I ran out of exclamation marks.




Our Magic Rhombicube is one of the simplest magic figures. It has just 3 more cells than the classic 3x3 Magic Square.
It is a Normal magic figure, as it uses the series of consecutive, positive numbers 1 to 12.
By only using twelve numbers we obtain its magic constant, 26, in twelve different ways.
It seems that each number has its own magic place.
Our Magic Rhombicube helps us visualize the different ways in which we can group numbers from a series and how these groups intersect each other.
The Magic Rhombicube can be used in education to enhance the students’ ability to think, observe and recognize patterns. For more on this, please keep reading...
Identifying Groups
Let’s look at out rhombicube from a different prospective. It is filled with the series (1 + 2 + 3 … + 12) and the total sum of the numbers in this series is 12 x 13 / 2 = 78, or 26 x 3.
Since our figure has twelve cells, we can divide it into:
  1. Two groups of six cells, with their sum averaging 78 / 2 = 39,
  2. Three groups of four cells, with their sum averaging 78 / 3 = 26, or
  3. Four groups of three cells, with their sum averaging 78 / 4 = 19.5.
1. There is no way to subdivide our 12 cell figure into two symmetrical non-overlapping shapes, because our rhombicube has three axes of symmetry and three faces.
However, in some cases, we can divide our rhombicube into two groups with equal sums.
For example, the following is an unusual non-symmetrical subdivision of the rhombicube shown in the previous section, into two groups. The numbers of each group are not just averaging to, but adding up to 39.

Non-symmetrical two-group subdivision, with Sums = 39


2. If we divide our figure in three groups of four cells, the average sum of these groups must be 26.
In our rhombicube, we found nine (9) groups of four cells that are symmetrical, where not only the group sums average 26, but each group sums add up to exactly 26:
In our diagram “The Amazing Magic Rhombicube” included in the previous section, these are: the white face, the gray face, the yellow face, the 2 groups, or slices, marked by the horizontal “azure” ribbons, the 2 vertical “pink” slices, and the 2 vertical “green” slices.
Other groups of not exactly symmetrical cells must average 26, but not necessarily the sum of the four cell numbers in each group will be equal to 26.
For example, we can divide our figure into three groups of four cells as follows:
“Each central cell together with the three cells opposite its corners.”
In our diagram above, these groups would be:
  • The central cell 7, with its corner-opposites 2, 5, and 12, adding up to 26,
  • The central cell 10, with its corner-opposites 4, 6, and 3, adding up to 23, and
  • The central cell 9, with its corner-opposites 1, 8, and 11, adding up to 29.
The addition of the above totals must be 78, thus the average of the above totals must be 26. These three groups are not exactly symmetrical, and our sums are not all 26, but they just average 26.
3. Our figure cannot be divided in four groups of three cells with an equal sum, because the sum would have to be 78 / 4 = 19.5 (not an integer number).
However, in our solution, we found another symmetrical formation of cells: Two groups of three cells (central cells and corner cells), and one group of six cells (corner-sharing cells).
The sums of the numbers in each of these three groups is 26 and their totals, 3 x 26, is 78.





[PART 2] The Magic Rhombicube solutions and properties
However surprising the solution shown above may be, it is one of 72 possible solutions. We isolated four solutions that are not obvious transformations of each other:


These four particular solutions are representative of groups of analogous solutions.
All together, the solutions to our Magic Rhombicube provide us with only a subset of all the ways in which four numbers, out of the series 1 to 12, can add up to 26 without repetition.
Some combinations appear many times, while others (e.g., 1+5+8+12=26 or 3+4+7+12=26) never appear.
Nevertheless, our 72 solution variations include unexpected patterns and properties.
Slice and face group properties
In each of our solutions, there are nine groups of four cells with numbers adding up to 26.
We initially identified six groups of four cells with our colored ribbons. We called these slices. We then saw that there were three more groups of four cells, the rhombicube faces, also with numbers adding up to 26.
However, these groups have different properties, as they intersect with each other differently:
  • Face groups intersect only four slice groups and no other face group.
  • Slice groups do not intersect with the slice group in the same direction, but intersect the four slice groups in the other directions plus they intersect with two face groups.
The following properties are observed in every solution of our Magic Rhombicube:
The first solution we presented, which uncovered its first surprise, although itself remarkable, did not show us the Magic Rhombicube’s full potential.
After we rearranged the numbers and required the three central cells to add up to our magic sum, we discovered more groups adding to our magic sum.
Without raising our requirements our journey of discovery would have been cut short.
Now instead, we are about to uncover even more properties which would not have occurred had we stopped at our first solution.
Property 1:
The number in a cell-owning corner of the external hexagon forming the perimeter of our rhombicube is always the sum of the two numbers in the opposite, corner-sharing cells.
This property is quite easy to verify, but not easy to notice!
For example, in the solution below, 4+8=12, 6+5=11 and 1+2=3.

The three numbers involved (the two addends and their sum), in each of the three corner cells, are each in a different face of the rhombicube. They are also part of three slices in each of the three directions. That’s a lot of constraints being satisfied!
As we have seen, there are three cells forming the central order-1 rhombicube. A similar sum property is observed with each of the numbers in these central cells:
Property 2:
If we pick one of the central cells, then its number is the sum of two addends. These can be found in the cells opposite to the acute angles of the central cell we picked.




As with Property 1, this property involves numbers in three different faces and in three different slices. All numbers are so tightly connected!
Property 3:
Face groups always contain a number that is the sum of two out of the other three.
For example, looking at the last diagram above, the first solution’s gray face contains the numbers 2, 5, 7 and 12. Here we can see that 2+5=7.
If we pick the last orange face on the right, we have the numbers 2, 4, 9 and 11. Here 2+9=11.
We can verify this for all faces of every solution, but this property does not always apply to slice groups.




 The following property has to do with the group of six corner-sharing cells (black numbers).
Property 4a:
Either one or two numbers of the corner-sharing cells group are the sum of two other numbers within the same group, such that the three numbers, addends and sum, are in three different faces and symmetrically arranged with respect to the center of the rhombicube.
In the following solution example, only the first two sums, 10 and 3, comply with this property.

 When one complying sum total is found, this number can only be 3 or 6. When a complying sum total of 8, 9 or 10 is found, then a second complying sum occurs. The solutions with two complying sums are: 8 and 5, or 9 and 4, or 10 and 3 (as in the above example).
No other sum number or combination can be found that complies with this property.
The following is is a further observation about the group of six corner-sharing cells:
Property 4b:
When two sums complying to Property 4a are found, then the two sums are always in the same face and in opposite cells.
When only one sum is found complying with Property 4a, then
  • if the sum is 3, its opposite cell in the same face is 4.
  • if the sum is 6, its opposite cell in the same face is 8.

Property 5a:
In each solution, for each face, we can add its numbers in pairs in a way that the two results (demi-sums) are the same for all three faces. These demi-sums can only be
7 and 19, 12 and 14 or twice 13. In one face the numbers to be added are opposite
to each other and in the other two faces the numbers are adjacent to each other.
Obviously by adding these demi-sums 7+19, 12+14 and 13+13 we obtain the total for the face: 26. The following is an example of a solution with face demi-sums 12 and 14:




 The demi-sums 3&23, 4&22, 5&21, 6&20, 8&18, 9&17, 10&16, 11&15 never appear in our solutions’ faces, while all possible demi-sums, from 3&23 to 13&13 appear in slice groups.
You can see further examples of this property by looking at the set of four representative solutions presented earlier.
  • in the first solution we find: white face (adjacent): 6+1=7 and 10+9=19 - gray face:(adjacent) 5+2=7 and 12+7=19 – orange face (opposite): 4+3=7 and 11+8=19.
  • in the other three solutions we find both demi-sums of 13 for each face: in two faces by adding adjacent numbers and, in the gray faces, by adding opposite cell numbers.
The following property has to do with the orientation of the demi-sums of property 5a:
Property 5b:
The adjacent demi-sums observed in two faces according to Property 5a always occur in both faces in the same direction. Furthermore, this direction (in apparent 3D) is always the direction along the slice line parallel to the third face.
Property 6:
Face groups always contain two even numbers and two odd numbers.
The other combinations that could be valid to obtain 26 (four even numbers and four odd numbers) never occur in face groups, although they do occur in slice groups.
Property 7 (implied by 5a and 6):
Every solution has one face with odd numbers and even numbers opposite to each other. The other two faces have adjacent even and adjacent odd numbers.
Property 8:
The sum of the numbers of the main diagonal (the center cell and the corner cell) in each face equals the sum of the four numbers in the secondary diagonals on the other two faces.
This visually appealing property is possible (but not necessary) because the cells in the main diagonal (red and blue) are both part of three-cell-groups, while the secondary diagonals are part of a six-cell-group.
In the following diagram,
The main diagonals are the two cells corresponding to the stem of the arrow.
The secondary diagonals are the four black cells corresponding to the arrow-heads.

Property 8: Arrow stems equal arrow heads.
In the above diagram: 3+9 = 4+1+5+2; 11+10 = 8+6+5+2; 12+7 = 8+6+4+1.
Conversely, and necessarily implied by the above, and by Property 1, we recognize that:
The sum of the numbers in any two corner cells is the same of the sum of the numbers of the four corner-sharing cells that are not between them.


Necessarily implied by the above and by Property 1



The following property has to do with the two groups of three cells (the central rhombicube cells and the corner-owning cells) and the group of six corner-sharing cells.
Property 9:
The numbers 1 and 2 never occur in three-cell groups.
The numbers 11 and 12 always appear in three-cell groups.
Conversely,
The numbers 11 and 12 never occur in the group of six corner-sharing cells.
The numbers 1 and 2 always appear in the group of six corner-sharing cells.
Unfriendly number pairs
Most numbers do not have a problem with each other, but some number pairs cannot see each other. They can be on the same slice, but are never on the same face of our Magic Rhombicube.
These are: 1 and 3, 1 and 5, 2 and 3, 5 and 6, 7 and 8, 8 and 12, and 10 and 12.


The role of the number 3 in the Magic Rhombicube
Our Magic Rhombicube has a 3-dimensional imprint. Because of this, it can show us the properties of groups better than a 2-dimensional Magic Square.
You may have noticed how the number 3 plays an important role in our Magic Rhombicube.
  1. The rhombicube appearance is 3-dimensional.
  2. There are 3 groups of rhombi (tiles) oriented in 3 different ways, the only 3 ways possible to form a hexagon.
  3. The total sum of the series of numbers 1 to 12 is 78. This divided by 3 gives us our magic constant.
  4. The six groups identified by ribbons, or slices, are in 3 directions in space.
  5. 3 of the six corners of the whole hexagon are corner-owning cells, The other 3 are corner-sharing cells.
  6. The whole hexagon, containing all cells, can be divided into 3 faces of equivalent weight – each with numbers adding up to 26.
  7. The same hexagon, containing all cells, can be divided into 3 zones (central order-1 rhombicube, corner-owning cells and corner-sharing cells) of equivalent weight - each with numbers adding up to 26.
  8. Two of our groups (central rhombicube and corner-owning cells) are composed of
    3 cells.
  9. Each face contains 3 numbers where one is the sum of the other two.
  10. The group of corner-sharing cells contains at least 3 numbers where one is the sum of the other two and complies with Property 4. These numbers are in 3 different faces.
  11. Property 1 and Property 2 involve numbers that are in 3 corner cells or 3 central cells.
    The 3 numbers involved (the addends and their sum) are in all 3 different faces of the rhombicube. They are also part of 3 slices in each of the 3 directions.


[PART 3] The Weird Brother of the Amazing Rhombicube (See https://bit.ly/2wAJpg4 for parts 1 and 2)
We have seen how our Amazing Rhombicube solution is Normal. That is, the numbers used to fill our rhombicube are exactly the series of sequential numbers from 1 to 12.




If instead of starting from 1, we use another sequential series, for example starting from number 11 (i.e., 11 to 22), we can fill a rhombicube where all slices and all faces still add up to the same sum, by just replacing the numbers 1 to 12 in the Normal Rhombicube solution with the numbers 11 to 22 in the same positional sequence. For example:



However, this is not a complete solution.
As you may verify, it is not possible to find a solution for this series that maintains the sum of the three central cells, or the three corner cells, or the six corner-sharing cells to the common total of 66.
A New challenge
Is there any other series of numbers that admits complete solutions which satisfy all of our sums, including the three central cells, and the three corner cells, and the six corner-sharing cells ?
Think about it, before you continue reading, which are the most likely series?
Maybe we could find a solution for the series 0 to 11 that could satisfy all the sums of the Magic Normal Rhombicube described in Part 1 ?
It turns out that the answer is no.
But remarkably, there is one series that admits complete solutions. We call this the “Weird brother” of our Magic Normal Rhombicube.

By using the series 2 to 13. we find complete solutions that satisfy all the sums of the Magic Normal Rhombicube described in Part 1. For example, the following:



The obvious follow-up question is then: Would the weird brother’s solutions also maintain the properties of the Normal Rhombicube that we examined in Part 2 ?
We can quickly verify that the properties 1, 2, 3, 6, 7 and 8 are also verified !
For example, as per Property 1, each corner cell is the sum of the two cells sharing the opposite corner: 13 = 6+7, 5 = 2+3, and 12 = 8+4.
Furthermore, the preeminent role of number 3, as described at the end of Part 2, remains unchallenged.
[PART 4] More magic figures
Amazing Numbers for the Fruit of Life
I discovered another amazing disposition of numbers for the well known Fruit of Life figure. I described this in a separate article at: https://bit.ly/3aPmQmZo or https://bit.ly/2wIK0fS

 A relative of the simplest Magic Rhombicube: The simplest Magic Star
In classic Magic Stars, as described in the literature, numbers are associated with the intersections between two lines and are not not written inside cells.
The simplest possible normal Magic Star is a hexagon-star of order-6, using the star of David basic shape. This was solved by H. E. Dudeney in 1926. It has 80 solutions.
Like our Magic Rhombicube, it also uses the numbers 1 to 12. Its magic constant is also 26:




 Relation between the Rhombicube and the simplest Magic Star
Each segment in this Magic Star identifies a groups of four numbers. Since these groups add up to 26, we can see the relations between the simplest Magic Star and the simplest Magic Rhombicube.
The simplest Magic Star identifies six such groups, one for each of its segments. This grouping is analogous to the grouping defined by the six slices in our rhombicube.
However, each rhombicube solution is more fertile as it shows us, with its faces, three more four-number group combinations adding up to the magic constant.
Furthermore, each rhombicube solution presents us with the surprises of two groups of three numbers and a group of six numbers also adding up to the same magic constant.
Like all other groups in our rhombicube, these extra groups are visually highlighted by their symmetric arrangement.
Finally, our Magic Rhombicube can show us the amazing properties of its numbers arranged in many symmetrical and interesting patterns.


The Magic Pulsar
Unknown to me, this was discovered in 1991, and was called the Hexagram. It is based on the same shape, the Star of David, but filling the inside triangles, and filling numbers in the triangles.
This magic star has only two unique solutions (two groups of homologous solutions).
Furthermore, these two solutions have a unique property: They have two different magic numbers !
If we add up the five numbers of each row in all directions, for one solution (the one to the left in the diagram below) we observe a magic constant of 33, but for the other solution (the one to the right in the diagram), the magic constant is 32 !!
In our star analogy, this is not a Magic Star, but a Magic Pulsar:
It is one star figure exhibiting two normal solutions (using the numbers from 1 to 12), that are both magic (each row has the same sum), but differ in the value of their magic sums.




Both solutions use the same numbers and all our magic rows have the same number of elements (five), but the solutions’ magic constants differ.
This is a property we have never discovered in any other magic figure of any size and shape. It derives from the fact that the central cells are part of three sums (the three directions), thus have more weight than the star point cells, which are considered in only two sums.


Other Known Magical Figures
There are other, more complex magic figures (see links below). However, uniqueness and beauty are found in the simplest patterns displaying the most unexpected properties.
Interesting Links
For more on Magic Stars see:


For Magic Squares see:


The smallest, normal Magic Hexagon is the unique 19-cell figure below.
Here the five rows, in each direction, have a different number of cells.




The only normal Magic Hexagon
Numbers: 1 to 19. Magic constant: 38


_____
Giuseppe Gori has a doctorate degree in Computer Science from the University of Pisa, Italy.
He has been Assistant Professor of Computer Science at the University of Pisa and visiting professor of computer communication and networking at Western University, Ontario, Canada. He has worked for several large companies in the IT industry. He is the CEO of Gorbyte (https://gorbyte.com), an Ontario, Canada company pioneering in blockchain research, development and innovation.



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