Attempt to make these shapes with your Soma set

Cube Counting

This is your first impossibility proof

Identify the number of cubes in shapes that are impossible to build with a Soma set.

Have you realize that it is not possible to build any shape that contains the following total number of unit cubes.These total number are 

1, 2, 4, 5, 6, 9, 10, 13,  17, 18, 21, 22, 25, and 26.  Furthermore, any shape that requires more than 27 unit cubes could not be built.

Vetices Counting

Let’s focus on the Soma Cube. It contains 8 vertices. There are 7 Soma blocks to make these 8 vertices.

Block number 1 could be used to form only one vertex of the Soma Cube.

When Soma block 1 makes one vertex, it cannot make another corner (vertex) in a Soma cube.

However, the block 1 could be placed within the cube in a manner such it does not make any corner (vertex).

The following provides the possible vertices each block could form.

Soma block 3 can make zero or two vertices. If it does not make a vertex, all remaining blocks have to make 8 vertices. However, all other blocks can make at most 7 vertices. Therefore, Soma block 3 must form two vertices in any Soma cube.

Likewise, the Soma cube could not built where Soma block 2 does not make any vertex.

If Soma block 2 makes one vertex, all other blocks must contribute to make the other 7 vertices.

Alternative Coloring

Alternative Coloring is one of the Coloring techniques. See the pattern of a checker board.

See the pattern of a checker board.

Have you noticed the squares are alternatively colored with two different colors. In other words, each white square has black squares adjacent to it, similarly, every black square has adjacent white squares.

This coloring method is known as alternative coloring, and could be used to assist in proving impossibilities.

Before moving onto impossibility proof, let’s see how to apply alternative coloring techniques to individual Soma blocks.

In Chess boar or Checker Board, squares are alternatively colored. Here, individual cubes are colored instead of squares.

Alternatively, it is possible to color the cubes such that the maximum number of cubes is colored in black.

When a Soma block is coloured such that the maximum number of white cubes is obtained, the number of black color cubes becomes minimum since the sum of the black and white cubes is constant.

For your complete understanding, the following picture shows alternative colouring for both colors. Therefore, it depicts for each block the maximum number of white cubes as well as maximum number of black cubes.

The number of white cubes in a shape cannot exceed the number of white cubes in the Soma blocks that builds the shape. In the previous example, the shape is built with Soma blocks 1, 2, 3 and 4.

• Summing up the maximum white cubes in each block comes to 9. In other words, the total number of white cubes in Soma block 1, 2, 3 and 4 is 9.
• After applying the alternative coloring technique, the maximum number of white cubes in the given shape is 8

Therefore, this shape passes the alternative coloring test. This is due to the total number of white cubes in the shape is equal or less than the total number of white cubes in the individual blocks that builds the shape.

Thus, it is not possible to say that building of this shape is impossible. Keep in mind that the alternative coloring technique does not guarantee the building of any shape. It only tells the impossibilities.

Suppose a shape contains more than 9 white cubes, then surely it can be said that it is not possible to build that shape with Soma blocks 1, 2, 3 and 4.

Similarly, it can be said that any shape that contains more than 15 white cubes cannot be built with a Soma set.

However, it is not correct to say that having less than 15 white cubes in a shape guarantees the building of that shape with a Soma set. In other words, there are many shapes that require less than 15 white cubes which cannot be built with a Soma set. Emphasizing this point, it can be said that what is considered here is proving impossibilities.

The test criteria discussed under coloring and counting techniques are known as a necessity condition. The necessary condition states that in order to build a shape, it is necessary to fulfill certain conditions. Without fulfilling these conditions, it is not possible to build the shape. Two necessary conditions to build the shape are discussed under cube counting technique and alternative colouring technique.

A collection of conditions are said to be sufficient if fulfilling these conditions guarantees the formation of the given shape.

The given shape given has 9 white cubes and 6 black cubes.

This fulfills the necessity conditions.

However, it is not possible to build this shape using the Soma set.

Therefore, it is said that the considered necessity condition is fulfilled but the sufficient conditions is not fulfilled.

The number of white cubes in a shape cannot exceed the number of white cubes in the Soma blocks that builds the shape. In the previous example, the shape is built with Soma blocks 1, 2, 3 and 4.

Straight Coloring

Straight coloring is another impossibility proof technique. In this case, the blocks and shapes are colored horizontally, vertically or perpendicular to the plane on which the blocks are kept.

The following diagram shows the possible combination of white and black cubes when the straight coloring technique is applied to Soma blocks. If the color one is black then the color two is white vice versa.

As shown in above diagram Soma block 2 can be coloured horizontally such that three white cubes are obtained. Colouring vertically also gives 3 white cubes. Colouring Perpendicularly gives 2 white cubes. These instances are given as possibility 1 and possibility 2 in the following table.

As shown in the figure, the maximum number of white cube obtained is 11. There are 4 black cubes.

Soma block 1,2,3 and 4 are the only candidate Soma blocks since others have more than one layer.

The table on possibilities tells that these four blocks have in total a maximum of 10 white cubes when horizontal coloring technique is applied. Therefore, it can be said that the given shape does not meet the necessity condition. Thus, forming that shape with Soma block 1,2,3, and 4 is not possible

It will be evident that the given shape pass the necessity condition required in cube counting and alternate coloring technique. However this shape fails the straight coloring technique. Therefore it can be said that building this shape is impossible.

Recording your constructs

Number Grid

Number grid is recording instrument. It can be used to record your progress in making Soma cubes and other shapes.

For Soma cubes, each Number Grid is a 3*3 matrix corresponding to 9 unit cubes in each layer. Hence, for every Soma Cube there are 3 Number Grids.

Top Level

Middle Level

Bottom Level

* Invisible

Cordinating Systems

Soma blocks and cubes could be represented  in the binary number system.

First learn how we could represent the 7 Soma blocks in the binary number system. Table 7 provides the binary representation for each block.

In a Soma cube, there are three layers each with nine unit cubes. These nine positions and three layers can be represented in a matrix form.

If a particular block resides in a position on a layer, those positions are marked as one (1). If none of the position is occupied by a cube, these positions are marked as zero(0).

The following table shows the position of Soma block 1. The three unit cubes resides on 3 places out of 9.

The following table presents a built Soma cube in a binary numbering system. This table contains 7 columns and 3 rows.

• The seven columns represent the seven Soma blocks. The first column represents Soma block 1 and the second column Soma block 2 etc..
• A set of three (3) columns represent a single Soma Block. Each set of three (3) columns are presented in different colors to represent each Soma block.
• The first 3 rows represents the bottom layer of the Soma cube and the next 3 rows represents the middle layer, and the last 3 rows represents the upper layer of the Soma cube.
• The very first row represent the first row of the Soma block of the bottom layer. Likewise, Row 4, and 7 represent the first row of the Soma block of the middle and upper layer respectively.
• Each column contains a triplet ( 3 digit number). The leftmost bit of the every triplet represent the first column of the Soma block. The second column of Soma cube is represented in the second bit in the triplet, and the third bit of the triplet represents the rightmost column of the Soma cube.
• As you know, the seven triplets in Row 1 of the table represent the seven Soma blocks. The triplet in Column 1 and row 1 of the table, represent Soma block 1 (since Column 1 represents Soma block 1 ) and the first row of the bottom layer of the Soma cube. The three digits in the triplet represents left, middle, and the right positions of the first row of the bottom layer of the Soma cube. Since the all three digits of the triplet is “0”, Soma block 1 does not positioned in any place in the first row of the bottom layer of the Soma cube.

• Notice that for column 1 we have 3 instances of 1. For the every other columns there are 4 instances of 1s. That tells the first column is represented by a Soma block that contains 3 cubes and other Soma blocks contain 4 cubes.
• When the XOR operation is applied to positions across a row, each position gets exactly 1 as shown in the following table.

Cartesian Coordinatinate Systems

Another way of representing a Soma cube or shape is to use the Cartesian coordinate system.

In presenting a Soma block in Cartesian coordinate system, triplets of the integers are used.

A triplet is a set of 3 digits corresponding to X,Y,Z coordinate positions of a unit cube.

Three triplets are used to represent Soma block 1 and four triplets are used to represent the other 6 soma blocks.

• The unit at the origin is represented with the triplet 000 indicating zero distance from X, Y and Z axis.
• The triplet 010 represents a unit cube that situated one unit distance along the Y axis and 0 distance along both X and Z axes.
• Similarly, a unit cube residing one unit distance along both X and Y axes and 0 distance along Z axis is represented in a triplet as 110.

Notice that the last digit of all the triplets is 0. This indicates there is no distance along the Z axis. In other words, these triplets represent unit cubes on the bottom layer. All the unit cubes on the second layer (one above the ground layer) have integer 1 as the last digit of every triplet.

• For example, the unit cube immediately above the unit cube at the origin is represented by triplet 001.

Notice that when a block is kept on the XY plane, Z value is zero.

The Soma block 1 lies on XY plane. It can be also kept on YZ, and XZ planes.

The positions on the YZ plane can be derived by replacing X’s position with Z’s, Y’s position with X’s position, and Z’s position with Y’s position. In this case, values in the X plane become zero.

How do you represent the unit cube at the opposite end of the diagonal from the origin. This cube is shown in black in the figure on your right?This can be represented as 222

Have you noticed that Soma block 1 can be placed in 144 ways in the Soma cube.

Identify how many different ways in which other Soma blocks could be placed in a Soma cube. Compare your answer with the following table.

144

72

72

96

96

64