Summing the digits of the numbers in each cell can be a useful method of error checking. Note the numerical sequence of the pattern in the LH matrix.

Summing the digits and observing the patterns can reveal discrete information about errors and the numerical input pattern used to create the matrix.

almost blank 27 x27

When summing the digits a distinct pattern emerges. With the aid of conditional formatting an error results in a violation of the emerging pattern and visibly indicates an error.

Using the original pattern applied in a 3 part fill pattern the rows columns and diagonals are equal, each 3 x3 has an average value equal to its center and the areas radiating outward from the center have an average value of 365 the center cell.

This Pattern works just as well with decimals. The value was incremented by 0.271003 yielding these results. ( 0.271003 = 100/369)

- Not just Integers works with decimals.
- Columns Rows and Diagonals add up to 100.
- Averages around the center square.
- Center Square is 11.111 reduces to 5
- Top value 21.951 reduces to 9.
- All of the above holds true with the same pattern and different fill sequences.

This Pattern is another application of the single fill pattern on a 27x27.

- Rows Columns Diagonals all equal sums.
- The number 1 is at the bottom row center , 729 at the top row center.
- The average values of the square around the center square are equal.
- It is interesting to observe the patterning of the of the digital roots of the numbers as well.

- Even when the offset fill is applied the pattern still effectively balances the matrix demonstrating just how fractal this pattern is.

- The core matrix is the individual matrix which can be a 3x3 , 9x9, 27x27 etc. using any of the fill patterns strictly adhering to the central pattern. Copies of the core matrix when aligned comprise a layer.
- The core matrix above is a 9x9 matrix. The 9x9 matrix is positioned in 9 layers on the Z Axis.
- A 3x3 matrix will have 3 layers on the Z-Axis, a 9x9 has 9 layers , an 81x81 has 81 layers etc.
- Each successive layer must be transposed by 1 position on the x- axis or on the y axis.
- The shifting of the layers is always performed in the same direction and each layer is shifted progressively by 1 position per layer.
- There is no limit to the maximum size of the x, y, or z axis. (The z-axis is based on the core matrix size, the core matrix size is not limited.)
- The core matrices are aligned side by side and end to end and extend indefinitely. The area or size of the core matrix can be focused anywhere, the sums in the x, y and z axis in that area of focus will be equivalent just like the Magic Square.
- If the outside edges of a column or row were brought together (producing a round tube ) the sums on the X, Y , and Z axis surrounding that connecting line would be seamless as no line would exist there at all.
- An Excel Spreadsheet with a sample 9x9x9 Magic cube is located in the Download section.

Not only can the magic squares be nested around themselves , they can be inserted within a frame work of a magic square of a different order. The 5x5's nested within a 4x4 frame work as an example. Not only do the columns, rows and diagonals sum equally , but the average about the center also applies.

Dimensionally it would seem the only limitation of size may lie within matrix that have a size involving prime numbers?

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