Examples including Digital Roots
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.
Decimals as well
Things to consider here.
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.
One of many differant variations
Things to consider here.
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.
27x27 with a row column fill variation
Things to consider here.
- Even when the offset fill is applied the pattern still effectively balances the matrix demonstrating just how fractal this pattern is.
5x5's arranged in a 4x4 creating a 20x20
Additional Information
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.
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Dimensionally it would seem the only limitation of size may lie within matrix that have a size involving prime numbers?