An interesting point about this table is that the center number always has a digital root of 5 and the maximum number always has digital root of 9. ( The values are based on a starting value of 1 and an incrementing value of 1.)
To explain the digital root a 243 x 243 matrix has a center square value of 29525 If you add those digits you get a value of 23 and if you add those digits you get a digital root of 5. So 29525 reduces has a digital root of 5. The maximum value for the 243x243 is 59049 the sum of the digits is 27 and the sum of 27 is 9 and the digital root of 59049 is 9.
The sequence and position is always maintained in each size matrix
Notice also that all 9 numbers reduce to their respective cell position address interesting coincidence?
( example :The value 729 reduces to 7+2+9 = 18 which reduces to 1+8= 9 and is located in address 999. )
Fill Pattern Description
The number indicates how many cells within the matrix are populated per cycle.
There are always 9 cells per matrix in this type of Magic Square.
The 3x3 has 9 individual cells
The 9x9 has 9 - 3x3 cells
The 27x27 has 9 - 9x9 cells
The 81x81 has 9 - 27x27 cells
The fill sequence ID starts with the left digit being the 3x3, the digit will be 1,3,or 9 indicating the number of cells filled per cycle.
Each digit extending from the left indicates the order of magnitude for the matrix.
1 digit is a 3x3 ex: 3
2 digits is a 9x9 ex: 19
3 digits is a 27x27 ex: 139
4 digits is a 81x81 ex: 9191
Each digit extending from the left represents an increase of the power of 3 , 3 raised to the 4th power is 81 which is an 81x81 matrix.
A 13 fill pattern is:
a 9x9 matrix because it has 2 digits.
In the 13 fill pattern the first digit is a 1 , the 3x3 matrix has the starting number entered in cell 1 of the 3x3 because the first digit is a 1 meaning 1 cell in the 3x3 matrix is populated per cycle.
In the 13 fill pattern the 2nd digit 3 represents the 9x9 matrix. In the 9x9 matrix, three of the 3x3 matrixes has a single number entered because the second digit is a 3 meaning 3 cells of the 9x9 are populated per cycle.
Address and number constants
The numbers in the 1st position of each higher order matrix for example 1,1,1 , 5,5,5 or a 9,9,9 in a 27x27 , the values contained in those positions never changes regardless of fill.
The same can be said for any higher order matrix for example a 6561 x 6561 8th order matrix will have 1,1,1,1,1,1,1,1 and 5,5,5,5,5,5,5,5 and 9,9,9,9,9,9,9,9 with values that do not change the 1's will be the first or lowest number, the 5's will be the middle or average value and the 9's are the highest or maximum values.
The 2's, 3's,4's and the 6's, 7's and 8's positions will only change between a x3 fill or x9 type fill otherwise do not change values with differing fill patterns.
A 10x10 Magic square with the rows columns and diagonals may be used to produce Magic squares of 10 raised to any power - 100,1000, 10000. The pattern may be applied in multiple ways so long as the fill pattern and sequence follow the original base pattern through the higher orders.
Primary Pattern filling each 10x10 in sequential order using the base 10 pattern as a reference. See the square in the upper left as the Base pattern.
The patterns may be down loaded in the Download Section of this website.
Another variation using the base pattern.
Another variation using the base pattern
Sequentially filling the cells, cell 1 of the first 10x10 then cell 1 of the second 10x10 , cell 1 of the 3rd 10 x 10 etc. Filling the cells using the base pattern as a reference.
The sums of the rows, columns , diagonals equal and the average value of the squares about the center square(s) all equal.
The 6x6 is populated with 2x2 's using the base pattern positions of the 3x3.
The 3x3 matrix used must be a legitimate Magic Squares where the sums of the rows , columns and diagonals are equal.
Then Populate a 4x4 where each position of the 3x3's contain the numerical identity of the cell position in the 4x4.
To verify the first step process you will see the sum values of the rows columns and diagonals are equal.
Multiply the corresponding cells of the 4x4's as shown in the picture. An Microsoft Excel file is available in the downloads section containing this worksheet.
This is an easy , yet imperfect Magic Square. The sums rows and diagonals are equal but the cell values can repeat and some values are missing.
Incrementally fill the 3x3's in the placement defined by the 3x3 into the 4x4 per the positions defined by the 4x4. The arrows in the picture demonstrate the order.
Note also the digital reduction at the right is perfectly ordered as a 3x3.
It is interesting to see that they both work using the same initial placement of numbers and furthermore the same processes works just as well for other combinations of Magic Squares.