Matrix Sizes and Values
Additional Information
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.
Location References
Additional Information
The sequence and position is always maintained in each size matrix
Numbers that always occur in the same location
in the 36 - X9 fill patterns of the 27x27 matrix these numbers occur in the same position.
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. )
100 x 100 Pattern 1
100 x 100 Pattern 1
100 x 100 Pattern 1
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.
100 x 100 Pattern 2
100 x 100 Pattern 1
100 x 100 Pattern 1
Another variation using the base pattern.
100 x 100 Pattern 3
100 x 100 Pattern 3
100 x 100 Pattern 3
Another variation using the base pattern
100 x 100 Pattern 4
100 x 100 Pattern 3
100 x 100 Pattern 3
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.
Differant Variations
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.
How to embed a 3x3 in a 4x4 Magic Square 2 differant ways
Create a 4x4 matrix consisting of 3x3's
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.
Sums of Rows Columns and Diagonals are equal.
To verify the first step process you will see the sum values of the rows columns and diagonals are equal.
The easy but not perfect Magic Square
Multiply the corresponding cells of the 2 4x4's
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.
Sums of Rows Columns and Diagonals are equal.
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.
Sequentially populating the magic square a better method
Incrementally populate the cells of the 3x3's per the numerical position of the 4x4.
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.
Both are Magic Squares containing equal sums for the rows columns and diagonals.
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.