Magic Squares 

Here to the right is the perfect size-5 square.  It is composed of numbers from 1 thru 25 = 5^2.

·  It has all its rows, columns and diagonals summing to 65, its characteristic number.

·  Its central number equals 13 = 65/5 = 325/25 and is the dimensional average of the sum of all the numbers in the square.

·  Every symmetric pair of numbers, each number opposite and equidistant from the center, equals  26 = 2 x 13. This is a property that holds for all odd-size squares but not for all even-size squares.

·  Every set of 4 numbers, called quadrals, corresponding to a rectangle symmetrically located about the center, including diamonds, sums to 52 = 4/5 x 65. Although that is an obvious corollary to the previously stated property, the previous property does not hold for half the squares of even-size, whereas this latter one here applies to squares of all sizes and is therefore a general property for 2-dimensional squares.

Moreover, this 5x5 square is perfect by reason of the fact that all the row, column and diagonal summations are equal to 65, and the sums of all rectangular-spaced numbers about the center, including diamonds, are equal to 52, a specific fraction, 4/5ths, of the characteristic number 65 for the size-5 square.

Now, observe that every "+" and "x" in the size-5 square also sums to the characteristic number 65. This is called a simultaneous dual complementary pattern in that two distinct patterns occur simultaneously. Not only that but each of these patterns interlock into a tiling pattern for the entire table. It can be dragged across the table and wrapped onto the opposite edges and still the patterns sum equally to the characteristic number. Every number in the table can function as the center of either tile pattern in the tiling pattern.

There are more of these characteristic patterns than there are row, column and diagonal equality summations combined by an order of magnitude. Whereas there are only 20 linear summations here, there are 50 pattern summations equaling the size-5 square's characteristic number. The book depicts these unique hidden tile patterns for every prime-number size square from size 5 thru 31. These hidden characteristic patterns have never been detected nor seen before this.

Not shown here but only in the book is a whole new world of geonometry discovered after any perfect number square has been reduced to its modulus table. All the prime-number size tables yield so many equal-summing string series running in four cross-directions that contain only the numbers from 1 thru n exactly once that the table takes on the role of a loom which provides for weaving many more perfect squares of the same size. And these too all have dual tile patterns containing all the numbers from 1 thru n exactly once regardless of where they are centered in their table!

Next is shown another amazing property, called the sub-additive property. Below is a table where each quadrant is just a duplicate of the the size-5 square. This arrangement has been shown so that it is easy for you to verify the block-square sums of sizes 2x2, 3x3 and 4x4 with overlap across the 5x5 table's borders. These sums are then placed in a 5x5 table relative to each other. Next, the tables are reduced by the smallest number in them less 1. Observe that after normalization, either the original 5x5 square or its reflection is recovered by this process for every size block-square sum between 2 and 4. Now that's how perfect this square is. There are only two other comparable perfect representations of the size-5 square with this property which are not derivations of the one shown here.

There's yet more: observe that all the numbers in each of the four quadrant triangles in the size 5 square above sum to 195. Further, this is 3 times 65 exactly. See the size 7 square below. The corner triangles each sum to 700 = 4 x 175. All odd-size squares greater than 3 in my book have the property that all corner triangles of the same size as the square sum equally. And that sum is always a specific multiple of its characteristic number !! For instance, the corner triangles of the size 9 perfect square sum equally to 5 times its characteristic number. Do you see the pattern of the multiple here: (n+1)/2?

And there's even still more.  If a table of differences between the numbers in each cell and it's adjacent cell is constructed in any one of the four directions: rows, columns and the two diagonal directions, employing wrapping with the extension to the other side of the table when encountering the edge cell, then each complementary tile applied to this difference-table will sum to 0 anywhere in the table, including edge-wrapping of the tile pattern !!!

This is labeled the zero differential property which holds for all perfect size squares. What this amazing property implies is that these perfect number squares are perfectly balanced; specifically, if a square table were loaded with weights equal to each cell's number, the table would balance if supported underneath by a thin strip across its center in any of the 4 directions.

And further, given that the tiles were connected together in a continuous fabric, each tile pattern would resist propagating any ripple effect induced by an impact anywhere beyond one row and column inside the table's border as it otherwise would if it were just floating on water because the inertia of all the surrounding weights would act as a cumulative counter-balancing force in all directions! So the table is planarly stable too.

Numeric squares with this level of perfection have never been seen before.

All squares of odd-size and even-size divisible by 4 are perfect.

Even-size squares not evenly divisible by 4 are only near-perfect (next) and lack these exceptional properties.

Near-perfect Numeric Squares