Some Wikipedia Articles on Group Theory

  I think part of the problem that I have been having in classifying the set of operations is that they do not comprise a simple set of objects. Mathematical subjects are partitioned and so we may only get a partial view of something that is by nature more complex. This list of Wikipedia articles may give a more complete picture.

Galois group

Group action (transformation group)

Class (set theory)

Classification theorem

simple groups

algebraic group

Coxeter group

But the articles in a sense seem to be talking around our transformation group for the magic squares. Part of the problem is abstraction or oversimplification or idealization rather than dealing with specific examples.

Reduced Abelian Groups

  I'm having a little trouble classifying my operators for the magic squares. For one thing the left and right products are "duals" of each other. Groups usually have only one product. In Combinatorial Group Theory they talk about words, factors and primitive elements but they might call FG a primitive element. Abelian groups can be reduced. Transpose is a complication but can be treated as second magic square. The "class" defined by F, G and transpose may be extensible under some unique circumstances. For instance one can go from one of the transformed simple magic squares to Dürer's by exchanging the two middle rows because the equal sums are 17. We first showed that the second diagonal flip could be represented by a transpose and multiplication by F. Then we showed that we could do something similar with the rotations. There may be room for improvement in group theory.

List of the Actions the Operators Perform

  This table might help to keep track of the operations that F, G and H=FG perform starting with the simple magic square.  Using M^T instead of M gives another set of 16 magic squares.

Doing the Simple Magic Square in Excel

  The procedure for creating the simple magic square in Excel is a little complicated so I decided to include some help for it. One needs to use the Define Name option in Formulas to label i, j, K, L and M. A curly brackets about a formula indicates you have to use ctrl-shift-enter 4x4 array selected to enter the formula. I've used formulatext( ) and offset( ) to indicate where to enter the formulas. The formula to enter at the highlighted location in an array is shown inside the curly brackets.

The first two numbers in offset( ) indicate a location relative to the upper left corner of an array and the second two the size of the selection in the array. The order is rows, columns. For K one selects the 4x4 array, types in "=4*(i-1)+j" (no quotes) and then presses ctrl-shift-enter. And likewise for F.

Primitive Operations Leave the Sums Unchanged

  We need to show that the primitive operations of transpose, vertical and horizontal reflection and the double shifts, acting on a magic square, leave the sums unchanged. To do this we perform the actions on a magic square and compare sums.

The sums for the rows, columns and the quarters after one of the primitive operations can all be found in the original magic square in rows, columns or quarters. For a transpose and a flip (multiplication by F) the sum for a diagonal is a sum for a diagonal in the original magic square and the same is true for the center and corners. For the double shifts (multiplication by G) sums for the diagonals, center and corners are evaluated above and remain unchanged. So all these primitive operations leave the sums for the magic square unchanged.

Supplemental (Dec 30): Had a little problem typing in the lower case i. It changed to what looked like an l. Retyping the i corrected the problem.

Turns out the lower case i was autocorrected to capital I. Decided to remove it from the autocorrection options.

Replacing Rotations With More Primitive Operations

  Since the rotations of the magic square can be defined in terms of the reflections they are all we need in order to find new arrangements. The group can be reduced to just the three primitive operations of multiplying a magic square M and its transpose M^T by the arrays F and G either on the left or right side. So each magic square belongs to a set or class of 32.

Even though we can exchange the order in which we multiply by F and G the order above needs to be fixed so we don't double count the number of magic squares. Using reflections and rotations in our group of operators risks double counting magic squares.

  Why can we exchange the order of F and G and the order of the transpose? F and G only exchange rows with rows and columns with columns. Transpose exchanges rows and columns. There appears to be something fundamentally different about doing this. Rotations exchange rows and columns too.

Summary of Magic Square Operators

  It might help for review purposes to keep the formulas needed in one place. F, G and H do the various flips and shifts. XM and MX with X ∈ {F, G, H} need to be added to the formulas below for completion.

F, G and H can be exchanged with each other but not necessarily with M, R and the diagonal flips.

Repeated Rotations

  We can repeatedly use or "nest" the expressions for counterclockwise and clockwise rotations but the expressions that result can be simplified and we are left with just four unique expressions.

Simple Rotations

  We can't treat the cells of an array as points and use matrices to rotate the array but it can be done with flips and transpositions. Here the superscript T indicates that M is transposed.

So it looks like we can examine these operations in Excel to see if they leave the sums for the magic squares unchanged.

Using Formulas To Create Arrays In Excel

  In Excel 2016 one can use a constant such as {1,2,3,4;5,...} to define an array but functions provide an easier method. Below the numbers 1-16 are placed in order in the array M. Also, the two operators F and G are used to produce two more operators, H and E, by multiplication. The multiplication table for the set of four operators shows that it forms an Abelian group and so the order of the multiplication among themselves doesn't matter.

Left and right multiplication of F, G and H with M produces a set of 16 distinct rearrangements including the original. In FM all the columns are inverted and MF has its rows reversed. GM does a double shift upwards with replacement. In MG there's a double shift to the left. One can see the right multiplication produces different results left multiplication does.

To show how this works we can look at the action of an array with just one nonzero value which allows shift a row or column depending on the filled cell's location.

These are some of the operators kinds of operators which leave the sums in the magic square unchanged. The situation is a little more complicated than that of the dihedral group of a square with has only reflections and rotations.

Using Matrix Operators For Flips in Excel 2016

 I've been using my free 30 day evaluation for Office 2016 to see what Excel 2016 is capable of doing. It was a little cumbersome at first but I'm starting to get the hang of it. Working with magic squares has proven to be a good motivator. I was able to get it to use matrix operators to some of the basic flips.

The basic tools are the matrices which I labeled Flip and HalfFlip. To show what these matrices do we start with the source matrix, S, and first multiply it on the left side by Flip with MMULT. The result is a matrix that is flipped along the columns. Multiplying S on the right flips the source along the rows. When HalfFlip multiplies S on the left the top and bottom halves are flipped along the columns. When it multiplies S on the right the left and right haves are flipped along the rows. The transpose function flips S leaving the 1-16 diagonal unchanged. The result is represented by S^T. The flip leaving the 4-13 diagonal unchanged is accomplished by the series of operations, (S·Flip)^T·Flip.

With Excel 2016 I find that it helps to keep track of what one is doing by labeling the matrices and displaying the formulas used with the FORMULATEXT( ) function.

P.S. Merry Christmas!

17s Complement Proof

  One can easily prove that if the sum of four numbers is 34 then the sum of their 17s' complements is also 34.

Another Example of These Operations

  These results appear to be general but the conclusions should be proven to not to change the sums. Here are the last two operations at on the Indian magic square.

For the simple magic square inversion resulted in replacement of a number by its 17s' complement. As seen above inversion doesn't always do this. Replacing a number by its 17s' complement does appear to leave the sums unaltered.

A Couple More Magic Square Invariants

  The magic square also appears to be invariant with respect to an inversion through its center.   Another operation exchanges the contents of a quarter diagonally.

Given that one magic square allows one to determine a number of others by using the invariant operations one wonders how many unique magic squares there actually are. Each magic square is associated with a class of them.

Some Shifts Don't Alter the Magic Square Sums

 There are some but not all shifts of the contents of a magic square which do not alter the sums. Below the first magic square is shifted one position left in the second square and the first column is moved to the last. The sums for the rows and columns are unaltered but sum for the diagonals and the center and corners have been changed. Repeating the operation produces a square in which all these sums are unchanged from the original. Another way of looking at the result is that the left and right halves can be exchanged. The same is true for vertical shifts since they are equivalent to a diagonal flip followed by a horizontal shift and a second diagonal shift back to the original position.

So we can say that the sums of magic squares are invariant with respect to reflections about an axis and translations including rotations. So from any given magic square we can easily find a number of others.

Operations on Magic Squares that Don't Change the Sums

  There are a number of operations which do not change the relative positions of the numbers in a magic square and so do not alter the sums. These include the operations that project a square onto itself like flips about a line through the the square's center. We can choose either horizontal or vertical flips, diagonal flips or any combination.

The first magic square above is the simple one found earlier. The second one is the result of flipping the first about a vertical line through the center. The third is the first square flipped about the 1-16 diagonal. It doesn't matter what the values since only their position or coordinates are changes. The same is true for a 90° rotation.

Reconstructing the Dürer Magic Square

  A procedure similar to that in the last blog seems to have used to construct the Dürer magic square.

A Very Simple Magic Square

  There's a very simple procedure resulting in a magic square. One starts by filling in the squares with the numbers in order. If one sums the rows and columns one finds that the sums are not all the same. However, upon flipping the two middle columns vertically, one gets a square will all the sums equal. Next, flipping the middle rows horizontally, the square ends up with all the sums equal.

The other sums also end up being correct.

Magic Square Creation Example

  How one goes about creating a magic square is best illustrated by a specific example. The magic square below is the first one that I found (on Dec 4).

I started by filling the rows of the bottom left quarter with the sums at the top of the lists for 9 and its complement 25. The order was arbitrary but I switched the 9 and 16 about so that the vertical sums had the lowest maximum. The bottom right quarter also needed row sums which added up to 9 and 25 so my 3rd and 4th choices were 7 + 2 and 14+11 again arranged to product the lowest maximum vertical sum. The horizontal, vertical and diagonal sums were then computed for the bottom half of the magic square. The remaining independent variable in the upper left quarter has to have a vertical sum of 20 and a diagonal sum of 13 as indicated in the lower half . The arrows in the tables for sums which equal 13 and 20 have only 12 as a common number so that was selected. The remainder of the magic square can be computed using the equal pairs of sums.

Magic Square Sums & Complements

   Here's a graphic to illustrate the pattern that the sums and their complements follow.

Discovering Magic Squares

  If one looks at the Indian and Dürer's magic one can see how pairs of sums are identical. I used Excel 2016 to compute these magic squares using the 6 unknowns and show the vertical, horizontal and diagonal sums for the partitioned magic squares.

It turns out that there are a dozen sums but half of them are 34's complements of the other half, i.e., a sum and its complement add up to 34. In the Indian magic square at the bottom left we see that 4+5=19 and the same is true for the upper row of the bottom right where 7+12=19. One can fill in a magic square by selecting pairs of numbers from a table of sums and also from the sum's complement.

In first magic square below the first sums 16=1+5 and 18=8+10 were used as a starting point. By changing the order of these pairs on can change the diagonal sum for the bottom left quarter. When filling the bottom right and upper left quarters one has to pick sums for which the sums are 16 and 18 and their diagonals also match. One can place an x next to a sum that was used to eliminate it for a later selection. One can also place x's in the table to note which of the 16 numbers have already been used to eliminate sums for the diagonals. Once the 6 unknowns (marked in green) are chosen one can compute the remainder of the magic square.

The same procedure was used for the magic square on the right. Here are a few more which I did by hand a couple of weeks ago.

One does not have to be able to solve a system of linear equations to find a magic square what follows the chosen rules.

Independent and Dependent Variables for a Magic Cube

The sums found in the last blog can be used to eliminate dependent variables from the magic square. The remaining independent variables can be reduced to the following set.

The magic square can be reconstructed with the following formulas.

In a similar manner the known sums can be used to fill in the unknown boxes if one initially fills in the magic square with some arbitrary numbers. The selections that can be made are limited by the known sums.

Supplemental (Dec 17): The set of independent variables is ordered so there are
16·15·14·13·12·11 = 16!/10! = 5,765,760 ways of assigning 6 out of the 16 possible numbers to them. Not all combinations will yield a magic square consistent with the rules.

More on Magic Squares

  The lack of recent posts is due to a fatal computer crash which prevented me from going online. However I had the opportunity to look at magic squares more closely. There is an Indian magic square from the 11th or 12th century in which there are more sums that turn out to be the same and only the integers 1 through 16 are found in the square.  Dürer's Melancolia from 1514 contains a similar magic square. Following some simple rules I was able to deduce another one that is similar to Dürer's along the bottom row but with some of the other numbers rearranged.

There are at least 16 ways in these magic squares in which one can arrive at a sum of 34.

If one looks at the list of sums one will notice that a pair of sums occurs in more than one equation so one can therefore subtract these equations to find two pairs whose sum is equal.

I found 18 pairs with identical sums.

These pairs of sums add structure to the magic squares and limit the choices for filling a magic square.

Supplemental (Dec 18): Corrected misstatement about my magic square being Dürer's flipped.