Siamese and Uniform Step Squares (Part III)

Loubère and Méziriac Squares-Background

The Siamese method which includes both the Loubère and Méziriac magic squares have the property that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n² + 1). In addition, the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S. Both squares also require an upward stepwise addition of consecutive numbers, i.e., 1,2,3... It's also a fact that only one Loubère square per order n has been handed down thru the centuries.

This article, a continuation of the Siamese method of Part I will show how a grid made up of congruence class modulo n numbers can be used to construct Uniform Step squares, specifically the Siamese Staircase squares, which little has been said about. The 1929 article by D. N. Lehmer gives the math and theory of the step squares in a highly abstruse form and this page will show how the alternative grid method can be used to generate these ancient magic squares by simple mathematical means.

The Theory Behind the Grid

As shown in Part I the method for generating Staircase squares consists of setting up a grid of variable (k,j) numbers which are used to determine the break step at the end of the filling up a stepwise diagonal. The method involves placing the initial 1 in a cell position and reading the break step right off the the grid to determine the move to make after filling a diagonal with consecutive numbers. This differs from the Uniform Step square method employed by Lehmer where six variables are used for making the desired moves and the break step move is strictly non gridlike.

To create the grid we first generate a sequence of odd numbers from 1 to 2n − 1 where n is the order of the square. We then reduce any number greater than or equal to the order n, by n to form a sequence of k,j break moves from 1 to n − 1 as shown for n = 5:

Thus, we partition all the integers into the congruence class, modulo 5, i.e. all numbers are listed from 0 to 4 in the specified order shown above. For example, if we connect a magic square at its ends to make a cylinder and count across a row, any numbers ≥ 5 on the left hand list above means that we have reached or circumnavigated the cylinder once. We can, therefore, apply clock or mod arithmetic to reduce the numbers on the left to those listed on the right, what we call residue classes. And we can similarly do the same for the 7^th order squares, converting the left 7 numbers into their right residue classes, mod 7:

The method of Uniform Step Squares (Lehmer, D. N. “On the Congruences Connected with Certain Magic Squares.” Transactions of the American Mathematical Society, vol. 31, no. 3, 1929, pp. 529–551) discussed in the Article shows the math behind the theory of Uniform Step squares. The method entails placing the numbers in a cell (p,q) of a square using a knights move for each move where α, for example, signifies the number of moves to the right and β the number of moves up.

When one can no longer make a move, i.e., (reached the end of a full diagonal) one makes a break move (a,b) where a is the number of moves from the leftmost digit on the last row to the column one desires and counts b moves up to the row one desires to place the starting 1. It uses an (a,b) break step notation similar to but different to the (k,j) method and without the gridlike simplicity. For a layman and non mathematician the rest of the article is profound and not for the faint of heart and the best advice I can give is to employ Occams Razor.

To simplify matters and retain the gist of the original, I've retrofitted the Uniform Step method so as to compare it to the grid constructed Siamese Staircase method of Part I and use the grid on both methods so as to avoid having to plow thru the actual (a,b) break steps of Lehmer's method. If we set α=1 and β=4 then these numbers should produce the Siamese Staircase squares (which in a sense would make them a subset of the Uniform Step squares). In the rest of this article we will construct Uniform squares where α=1 and β=2 and compare them to the Siamese squares to determine the differences and similarities between them.

Although, these α and β values were given as variables that could be used to generate Uniform Step squares, no mention was made of using the (α=1,β=4) numbers to generate Siamese Staircase squares having the Loubère and the Méziriac structures. During the 90 years since the article was published it was the assumption that the Siamese method still consisted mainly of the two known old methods. No attempts were made to generalize the method although I can see that reading the Lehmer article could discourage people not to try especially if one thinks they are reinventing the wheel, however square the wheel was.

Tables of Magic and non Magic Squares

This section will tabulate which Staircase and Uniform Step squares are magic and which are not. The tables will be presented in Soduko fashion with four symbols represented by:

in which one of these symbols is placed into the same cell of a square where the initial number 1 is to be placed. A diabolic square, labeled here as an O with horns (el Diablo), is a magic square in which every row, column, two major diagonals and its negative diagonals (all the diagonals going from the left to the right) sum to the magic number. Comparing Staircase 5 and Uniform Step 5 squaretype tables it can be seen that the the cells in these two tables do not necessarily produce the same type of magic/non magic square. In addition, although the first column of both tables do not produce any magic squares, they also differ across rows 2 and 5.

Importantly an axis of symmetry runs along the right diagonal of squaretype table Staircase 5, so that the cells below the axis show a mirror image symmetry to the cells above the axis. Though a symmetry exist between the symbols above and below the axis, I will show that the squares to which these symbols belong to are indeed mirror images of each other. This, however, is not possible for the squaretype Uniform Step 5 table without a similar axis of symmetry. Moreover, for composite order squares such as the 9^th the tables generated differ from the prime order squares 5 and 7 and will be discussed below in the next section.

For example, for two 5^th order Staircase squares placing the starting 1 in cell (3,1) or in cell (3,3) produces, respectively, a magic square S1 for the former and a diabolic S2 for the latter. For two 5^th order Uniform Step squares placing the starting 1 in cell (3,1) or in cell (3,0) produces, respectively, a diabolic square U1 for the former and a magic square U2 for the latter.

9^th Order Composite Squares

Composite orders such as the ninth give rise only to non diabolic magic squares for both the Siamese and Uniform Step. The squaretype tables for both of these type of squares appear to be dissimilar as shown in the two ninth squaretype tables below. However, rotating the squaretype Uniform Step 9 by 90^o degrees counter clockwise shows the table to be identical to the squaretype Staircase 9 even though both tables have an axis of symmetry. However, in spite of this similarity, it is the squaretype Siamese Staircase squares that have both symmetrically positioned magic/nonmagic symbols as well as identical squares (mirror images) on both sides of the axis of symmetry.

Actual construction of every square in the two tables below was done to determine the actual magic state of each square. From the 9x9 tables it may be seen that both squaretype Staircase 9 and the squaretype Uniform Step 9 tables are built up, respectively, from two smaller squaretype 3x3 squares SC 3 and US 3, where, for example, the 9x9 square tables are actually nine smaller 3x3 squares having identical 3x3 units. So for this type of odd composite square (8n + 1, for example 9,25,49,81,...) one can surmise the magic state each square in the larger table just from the smaller table:

For example, the Staircase 9 table shows that the two squares produced at cell positions (3,4) and (5,6) are magic but not diabolic and that construction of the squares having an initial 1 in these two positions produces identical squares, S4 and S5, differing by 180^o degrees. For the Uniform Step 9 table, the cell positions corresponding to the Staircase 9 positions, (5,3) and (3,5) produce magic squares that are non mirror, non identical, U4 and U5.

This completes this section (Part III). Part III for composite order 25 and 49 squares is continued in Part III Con't. To go to Part I. To return to homepage.