General Method and Rules for Staircase Squares (Part I)

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. In addition, construction of the square requires a one down shift after filling of a diagonal to move to the next diagonal until the square is filled. This article will show that one can place the initial 1 in any cells of the square except main diagonal.

It was shown previously that one could generate stepwise squares using first row, middle column and left diagonal methods and that one could predict which squares are magic and which are not. It will be shown here that those methods are actually specific methods for those particular squares and that each of these methods can be combined into a general method that subsumes every position on the square. With the exception of the main diagonal and the first column and last row every cell in the square can be starting point for the construction of the square. The variable knight move for switching from the end of one step diagonal to the start of the next step diagonal will now depend on two values, k, a move across the square, and j, a move down the square.

The General Method

As mentioned above the starting 1 can be placed in any cell of a square that is being constructed except for the main diagonal, where placement of the number 1 results in an inconstructible square. In addition, the first column and last row, although the squares are constructible, produce squares that are not magic. The main diagonal is reserved only for those numbers from ½ (n²-n+2) to ½(n²+n) where n is the order of the square. The shift values, the end of one diagonal to the start of the next diagonal, consists of two variable numbers (k,j) where k and j are listed on the first row and last column, respectively, as depicted in the examples.

The values of k and j are listed in the following table where the ellipsis corresponds to larger and larger order squares. The numbers in this table will start originally with with 1 and end with n. Although the table consists only of odd numbers the even numbers will be incorporated during the construction of the squares in a revised table.

Two types of squares will be constructed, one based on prime numbers the other on composite numbers. In addition, a comparison between this Staircase Grid method and the known Uniform Step Method for squares formed via what appears to be a variable knight group move is fully discussed in Part III where the methods appear to be intertwined. Though not explicitly stated the Siamese staircase squares may be produced from a series of six variables within the Lehmer Article in which the Uniform Step squares are discussed. But would this give someone the incentive to to go out and try to find if the Siamese method can be generalized. Since the article was published 90 years ago, the Siamese method consisted only of the original Loubère and Méziriac magic squares, however, all that has changed. I would also like to thank the reader who brought to my attention the Lehmer article which forced me to look into the differences and similarities between the Siamese and the Uniform Step methods discussed in Part III .

Part III is an addendum to this page and describes the Uniform Step Method as well as the type of magic squares formed, magic and magic diabolic, alike, as well as constructible and inconstructible non magic squares.

5^th Order Squares

The following are examples of five order squares A1-A4. Only A1 and A2 produce magic squares, while A3 and A4 do not. It appears that when both the the starting 1 and the next number in the new diagonal, in this case 6, fall in the same column or row no magic occurs. This has verified when the starting 1 is placed in every cell of the first and last row and occurs with the larger squares. The first colored row and last column are the k/j knight moves used in constructing the squares. What the significance and meaning of these k/j numbers is explained just below.

7^th Order Squares

The 7^th order squares have been constructed on the previous specific methods stated above. This section will use the new general method to construct squares where the initia1 starting 1 is on the second or sixth row. Then we construct the k/j table as follows where the same values are used for both k and j. The k/j 2-tuple (k,j) is the ordered pair of numbers obtained at the starting position of 1 and are used in all knight moves during the construction of the square. In addition, the first knight move is depicted as green followed by blue.

However, since numbers equal or greater that 7 mean that we have made a knight move across the entire breadth of the square, 7 is subtracted from all those numbers greater than 7 to generate the revised table:

and these are the new values of k and j posted above and to the side of each square where k is a move to the right (→) and j a move down (↓). These four squares G1, G2, G3 and G4 and their symmetrical counterparts, obtained by 180^o rotation along the main (the right) diagonal and identical to the the four mentioned, are all magic.

9^th Order Composite Squares

Construction of squares with composite numbers however, generates a much smaller number of magic squares besides the known Loubère and Méziriac squares. Squares G5 and G6 are not magic but G7 and G8 are. I have found that a square is not magic if the starting number in one of the secondary rising diagonals, not the starting one, falls on one of the cells in the first column, last row or right principal diagonal and where that number is a multiple of C×n + 1. C being the squares composite order and n being greater than 0.

For example, for C = 9 that multiple number is M = 10. See Squares G5 and G6 which follow the rule and, thus, are not magic. Table C/M shows the multiples that can occur for six composite odd numbers found so far. It appears that when the composite number is composed of one prime that one multiple, M, is found. For example, in the 49^th order square the multiple appears at the 10^th diagonal.

Squares G8 and G9 where the positions of the initial 1 in the middle column differ only by position below or over the central cell. G8 is magic but not G9.

This completes this section (Part I). To go to Part II or Part III. To return to homepage.