New Method and Rules for Left Diagonal 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. The move appears to have been chosen by luck. This page will show that placing the initial 1 in any cells of the left diagonal except for the central cell can lead to n−1 squares which may or may not be magic. Depending on the position of the starting number 1 each cell in the diagonal will use a variable knight move for switching from the end of one step diagonal to the start of the next step diagonal.

The New Staircase Method

This new method is similar to the previous Loubère method where the initial number 1 could be placed in any cell of the first row of the square.

It will be shown that the initial number 1 can be placed on any cell of the left diagonal with the requisite knight break moves listed in the row preceding the table. The moves are only required at the end of one diagonal to the start of the next diagonal. Non similar methods have been employed in the past where the knight moves are used throughout the construction of the square, i.e., after each number a knight move is used to place the next number into the following cell. In addition, the length of the move is 2 units in a direction followed by 1 unit unit as in a chess move.

It has been found that the the knight moves in this new method only occur at the end of a filled diagonal to the start of the next diagonal. Furthermore, the moves are no longer of constant value but variable where the initial knight move may be either a length K always followed by a down move of K . The value of K is determined by the value in the following table where, for example, the K value of 1 is used as the knight move where the starting number 1 is placed in cell position 1 of the diagonal, 3 for the one in cell position 2 of the diagonal, etcetera, excetera, excetera. And where the ellipsis corresponds to larger and larger order squares.

The 5×5 Squares

Let's take the order of 5 squares. Rule number 1: All prime number squares will generate two squares that are not magic, those where the initial number 1 is in the first and last cell of the diagonal (see A1 and A2 below). Rule number 2: There is no square with the starting number 1 in the center cell. Therefore the number of magic squares for a prime number order is n − 3. There should be four constructible squares for a 5^th and of these only A3 and A4 below are magic. However, because of the symmetry of the central cell on the diagonal the two complementary squares are identical only rotated 180^o about the right diagonal. Therefore, construction of prime squares only requires only half the number of squares. Rule number 3: Construction of non prime, i.e., composite order squares, however, generates less than the number of requisite squares and produce a really fascinating and novel phenomenon, something really odd which will be covered soon.

First we construct the K table for this order. Then we subtract 5 from each number greater than 5 since this means a shift greater than the size of the square has been performed (just like clock arithmetic). These are the revised shifts. We then set up the shifts as 2 complementary strands linked at a central hinge with 0 as the hinge and all other as complementary pairs summing up to n. From this setup it can easily be seen which Ks are complementary to each other and if one is deleted, automatically its complement is also deleted. Those cells in white mean that the squares are constructable and magic. Those in green or yellow/green are non magic. Note that a good way remembering how to set up the K table after revision is that it consists of a series of odd numbers starting with 1, followed by a series of even numbers starting with 0:

and applying this method to the 5^th order square:

The following four squares were constructed by placing the initial 1 in the appropriate cell with the number of knight K moves for that particular cell listed above row one. This top row is actually the K shift values of the complement table constructed previously. Furthermore, the squares are labelled with a letter and right or left knight moves where, for example, 1→ means 1 move to the right and one move down while 4← means 4 moves to the left and 4 moves up. Both right and left moves are equivalent.

Again A1 and A2 are non magic while A3 and A4 are identical and magic.

7^th and 9^th Order Squares

As for the 7×7 squares first we construct the complement table:

Then from these K values we construct the squares crossing out the 7 and its complement as was done above (since 7 is divisible by the order of the square) and all the other K values are relatively prime to 7.

As can be seen all squares excluding the B1 and its complement, are magic as was predicted by the complement table.

What about the composite n. When n is composite the number of Ks that are divisible by any of the primes which make up the composite n increases and, therefore, a larger number of Ks must be struck out from the complement table. Since n is equal to 3×3 then 3, 6 and 0 (whose complement is 9) are automatically crossed out along with their complements. These squares are inconstructible and, therefore, not magic. Of the three magic squares left over, the C1, C4, and C7 the C1 is not magic according to Rule 1.

C4 and C7, however, should be magic since they are relatively prime to 3. However, this is where the odd phenomenon occurs, C2 and C3 though prime are no longer magic. The assumption that I am about to make is that the even K multiple of 3 (in this case 6) situated a distance of ±1 from the two Ks must be having a confounding effect on these squares. The normal meaning for a confounding variable is that of a variable that is influencing two other variables (dependent and independent) causing a spurious association. I define confounding here as a variable that appears to influence two other variables adjacent to it, i.e., in close proximity.

In fact it will be shown that this occurs even for for larger and larger squares and that the confounded squares can have K values that are prime or composite. Here I introduce Rule number 4: All composite order squares divisible by three are non magic. Order 9 squares are the first squares with this property. In the complement table for order 9 squares the confounded K values are shown in light yellow/green to differentiate them from those in green, viz, those that are constructible but not magic and the inconstructible.

The following shows the complementary tables for the nth order 15 and 21. Each and every square generated from these tables were constructed to verify the rules and all were found to be non magic or inconstructible. Note that both order 15 and 21 are divisible by three and follow Rule number 4. The confounded numbers (even multiples of three) are listed in the table below for the 15th and 21st order squares. Note that the prime factors 5 in the former and 7 in the latter also lead to inconstructible squares and, therefore, retain their green color.

The complements of the K values give rise to identical squares and thus if one square is non magic then automatically so is its complement. So far if we don't count 15 all the confounded K values are prime and are shown in light yellow/green.

A Look at Larger Squares

Three orders, the series of squares 25^th, 35^th and 49^th order squares were constructed to determine if they follow the rules invoked above. While the squares of different K values confirmed Rules 1-4, they are too big to depict and only the complementary tables are shown here for visualization.

Again all squares whose K move values are in white cells are magic. For the 35th order the even multiples of 5, viz., 10, 20 and 30 and the even multiples of 7, viz., 14 and 28 confound the light yellow/green cells. For the 49th order the even multiples of 7, viz., 14, 28 and 42 confound the light yellow/green cells (non magic) and the odd multiples of 7 and 7 itself form inconstructible squares. This leaves 14 magic squares and 14 complementary identical squares that can be constructed. In addition, the even multiples of 7 must be in the bottom part of the complementary table. Let it be said that these large squares are no easy matter to construct since each square requires a different K value.

An Alternative Method: Using a Little Math

Again let me reiterate the numbers in the complement table are Knight moves applied to the square when the initial number 1 is placed in that particular cell on the diagonal. I have found that two simple equations may be applied to obtain the same results using the K moves. These equations include:

B is a number subtracted from the last entry in a step up diagonal. Go to that number and go down or up the number of cells (UD) to where the next number is to be placed. A negative number for UD means go up the column while a positive number means go down the column. Take square C3 when only the first step up diagonal has been entered:

Both methods may be used and is a good check against errors introduced inadvertently.

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