New Method and Rules for Loubère Type 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 why the Loubère knight move is the mathematically correct one and that many more Loubère type squares are possible, as many as of order n, all situated on row one of the square and each one having a variable knight move consistent on the position of the starting number 1. This page consist of Part I followed by Part II on a separate page.

The Loubère Square Revisited

The 5×5 A1 and the 7×7 B1 regular Loubère squares are shown below each showing the first requisite down move of 1 from the end of one diagonal to the start of the next diagonal:

A1
17	24	1	8	15
23	5	7	14	16
4	6	13	20	22
10	12	19	21	3
11	18	25	2	9

B1 (1→,1←)
30	39	48	1	10	19	28
38	47	7	9	18	27	29
46	6	8	17	26	35	37
5	14	16	25	34	36	45
13	15	24	33	42	44	4
21	23	32	41	43	3	12
22	31	40	49	2	11	20

It will be shown that the initial number can be placed on any cell of row one with the requisite number of knight moves 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 method, as stated, 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 − 1 always followed by a down move of 1 or a 1 followed by the variable move. The value of K is determined by the value in the following table where, for example, the K value of 2 is used as the knight move for cell position 1, 4 for cell position 2, etcetera, excetera, excetera. And where the ellipsis corresponds to larger and larger order squares.

Knight Down Shifts (K)
K	2	4	6	8	10	12	14	16	18	20	22	24	26	28	...

The 5×5 Squares

Let's take the order of 5 squares. There should be five squares some of which will be constructible and some inconstructible. Those that are constructible squares may or not be magic. 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 the 1 (the regular Loubère shift) as the hinge and all other as complementary pairs summing up to n + 2. 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. Note that a good way remembering how to set up the K table after revision is that it consists of a series of even numbers starting with 2, followed by a series of odd numbers starting with 1:

2.. n − 1, 1.. n

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

K Shifts
K	2	4	6	8	10

→

K Shifts (rev.)
K	2	4	1	3	5

→

2	4
		1
5	3

→

2	4
		1
5	3

Thus, whenever K is divisible by n or its composite primes the number is crossed out and colored green. An initial K value (see the above complement table) will produce an inconstructible square upon attempted construction. Furthermore, the complement of this K value is also crossed out in the table since this complement will give rise to either an inconstructible or non magic square upon construction. The Ks that are left uncrossed (in white) are relatively prime to the order n, i.e., having no common factors except for 1 and, therefore, constructible and magic. This rule has been verified by the actual construction of every square in Parts I and II and every 15^th, 21^st and 25^th orders as mentioned at the end of this paper. Its easy to set up the complement table but tedious to actually construct the squares especially large order squares as I can attest.

The following five 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.

A2 (2→,5←)
2	4	1	3	5
1	17	8	24	15
16	7	23	14	5
6	22	13	4	20
21	12	3	19	10
11	2	18	9	25

A4 (4→,3←)
2	4	1	3	5
8	1	24	17	15
5	23	16	14	7
22	20	13	6	4
19	12	10	3	21
11	9	2	25	18

A1 (1→,1←)
2	4	1	3	5
17	24	1	8	15
23	5	7	14	16
4	6	13	20	22
10	12	19	21	3
11	18	25	2	9

A3 (3→,4←)
2	4	1	3	5
24	8	17	1	15
7	16	5	14	23
20	4	13	22	6
3	12	21	10	19
11	25	9	18	2

A5 (5→,2←)
2	4	1	3	5
				1
			5
		4
	3
2

Although it is impossible to construct a square from A5, it is possible to construct one from its complement A2. However, the square is not magic. We may regard A2 in a sense as coupled to A5 since A2 has a K move opposite to that of A5. Thus, if a square is inconstructible then its complement is non magic whether its constructible or not. And finally for this set, the other three squares, including the regular Loubère A1, are all magic.

At this point we can propose why the Loubère is a down move of 1. All the knight moves involve a right shift of 1 to n − 1 moves while the down move is always a 1. The Loubère is therefore a right or left move of 0 followed by a down move of 1 equivalent to a down move of 1 as was historically given to this type of square.

7^th and 9^th Order Squares

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

2	4	6
			1
7	5	3

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. The Loubère, B1, for comparison is already listed at the start of this paper:

B2 (2→,7←)
2	4	6	1	3	5	7
1	30	10	39	19	48	28
29	9	38	18	47	27	7
8	37	17	46	26	6	35
36	16	45	25	5	34	14
15	44	24	4	33	13	42
43	23	3	32	12	41	21
22	2	31	11	40	20	49

B4 (4→,5←)
2	4	6	1	3	5	7
39	1	19	30	48	10	28
7	18	29	47	9	27	38
17	35	46	8	26	37	6
34	45	14	25	36	5	16
44	13	24	42	4	15	33
12	23	41	3	21	32	43
22	40	2	20	31	49	11

B6 (6→,3←)
2	4	6	1	3	5	7
19	10	1	48	39	30	28
9	7	47	38	29	27	18
6	46	37	35	26	17	8
45	36	34	25	16	4	5
42	33	24	15	13	4	44
32	23	21	12	3	43	41
22	20	11	2	49	40	31

B3 (3→,6←)
2	4	6	1	3	5	7
10	48	30	19	1	39	28
47	29	18	7	38	27	9
35	17	6	37	26	8	46
16	5	36	25	14	45	34
4	42	24	13	44	33	15
41	23	12	43	32	21	3
22	11	49	31	20	2	40

B5 (5→,4←)
2	4	6	1	3	5	7
48	19	39	10	30	1	28
18	38	9	29	7	27	47
37	8	35	6	26	46	17
14	34	5	25	45	16	36
33	4	24	44	15	42	13
3	23	43	21	41	12	32
22	49	20	40	11	31	2

B7 (7→,2←)
2	4	6	1	3	5	7
						1
					7
				6
			5
		4
	3
2

As can be seen all squares (along with the B1), excluding the B2 and B7, are magic as was predicted by the complement table.

What abrout 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 9 are crossed out along with their complements giving rise to three magic squares, the C4, C7 and the C1. C4 and C7 are shown below along with the C5 an example of a square whose initial position of the starting 1 generated a square which is non magic as was predicted by the complement table.

2	4	6	8
				1
9	7	5	3

C4 (4→ 7←)
2	4	6	8	1	3	5	7	9
23	1	69	47	34	12	80	58	45
9	68	46	53	11	79	57	44	22
67	54	32	10	78	56	43	21	8
53	31	18	77	55	42	20	7	66
30	17	76	63	41	19	6	65	52
16	75	62	40	27	5	64	51	29
74	61	39	26	4	72	50	28	15
60	38	25	3	71	49	36	14	73
37	24	2	70	48	35	13	81	59

C5 (5→ 6←)
2	4	6	8	1	3	5	7	9
58	80	12	34	47	69	1	23	45
79	11	33	46	68	9	22	44	57
10	32	54	67	8	21	43	56	78
31	53	66	7	20	42	55	77	18
52	65	6	19	41	63	76	17	30
60	5	27	40	62	75	16	29	51
4	26	39	61	74	15	28	50	72
25	38	60	73	14	36	49	71	3
37	59	81	13	35	48	70	2	24

C7 (7→ 4←)
2	4	6	8	1	3	5	7	9
80	34	69	23	58	12	47	1	45
33	68	22	57	11	46	9	44	79
67	21	56	10	54	8	43	78	32
20	55	18	53	7	42	27	31	66
63	17	52	6	41	76	30	65	19
16	51	5	40	75	29	64	27	62
50	4	39	74	28	72	26	61	15
3	38	73	36	71	25	60	14	49
37	81	35	70	24	59	13	48	2

The following three complement tables show what squares are possible and not possible when n is of order 15, 21, and 25. All the squares generated from these tables were constructed to verify the rules but only the two (other than the Loubère which we know is magic) of order 15 will be depicted in Part II. Those of order 21 and 25 will not be shown due to size constraints. Let me repeat the cells in white are those which are relatively prime to their order and thus, are the ones that produce magic squares. Half of those which are divisible by the primes (in green), making up the composite order, are inconstructible but the other complementaty halves (also in green), may be non magic or inconstructible. Note that when n is of order 21 the n is composed of primes 3 and 7 both of which act as divisors to the K shift table. Furthermore, the 15^th order squares are discussed in Part II.

15×15 K moves
2	4	6	8	10	12	14
							1
15	13	11	9	7	5	3

21×21 K moves
2	4	6	8	10	12	14	16	18	20
										1
21	19	17	15	13	11	9	7	5	3

25×25 K moves
2	4	6	8	10	12	14	16	18	20	22	24
												1
25	23	21	19	17	15	13	11	9	7	5	3

As an interjection this method is viable for row one or for column one, i.e, the square is rotated about the main diagonal so that row one becomes the last column. Moreover, the last row and first column were shown by construction, not to produce any magic squares using the same knight moves as in row one or column n.

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

New Method and Rules for Loubère Type Squares (Part I)

Loubère and Méziriac Squares-Background

The Loubère Square Revisited

The 5×5 Squares

7th and 9th Order Squares

7^th and 9^th Order Squares