Welcome to Oddwheel the home of new Magic Square methods

HOME to THE WHEEL, LOUBÈRE, MÉZIRIAC, KNIGHT, de la HIRE, Cross and Mask METHODS

Table of Contents I

Loubère, Méziriac, Wheel and Knight Section

1. Loubère Block Modified Squares I
2. Loubère Block Modified Squares II
3. Méziriac Block Modified Squares III
4. Loubère and Méziriac Knight Block Modified Squares IV
5. Loubère and Méziriac Knight Block Modified 7x7 Squares V
6. New Procedure for Loubère and Méziriac Knight Squares with Transposition VI
7. New Procedure Loubère Knight Squares with Transposition VII
8. New Procedure for Méziriac Knight Squares with Transposition VIII
9. New Procedure for 9x9 Magic Squares IX
10. New Reverse Wheel Method X
11. New Reverse Wheel Method XI
12. New Reverse Wheel Method XII
13. New Unbalanced Reverse Wheel Method XIII
14. New Unbalanced Reverse Wheel Method XIV
15. New Color-Coded Wheel Method
16. New Non-consecutive Wheel Method (Part I)
17. New Non-consecutive Wheel Method (Part II)
18. New Non-consecutive Wheel Method (Part III)
19. New Non-consecutive Wheel Method (Part IV)
20. New Non-consecutive Wheel Method (Part V) >
21. New Loubère type (slant break) Squares

Table of Contents II

Mask, Cross and de la Hire Section

1. 5x5 Cross and Mask Generated Squares (Part I)
2. 7x7 Mask Generated Squares (Part II)
3. 9x9 and 13x13 Cross Squares (Part III)
4. 7x7 and 11x1 Cross Squares (Part IV)
5. 9x9 Mask Generated Squares (Part V)
6. 5x5 Zig Zag Cross and Mask Squares (Part I)
7. 9x9 Zig Zag Cross and Mask Squares (Part II)
8. 7x7 and 11x11 Zig Zag Cross Squares (Part II)
9. 5x5 and 9x9 Zig Zag Consecutive Mask Squares (Part I)
10. 7x7 Zig Zag Consecutive Mask Squares (Part II)
11. 5x5 Mask Generated Consecutive Squares (Part I)
12. 7x7 Mask Generated Consecutive Squares (Part II) NEW (3/10)
13. 9x9 Mask Generated Consecutive Squares (Part III) NEW (3/10)
14. 9x9 Mask Generated Consecutive Boustrophedonic Squares (Part IV) NEW (3/10)
15. 5x5 and 7x7 Mask Consecutive Generated Boustrophedonic Squares (Part I) NEW (3/10)
16. 9x9 Mask Consecutive Generated Boustrophedonic Squares (Part II) NEW (3/10)
17. 13x13 Mask Consecutive Generated Boustrophedonic Squares (Part III) NEW (3/10)
18. 5x5 and 7x7 Consecutive Boustrophedonic Mask-Generated Squares (Part IA) NEW (4/10)
19. 7x7 Consecutive Boustrophedonic Mask-Generated Squares (Part IB) NEW (4/10)
20. Consecutive Boustrophedonic 9x9 Mask-Generated Squares (Part II) NEW (4/10)
21. Consecutive Boustrophedonic Knight Break Mask-Generated Squares (Part IA) NEW (4/10)
22. Consecutive Knight Break Mask-Generated Squares (Part IB) NEW (4/10)
23. Consecutive Knight Break Mask-Generated Squares (Part IC) NEW (4/10)
24. Consecutive Knight Break Mask-Generated Squares (Part ID) NEW (4/10)
25. 5x5 Centered Sequential Mask-Generated Squares (Part IA) NEW (6/10)
26. 9x9 Centered Sequential Mask-Generated Squares (Part IB) NEW (6/10)
27. 7x7 Centered Sequential Mask-Generated Squares (Part IC) NEW (6/10)
28. 11x11 Centered Sequential Mask-Generated Squares (Part ID) NEW (6/10)
29. 15x15 Centered Sequential Mask-Generated Squares (Part IE) NEW (6/10)
29. 5x5 Equation Generated Centered Sequential Mask-Generated Squares (Part IA) NEW (7/10)
30. 9x9 Equation Generated Centered Sequential Mask-Generated Squares (Part IB) NEW (7/10)
31. 13x13 Equation Generated Centered Sequential Mask-Generated Squares (Part IC) NEW (7/10)

Table of Contents III

Wheel Section

1. Introduction and Discussion
2. Magic Square Wheel Method A-1:Variant 1
3. Magic Square Wheel Method A-1:Variant 2
4. Magic Square Wheel Method A-1:Variant 2 7x7
5. Magic Square Wheel Method A-1:a 9x9 Variant
6. Magic Square Wheel Method A-2 Template Inversion
7. Method B Wheel Expansion
8. Method C: Expansion of Wheel Squares Part IIIB
9. Method C: Continuation of Part IIIB
10. Method D: Wheel Spoke/Anti-Spoke Method Part IV

Table of Contents IV

Loubère and Méziriac Section

1. Modified Loubère and Wheel Methods
2. Expansion of Modified Loubère and Wheel Methods Part IA
3. Expansion of Modified Loubère Method Part IB
4. Expansion of Modified Loubère Method Part IC
5. Expansion of Modified Loubère and Wheel Methods Part IIA
6. Expansion of Modified Loubère Method Part IIB
7. Expansion of Modified Loubère using the Wheel Method Part IIIA
8. Expansion of Loubère Squares using Wheel and Sweep Methods
9. The Full Monty I Loubère Method (Part I)
10. The Full Monty II Loubère-Knight Method (Part II)
11. The Full Monty III Loubère-Knight Method (Part IIIA)
12. The Full Monty II Loubère-Knight Method (Part IIIB)
13. New regular and non-regular Bachet de Méziriac Squares and Method
14. New regular and non-regular Bachet de Méziriac Semi-magic Squares and Method
15. A New regular and non-regular Bachet de Méziriac Method (Part I)
16. A New regular and non-regular Bachet de Méziriac Method (Part II)
17. A New regular and non-regular Bachet de Méziriac Method (Part III)
18. A New regular and non-regular Bachet de Méziriac Method (Part IV)
19. A New regular and non-regular Bachet de Méziriac-Knight Method (Part V)
20. A New True Bachet de Méziriac Variable Knight Method and Squares (Part I)
21. A New True Bachet de Méziriac Variable Knight Method and Squares (Part II)
22. A New Loubère Complementary table Method and Squares (Part I)
23. A New Bachet de Méziriac Complementary table Method and Squares (Part II)
24. A New Loubère Complementary table Method and 7x7 Squares (Part IIIA)
25. A New Loubère Complementary table Method and 7x7 Squares (Part IIIB)
26. New Loubère Two Step Staircase and Knight Methods Squares (Part I)
27. New Méziriac Three Step Staircase and Knight Methods and Squares (Part II)
28. New Loubère Knight- step Methods and Squares (Part I)
29. New Loubère Knight- step Methods and Squares (Part II)
30. New Loubère Knight-step Methods and Squares (Part III)
31. New Méziriac Knight-step Methods and Squares (Part I)
32. New Méziriac Knight-step Methods and Squares (Part II)
33. New Méziriac Knight-step Methods and Squares (Part III)
34. New Loubère Pendulum Regular break and Knight-Break Methods and Squares (Part I)
35. Continuation of New Loubère Pendulum Regular break and Knight-Break Methods and Squares (Part II)
36. New Méziriac Pendulum Regular Break Methods and Squares (Part III)
37. New Méziriac Pendulum Knight-Break Methods and Squares (Part IV)
38. New Loubère Full Pendulum Regular break and Knight-Break Methods and Squares (Part I)
39. New Méziriac Full Pendulum Regular break and Knight-Break Methods and Squares (Part II)
40. New Generalized Procedure of Loubère Regular and Knight break Methods and Squares (Part I)
41. New Generalized Procedure of Méziriac Regular break and Knight break Methods and Squares (Part II)
42. New Generalized Procedure of Loubère and Méziriac Methods and Squares (Part I)
43. New Generalized Procedure of Loubère and Méziriac Methods and Squares Continuation (Part II)
44. New Generalized Procedure of Loubère and Méziriac Methods and Squares (Part III)
45. New Generalized Procedure of Loubère and Méziriac Methods and Squares Continuation (Part IV)

Introduction

A magic square is an arrangement of numbers 1,2,3,... n² where every row, column and diagonal add up to the same magic sum S and n is also the order of the square. A magic square having all pairs of cells diametrically equidistant from the center of the square and equal to the sum of the first and last terms of the series n² + 1 is also called associated or symmetric. In addition, the center of this type of square must always contain the middle number of the series, i.e., ½(n² + 1).

This site introduces two new methods used for the construction of semi-associated odd magic squares. The method consists of pairing numbers in complementary fashion, partitioning these complementary pairs into groups and placing these groups into a square in a certain order using parity as an aid.

New previous approaches to magic squares, stars, tesseracts have been the subject of the web especially to those interested in recreational mathematics. A good place to start is at harvey heinz webpage which gives examples of these new approaches.

A Discussion of the general Wheel methods

To construct a magic square by these new methods the series 1..n² is paired up in complementary fashion, for example 1 is paired with n², 2 with n²-1, etc. These pair of numbers are then partitioned into n+1 groups of ½(n-1) pairs including the unpaired middle number of the series. This is depicted below using n = 5 complementary table example. We start at n = 5 since the known n = 3 consists totally of "spoke" numbers (explained below).

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

Methods A-1 and A-2

Using method A-1 a magic square is first constructed by filling in the left diagonal with the group of numbers ½ (n²-n+2) to ½(n²+n) in any combinatorial order, as shown below, from the numbers listed in the complementary table above. This produces a template that can be used to (1) fill in the right diagonal with n - 1 numbers in reverse order from bottom left corner to the right upper corner (2) fill in the central column by a select group of n - 1 numbers and (3) fill, in reverse order, the central row by a select fourth group of n - 1 numbers. Method A-2, on the other hand, while being similar to A-1 uses a mix of templates, the normal and its invert, and is discussed under template inversion.

These special numbers are labeled the "spoke" numbers. The remainder cells are labeled the "non-spoke" numbers and these are subsequently filled using adjacent complementary pairs using parity to choose the pairs. These may be added semi-associatively as opposed to the "spoke" pairs which are always associative, i.e diametrically equidistant from the center cell. The following shows an illustration of a partial magic square (n = 5) generated using this method along with its complementary table.

11		5		23
	12	6	22
25	24	13	2	1
	4	20	14
3		21		15

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

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To fill in the "non-spoke" numbers, the sum of the "spoke" numbers on every row are subtracted from S producing pairs of numbers. These pairs are added to the magic square using the complementary table as a guide. Five variant examples on subsequent pages will be shown step by step in Method A-1:Variant 1, three other 5x5 variants Method A-1:Variant 2, three 7x7 variants Method A-1:Variant 2 a 7x7, a 9x9 square Method A-1:a 9x9 Variant and two 7x7 variants Method A-2:Template Inversion a mixed square.

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Method B

Using this method a magic square is first constructed by filling in the left diagonal with the group of numbers ½ (n²-n+2) to ½(n²+n) in any combinatorial order (top left corner to the right lower corner) from the numbers listed in a n x n complementary table. A group of three adjacent pairs, as opposed to three ½(n-1) adjacent pairs used in Method A, are used to generate a partial wheel. This is followed by placing the requisite numbers into the middle column and rows to complete each concentric square until the entire wheel is built up or "expanded". Then using parity, the "non-spoke" numbers are filled similarly to Method A-1 or A-2 as shown in Method B:Wheel Expansion.

Modified Loubère, Bachet de Méziriac and Wheel Methods

These new methods are constructed using complementary tables of (n+2)x(n+2) or greater than the magic square itself (n_s x n_s). This generates a series of De La Loubère or wheel magic squares that are related to one another via the main diagonal within the same complementary set. These n_s x n_s consists of a smaller subset of complementary numbers chosen from the larger 1..n² complementary table as well a five other methods based on Loubère and wheel magic squares. These are included in the indices.