Magical Squares (The Science of ...)	by Peter Loly
loly (at) cc (dot) umanitoba (dot) ca	home.cc.umanitoba.ca/~loly/
Design: The 3-by-3 motif (9 cells) of the ancient Loshu magic square link to topics through cells: I,II,III,IV,V,VI,VII,VIII,IX.	Revised 12,22 April 2012. More design: fills Safari in landscape on iPad2 for B&W printing.

IV. Dϋrer (1514) S=34 (r=3) 0ne of 880.

16	3	2	13
5	10	11	8
9	6	7	12
4	15	14	1

Myers-Briggs Type Table:

ISTJ	ISFJ	INFJ	INTJ
ISTP	ISFP	INFP	INTP
ESTP	ESFP	ENFP	ENTP
ESTJ	ESFJ	ENFJ	ENTJ

Quarto! – a board game.

IX. Abul Wafa al-Buzani (10CE) before Sudoku! Compound magic square (r=5) [Chan and Loly]

31	36	29	76	81	74	13	18	11
30	32	34	75	77	79	12	14	16
35	28	33	80	73	78	17	10	15
22	27	20	40	45	38	58	63	56
21	23	25	39	41	43	57	59	61
26	19	24	44	37	42	62	55	60
67	72	65	4	9	2	49	54	47
66	68	70	3	5	7	48	50	52
71	64	69	7	1	6	53	46	51

II.n=2: NO NATURAL MAGIC SQUARE

Latin

Σ	©
©	Σ

2 symbols, e.g. 0,1 or 1,2

Natural

1	2
3	4

Diagonal sum 5

Cyclic

1	2
4	3

Column sum 5

Crossover

1	3
4	2

Column sum 5

The population problem with magic squares.

Order 12.544 magic square.

3D cubes,

4D hypercubes

Pandiagonal logic squares.

Moment of inertia theorem for magic squares and for magic cubes.

NKS2006 paper.

Ho-t’u

Tetraktys: 1+2+3+4=10

III. Lo-Shu magic square S=15 (r=3)

4	9	2
3	5	7
8	1	6

This prototype magic square has all Row, Column and main Diagonal sums equal to 15. RCD symmetry. Generally for order n with elements: 1..n² the linesum is S_n=(n/2)(1+ n²).

V. 275,305,224 distinct magic squares

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

S=65 (r=4) [Schroeppel

1972]

VII. Order 7: the magic square of Venus

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

Singular Values (σ_i) (SVs)

Rank (r)

Clan

Index

VIII. magic Franklin 2006 (r=3) BF151, BF81

1	32	38	59	5	28	34	63
46	51	9	24	42	55	13	20
27	6	64	33	31	2	60	37
56	41	19	14	52	45	23	10
11	22	48	49	15	18	44	53
40	57	3	30	36	61	7	26
17	16	54	43	21	12	50	47
62	35	25	8	58	39	29	4

Author’s postage stamp

I. A square has 4 corners, 4 edges and 2 diagonals: the edges point N-S and E-W, the diagonals point NW-SE and NE-SW, for the “8” directions:

VI. Order 6 magic squares estimated by Pinn & Wieczerkowski (1998) AND Trump (c.2001) to number ~5.2210(70)*10^54

1	35	34	3	32	6
30	8	28	27	11	7
24	23	15	16	14	19
13	17	21	22	20	18
12	26	9	10	29	25
31	2	4	33	5	36

The magic square of the Sun.

REFERENCES

REPRINTS

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