Varino Magic Square

Varino Magic Square

Armahedi Mahzar (c) 2016

In my last blog I presented a new kind of magic square built up from monominoes each containing 1 to 16 dots. In this blog I will present you another form of magic square  containing varinoes magic as its elements. A varino is a single dot monomino with the the color of the dot and the the color of the background are varied. What are the puzzles around varinoes ?

Varino Magic Squares

If we have 4x4 square, can we distribute all possible 4-colors varinoes such that the dot colors and its background colors are not repeated in the same row, column and diagonal?
The following is one of the possible solutions:



Well, since every row column and diagonal has 4 different colored square and 4 different colored dots, they have the same set colored squares and the same set of colored dots, we will called the set pair as the magic set pair.

So, we have generalized the numeric magic square to a general magic square by replacing number with any similarly type element, and replacing the numeric addition with the set union. In this case the elements is the varinos.

Relation to Numeric Magic Square

In fact, the varino magic square is equivalent to numeric magic square, as it will be seen if we proceed the following replacement procedure:
If we replace
•    the red square by the numeral ‘0’ and the red dot by the numeral ‘0’
•    the blue square by the numeral ‘1’ and the blue dot by the numeral ‘1’
•    the green square by the numeral ‘2’and the green dot  by the numeral ‘2’
•    the yellow square by numeral ‘3’ and the yellow dot by the numeral ‘3’.
then we get the following numerical magic square

02	11	23	30
33	20	12	01
10	03	31	22
21	32	00	13

with the magic number 66 as the sum of all four numbers in any row, column and diagonal.
This numerical magic square can be transformed to the following numerical magic square

  2    5   11    12
15    8    6       1
  4    3   13    10
  9  14     0      7
with magic number 30, if we read the previous square as quaternary number base 4.
If we add 1 to every number in the last magic square, then we get the ordinary magic square of order 4

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

which isomorphic to the monominoes in the last blog

Afternote

In the next blog we will replace the varinoes with the combinoes which symbolize subsets of a set to get the combino magic square. This is another generalized magic square. Hopefully you will enjoy it as a mathematical recreation.

Unomino Magic Square

Unomino Magic Square
Armahedi Mahzar (c) 2015

A domino is two connected squares, each containing many dots or none. A unomino  is a single square containing dots like dominos. We can rearrange the little black square places so each column, row and diagonal is containing exactly the same numbers of dots to get


The unonimo magic square is isomorphic to the 3x3 magic square known as Lo Shu was discovered thousand years ago in the back of mythical turtle by Fuh-Shi,  the mythical founder of Chinese civilisation in around 2400 BC.


Before they invented the zero numeral, the Arabs used alphabets as the written symbols of numbers. Here is the 3×3 Magic Square



In fact there are bigger and bigger Magic Squares.

For example, the earliest 4X4 Magic Square is discovered in Khajuraho India dating
from the eleventh or twelfth century.



Later, another 4X4 Magic Square  was found in Albert Dürer’s engraving ” Melencolia”, where the date of its creation, 1514 AD. See it under the bell.



The previous nine monominoes is only part of larger set of monominoes containing dots from 1 up to 16. These are the sixteens monominos arranged is 4x4 square

We can rearrange the unominoes such that each column, row, diagonal and little 2×2 square is containing exactly the same number of dots. For example the following is the unomino magic square of order 4.
 


This magic square  is wonderful. Because all the collumn. rows and diagonals are containing 34 dots. The 2×2 center square is also containing 34 dots. The dots in each corner 2×2 squares are 34.

This is only one solution of the Puzzle. The French mathematician Frenicle de Bessy   in 1693 enumerated the number of all possible 4×4 Magic Square and get the number  880. 

Magic Square of the Sun

Magic Square of the Sun,
The Sun's I-Ching.



If you draw a line sequentially from 1 to 64 you will see the double pattern of the permutations of life, represented by each kua, in The Mayan Factor by Jose Arguelles. It is also called The Mind of the Earth, The Van Allen Belts, Psi Bank, Oversoul, World Soul of Plato, Psychic Bank and Sensorium of Harold.
This magic square pertains to life (I-Ching), not light (Tzolkin), 

If the Magic Square of 8 x 8, was found by the Mayans, to represent the mind of the earth, then the Sun, being a more massive, hence, more organized form, should be found to represent an even larger, more organized Magic Square as indicated in the 7 levels of Magic Squares in File, "14 Space Patterns".

If you will connect straight lines between each two numbers of this magic square, sequentially, beginning with 1, you will discover a very ornate, 3-part pattern that should describe the Suns 256 (16 x 16) mind permutations (I-Ching).

This 16 x 16 Magic Square corresponds to the 8 x 8 of the earth, therefore I concluded that it is embedded in what I call a Sun Tzolkin of unknown number. To find that number I divided the earth Tzolkin (260) by the I-Ching 64 (8 x 8) = factor 4.0625, then multiplied factor 4.0625 x 256 (16 x 16), the number of squares in the Sun Magic Square, representing the Suns I-Ching.

This gives 1040 squares in the Suns Tzolkin, but, upon taking the square root of 1040 I expected a round number like 32, 32 being the assumed next larger Magic Square (32 x 32), following the binary progression of nature namely 2, 4, 8, 16, 32, 64, but to get 32 as an expected number one has to take the square root of a lesser number, 1024, as the square root of 1040 = 32.24903.

Summary:
The square root of:
The I-Ching, 64, of the earth = 8
The I-Ching, 256, of the Sun = 16
The Tzolkin, 260, of the earth = 16.124515
The Tzolkin, 1040, of the Sun = 32.24903

So it seems these ratios following are appropriate to describe earth and Sun mind, remembering that the I-Ching is embedded within the Tzolkin:

64 : 260 :: 256 : 1040 or in words:
64, the earths I-Ching is to 260, the earths Tzolkin, as
256, the Suns I-Ching, is to 1040, the Suns Tzolkin.

Note that the number 52 does not occupy the precise center, but shares it with three other numbers, 77, 181 and 204 and is fractally involved with the Mayan 52 years, 5,200 year cycle and Bodes Law of Sun / Planet distances (5.2), for Jupiter.

200	217	232	249	8	25	40	57	72	89	104	121	136	53	168	185
58	39	26	7	250	231	218	199	186	167	154	135	122	103	90	71
198	219	230	251	6	27	38	59	70	91	102	123	134	155	166	187
60	37	28	5	252	229	220	197	188	165	156	133	124	101	92	69
201	216	233	248	9	24	41	56	73	88	105	120	137	152	169	184
55	42	23	10	247	234	215	202	183	170	151	138	119	106	87	74
203	214	235	246	11	22	43	54	75	86	107	118	139	150	171	182
53	44	21	12	245	236	213	204	181	172	149	140	117	108	85	76
205	212	237	244	13	20	45	52	77	84	109	116	141	148	173	180
51	46	19	14	243	238	211	206	179	174	147	142	115	110	83	78
207	210	239	242	15	18	47	50	79	82	111	114	143	146	175	178
49	48	17	16	241	240	209	208	177	176	145	144	133	112	81	80
196	221	228	253	4	29	36	61	68	93	100	125	132	157	164	189
62	35	30	3	254	227	222	195	190	163	158	131	126	99	94	67
194	223	226	255	2	31	34	63	66	95	98	127	130	159	162	191
64	32	33	1	256	225	224	193	192	161	160	129	128	97	96	65

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Codon Anti-codon Magic Square

 CODON ANTI-CODON MAGIC SQUARE

  The original name was: "The sequence is an anti-diagonal of the decimal of a mapped 4-ary Gray code matrix as a triangular sequence."

  Gary W. Adamson's explanation of the sequence: Here's the conversion rules for the codons, 4-Ary gray code, which "turns out" to be the most appropriate format for mapping the Codons on a gray code Karnaugh map. The "why" this is the appropriate format relates to a degree of trial and error to find the proper fit in terms of the numbers of hydrogen bonds per codon- anticodon.

(Antti Karttunen's comment: obscure definition. The "degree of trial and error" should be defined transparently.)

  The "H-bond codon-anticodon magic square" map by Gary Adamson, published on page 287 of Cliff Pickover's book "Zen of Magic Squares..." looks like this:
  

CCC CCU CUU CUC UUC UUU UCU UCC

CCA CCG CUG CUA UUA UUG UCG UCA

CAA CAG CGG CGA UGA UGG UAG UAA

CAC CAU CGU CGC UGC UGU UAU UAC

AAC AAU AGU AGC GGC GGU GAU GAC

AAA AAG AGG AGA GGA GGG GAG GAA

ACA ACG AUG AUA GUA GUG GCG GCA

ACC ACU AUU AUC GUC GUU GCU GCC

Using the conversion rules:
0 = C, 1 = A, 2 = G, 3 = U,
we convert to 4-ary gray code:
 

000 003 033 030 330 333 303 300

001 002 032 031 331 332 302 301

011 012 022 021 321 322 312 311

010 013 023 020 320 323 313 310

110 113 123 120 220 223 213 210

111 112 122 121 221 222 212 211

101 102 132 131 231 232 202 201

100 103 133 130 230 233 203 200

To convert back to decimal:
 

+0 +3 14 15 58 57 62 63

+1 +2 13 12 59 56 61 60

+6 +7 +8 11 54 55 50 49

+5 +4 +9 10 53 52 51 48

26 25 30 31 32 35 46 47

27 24 29 28 33 34 45 44

22 23 18 17 38 39 40 43

21 20 19 16 37 36 41 42

... and that's it! Notice how the 1,2,3... jumps around, somewhat like a Peano curve, from one 4-unit cell to the next.

  Antti Karttunen's notes: The steps 1 & 2 are clear, but the step 3 would not produce the array given here, but instead the array A163239. Furthermore, in Pickover's book the conversion rules C=0, A=1, U=2, G=3 are used, in which case we get the array A163235. 

 

Also, the path taken by the terms does not form a continuous Peano curve (Hamiltonian path), because there are discontinuities, e.g. when going from 3 to 4, or from 15 to 16. See A163357/A163359 & A163334/A163336 for examples of continuous Peano/Hilbert curves/paths in an NxN grid. However, this sequence is uniquely defined by the formula a(n) = A163485(A057300(A054238(n))). The 8x8 array given at the step 3 is the top left corner of the infinite square array whose antidiagonal gives this sequence.

REFERENCES

Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002, pp. 285-289.

 

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