Multimagic square - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a P-multimagic square (also known as a satanic square) is a magic square

Magic square

In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...

that remains magic even if all its numbers are replaced by their kth power for 1 ≤ k ≤ P. Thus, a magic square

Magic square

is bimagic if it is 2-multimagic, and trimagic if it is 3-multimagic; tetramagic for 4-multimagic; and pentamagic for a 5-multimagic square.

Constants for normal squares

If the squares are normal, the constant for the power-squares can be determined as follows:

Bimagic series totals for bimagic squares are also linked to the square-pyramidal number sequence is as follows :-

Squares 0, 1, 4, 9, 16, 25, 36, 49, ....

Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... )number of units in a square-based pyramid)

The bimagic series is the 1st, 4th, 9th in this series (divided by 1, 2, 3, n) etc so values for the rows and columns in order-1, order-2, order-3 Bimagic squares would be 1, 15, 95, 374, 1105, 2701, 5775, 11180, ...

The trimagic series would be related in the same way to the hyper-pyramidal sequence of nested cubes.

Cubes 0, 1, 8, 27, 64, 125, 216, ...

Sum of Cubes 0, 1, 9, 36, 100, ...

Value for Trimagic squares 1, 50, 675, 4624, ...

Similarly the tetramagic sequence

4-Power 0, 1, 16, 81, 256, 625, 1296, ...

Sum of 4-Power 0, 1, 17, 98, 354, 979, 2275, ...

Sums for Tetramagic squares 0, 1, 177, ...

Bimagic square

The first known bimagic square has order 8 and magic constant 260 and a bimagic constant of 11180.

It has been conjectured by Bensen and Jacoby that no nontrivial bimagic squares of order less than 8 exist. This was shown for magic squares containing the elements 1 to n² by Boyer and Trump.

However, J. R. Hendricks

J. R. Hendricks

John Robert Hendricks was a mathematician specializing in magic squares and hypercubes. He has published many articles in the Journal of Recreational Mathematics as well as other Journals.-Early years:...

was able to show in 1998 that no bimagic square of order 3 exists, save for the trivial bimagic square containing the same number nine times. The proof is fairly simple: let the following be our bimagic square.

a	b	c
d	e	f
g	h	i

It is well known that a property of magic squares is that

. Similarly,

. Therefore

. It follows that

. The same holds for all lines going through the center.

For 4×4 squares, Luke Pebody

Luke Pebody

Luke Thomas Pebody is a mathematician who solved the necklace problem. Educated at Rugby School, and competing three times in the International Mathematical Olympiad, Luke Pebody was admitted to Cambridge University at the age of 14 to read mathematics...

was able to show by similar methods that the only 4×4 bimagic squares (up to symmetry) are of the form

a	b	c	d
c	d	a	b
d	c	b	a
b	a	d	c

a	a	b	b
b	b	a	a
a	a	b	b
b	b	a	a

An 8×8 bimagic square.

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

Nontrivial bimagic squares are now (2010) known for any order from eight to 64. Li Wen of China created the first known bimagic squares of orders 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62 filling the gaps of the last unknown orders.

Trimagic square

Trimagic squares of orders 12, 32, 64, 81 and 128 have been discovered so far; the only known trimagic square of order 12, given below, was found in June 2002 by German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Walter Trump

Walter Trump is a German mathematician. He is known for discovering results on recreational mathematics.He has credits from square packing problem and magic tile problem...

1	22	33	41	62	66	79	83	104	112	123	144
9	119	45	115	107	93	52	38	30	100	26	136
75	141	35	48	57	14	131	88	97	110	4	70
74	8	106	49	12	43	102	133	96	39	137	71
140	101	124	42	60	37	108	85	103	21	44	5
122	76	142	86	67	126	19	78	59	3	69	23
55	27	95	135	130	89	56	15	10	50	118	90
132	117	68	91	11	99	46	134	54	77	28	13
73	64	2	121	109	32	113	36	24	143	81	72
58	98	84	116	138	16	129	7	29	61	47	87
80	34	105	6	92	127	18	53	139	40	111	65
51	63	31	20	25	128	17	120	125	114	82	94

Higher order

The first 4-magic square, of order

Magic square

512, was constructed in May 2001 by André Viricel and Christian Boyer.

The first 5-magic square, of order 1024 arrived about one month later, in June 2001 again by Viricel and Boyer. They also presented a smaller 4-magic square of order 256 in January 2003. Another 5-magic square, of order 729, was constructed in June 2003 by Chinese

China

Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Li Wen.

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

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

1	22	33	41	62	66	79	83	104	112	123	144
9	119	45	115	107	93	52	38	30	100	26	136
75	141	35	48	57	14	131	88	97	110	4	70
74	8	106	49	12	43	102	133	96	39	137	71
140	101	124	42	60	37	108	85	103	21	44	5
122	76	142	86	67	126	19	78	59	3	69	23
55	27	95	135	130	89	56	15	10	50	118	90
132	117	68	91	11	99	46	134	54	77	28	13
73	64	2	121	109	32	113	36	24	143	81	72
58	98	84	116	138	16	129	7	29	61	47	87
80	34	105	6	92	127	18	53	139	40	111	65
51	63	31	20	25	128	17	120	125	114	82	94

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

1	22	33	41	62	66	79	83	104	112	123	144
9	119	45	115	107	93	52	38	30	100	26	136
75	141	35	48	57	14	131	88	97	110	4	70
74	8	106	49	12	43	102	133	96	39	137	71
140	101	124	42	60	37	108	85	103	21	44	5
122	76	142	86	67	126	19	78	59	3	69	23
55	27	95	135	130	89	56	15	10	50	118	90
132	117	68	91	11	99	46	134	54	77	28	13
73	64	2	121	109	32	113	36	24	143	81	72
58	98	84	116	138	16	129	7	29	61	47	87
80	34	105	6	92	127	18	53	139	40	111	65
51	63	31	20	25	128	17	120	125	114	82	94

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

1	22	33	41	62	66	79	83	104	112	123	144
9	119	45	115	107	93	52	38	30	100	26	136
75	141	35	48	57	14	131	88	97	110	4	70
74	8	106	49	12	43	102	133	96	39	137	71
140	101	124	42	60	37	108	85	103	21	44	5
122	76	142	86	67	126	19	78	59	3	69	23
55	27	95	135	130	89	56	15	10	50	118	90
132	117	68	91	11	99	46	134	54	77	28	13
73	64	2	121	109	32	113	36	24	143	81	72
58	98	84	116	138	16	129	7	29	61	47	87
80	34	105	6	92	127	18	53	139	40	111	65
51	63	31	20	25	128	17	120	125	114	82	94