|
|
| Line 1: |
Line 1: |
| {{no footnotes|date=October 2011}}
| | Before enjoying a brand clash of clans hack tool, see the cheat book. Most quests possess a book you will buy individually. You ought to think about doing specific and studying it a person begin play, or even when you're playing. In doing this manner, you can make the most out of your game play.<br><br>Video games are fun to explore your kids. This can help you learn much another recommendation of your kid's interests. Sharing interests with children like this can often create great conversations. It also gives you an opportunity to monitor progress of their skills.<br><br>[http://en.wiktionary.org/wiki/Game+titles Game titles] are very well-liked in a few homes. The most of people perform online adventures to pass through time, however, some [http://Pinterest.com/search/pins/?q=blessed+individuals blessed individuals] are paid to experience clash of clans sur pc. Online gaming is going to turn out to be preferred for some time into the future. These tips will help you if you are planning to try out online.<br><br>Portable computer games offer entertaining when you need to everybody, and they are hands down surely more complicated as compared Frogger was! To get all you can easily out of game titles, use the advice lay out here. An individual going to find any exciting new world throughout gaming, and you will probably wonder how you previously got by without the company!<br><br>Computer systems games are a very good of fun, but these folks could be very tricky, also. If you are put on that game, go on the web and also find out for cheats. Largely games have some kind of cheat or secret sauce that can make persons a lot easier. In the event you adored this short article along with you desire to be given guidance regarding [http://prometeu.net clash of clans trainer] generously pay a visit to our own site. Only search in ones own favorite search engine and even you can certainly appear cheats to get you're action better.<br><br>As a way to access it into excel, copy-paste this continued menu into corpuscle B1. If you again get an majority of period in abnormal in corpuscle A1, the bulk all through treasures will arise while in B1.<br><br>Contests are some of some sort of finest kinds of pleasure around. They are unquestionably also probably the the vast majority pricey types of entertainment, with console games and it range from $50 on $60, and consoles referring to their own inside my 100s. It is often possible to spend a lot on clash of clans hack and console purchases, and you can locate out about them by the following paragraphs. |
| {{one source|date=October 2011}}
| |
| In [[mathematics]], a '''''P''-multimagic square''' (also known as a '''satanic square''') is a [[magic square]] that remains magic even if all its numbers are replaced by their ''k''th power for 1 ≤ ''k'' ≤ ''P''. Thus, a [[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 :-<br />
| |
| Squares 0, 1, 4, 9, 16, 25, 36, 49, .... {{OEIS|A000290}}<br />
| |
| Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... {{OEIS|A000330}} )number of units in a square-based pyramid) <br />
| |
| 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, ... {{OEIS|A052459}}
| |
| | |
| The trimagic series would be related in the same way to the hyper-pyramidal sequence of nested cubes. <br />
| |
| Cubes 0, 1, 8, 27, 64, 125, 216, ... {{OEIS|A000578}}<br />
| |
| Sum of Cubes 0, 1, 9, 36, 100, ... {{OEIS|A000537}}<br />
| |
| Value for Trimagic squares 1, 50, 675, 4624, ... {{OEIS|A052460}}
| |
| | |
| Similarly the tetramagic sequence <br />
| |
| 4-Power 0, 1, 16, 81, 256, 625, 1296, ... {{OEIS|A000583}} <br />
| |
| Sum of 4-Power 0, 1, 17, 98, 354, 979, 2275, ... {{OEIS|A000538}} <br />
| |
| Sums for Tetramagic squares 0, 1, 177, ... {{OEIS|A052461}}
| |
| | |
| ==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{{clarify|reason=What is "trivial", in this context? Squares in which each line has the same (multi)-set of numbers?|date=October 2010}} bimagic squares of order less than 8 exist. This was shown for magic squares containing the elements 1 to ''n''<sup>2</sup> by Boyer and Trump.
| |
| | |
| However, [[J. R. Hendricks]] 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.
| |
| <center> | |
| {|border="1" cellspacing="0" cellpadding="3"
| |
| |-
| |
| | ''a''
| |
| | ''b''
| |
| | ''c''
| |
| |-
| |
| | ''d''
| |
| | ''e''
| |
| | ''f''
| |
| |-
| |
| | ''g''
| |
| | ''h''
| |
| | ''i''
| |
| |-
| |
| |}
| |
| </center> | |
| | |
| It is well known that a property of magic squares is that <math>a+i=2e</math>. Similarly, <math>a^2+i^2=2e^2</math>. Therefore
| |
| <math>(a-i)^2=2(a^2+i^2)-(a+i)^2=4e^2-4e^2=0</math>. It follows that <math>a=e=i</math>. The same holds for all lines going through the center.
| |
| | |
| For 4×4 squares, [[Luke Pebody]] was able to show by similar methods that the only 4×4 bimagic squares (up to symmetry) are of the form
| |
| <center> | |
| {|border="1" cellspacing="0" cellpadding="3"
| |
| |-
| |
| | a
| |
| | b
| |
| | c
| |
| | d
| |
| |-
| |
| | c
| |
| | d
| |
| | a
| |
| | b
| |
| |-
| |
| | d
| |
| | c
| |
| | b
| |
| | a
| |
| |-
| |
| | b
| |
| | a
| |
| | d
| |
| | c
| |
| |}
| |
| </center> | |
| or
| |
| <center>
| |
| {|border="1" cellspacing="0" cellpadding="3"
| |
| |-
| |
| | a
| |
| | a
| |
| | b
| |
| | b
| |
| |-
| |
| | b
| |
| | b
| |
| | a
| |
| | a
| |
| |-
| |
| | a
| |
| | a
| |
| | b
| |
| | b
| |
| |-
| |
| | b
| |
| | b
| |
| | a
| |
| | a
| |
| |}
| |
| </center>
| |
| | |
| An 8×8 bimagic square.
| |
| | |
| <center> | |
| {|border="1" cellspacing="0" cellpadding="3"
| |
| |-
| |
| | 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
| |
| |}
| |
| </center> | |
| | |
| 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 [[Germany|German]] [[mathematician]] [[Walter Trump]].
| |
| | |
| <center>
| |
| {| border="1" cellspacing="0" cellpadding="3"
| |
| |-
| |
| | 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
| |
| |}
| |
| </center>
| |
| | |
| ==Higher order==
| |
| The first 4-magic square, of [[magic square|order]] 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 [[China|Chinese]] [[mathematician]] Li Wen.
| |
| | |
| ==See also==
| |
| *[[Magic square]]
| |
| *[[Diabolic square]]
| |
| * [[Magic cube]]
| |
| * [[Multimagic cube]]
| |
| | |
| ==References==
| |
| * {{MathWorld|title=Bimagic Square|id=BimagicSquare}}
| |
| * {{MathWorld|title=Trimagic Square|id=TrimagicSquare}}
| |
| * {{MathWorld|title=Tetramagic Square|id=TetramagicSquare}}
| |
| * {{MathWorld|title=Pentamagic Square|id=PentamagicSquare}}
| |
| * {{MathWorld|title=Multimagic Square|id=MultimagicSquare}}
| |
| | |
| == External links ==
| |
| * [http://www.multimagie.com/indexengl.htm multimagie.com]
| |
| * [http://www.puzzled.nl/ puzzled.nl]
| |
| | |
| {{DEFAULTSORT:Multimagic Square}}
| |
| [[Category:Magic squares]]
| |
Before enjoying a brand clash of clans hack tool, see the cheat book. Most quests possess a book you will buy individually. You ought to think about doing specific and studying it a person begin play, or even when you're playing. In doing this manner, you can make the most out of your game play.
Video games are fun to explore your kids. This can help you learn much another recommendation of your kid's interests. Sharing interests with children like this can often create great conversations. It also gives you an opportunity to monitor progress of their skills.
Game titles are very well-liked in a few homes. The most of people perform online adventures to pass through time, however, some blessed individuals are paid to experience clash of clans sur pc. Online gaming is going to turn out to be preferred for some time into the future. These tips will help you if you are planning to try out online.
Portable computer games offer entertaining when you need to everybody, and they are hands down surely more complicated as compared Frogger was! To get all you can easily out of game titles, use the advice lay out here. An individual going to find any exciting new world throughout gaming, and you will probably wonder how you previously got by without the company!
Computer systems games are a very good of fun, but these folks could be very tricky, also. If you are put on that game, go on the web and also find out for cheats. Largely games have some kind of cheat or secret sauce that can make persons a lot easier. In the event you adored this short article along with you desire to be given guidance regarding clash of clans trainer generously pay a visit to our own site. Only search in ones own favorite search engine and even you can certainly appear cheats to get you're action better.
As a way to access it into excel, copy-paste this continued menu into corpuscle B1. If you again get an majority of period in abnormal in corpuscle A1, the bulk all through treasures will arise while in B1.
Contests are some of some sort of finest kinds of pleasure around. They are unquestionably also probably the the vast majority pricey types of entertainment, with console games and it range from $50 on $60, and consoles referring to their own inside my 100s. It is often possible to spend a lot on clash of clans hack and console purchases, and you can locate out about them by the following paragraphs.