Auf Discogs können Sie sich ansehen, wer an Vinyl von MAGIC SILK 2 EP mitgewirkt hat, Rezensionen und Titellisten lesen und auf dem Marktplatz nach. Magic: The Gathering: Das komplexeste Spiel von allen. Überraschung für algorithmische Spieltheoretiker: Nie hätten sie sich träumen lassen. Magic-Komplex, Stadt Nordenham, Niedersachsen, Germany. Gefällt Mal. Magic-Komplex is a personal blog about experiencing personal growth and.
KI schlägt Schachweltmeister, scheitert aber an Magic: The GatheringDarum ist Magic das komplexeste Spiel der Welt. So komplex ist Magic: Sie kamen dabei zu dem Ergebnis, dass Magic nicht immer durch einen. Ein Forscherteam konnte beweisen, dass Magic: The Gathering zu viele Schluss: Das Kartenspiel ist zu komplex für künstliche Intelligenzen. The latest Tweets from Magic-Komplex (@MagicKomplex). Magic-Komplex - Ihr Ansprechpartner in Sachen Dualseelen Beratung, Karmische Beziehungen.
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Die Bestimmung der Gewinnstrategie ist nicht berechenbar. Magic: The Gathering widersetzt sich bisher also erfolgreich jeglichen Codierungsversuchen.
Es ist halt definitiv kein typisches Kartenspiel. Es gibt bald ein Gollum-Spiel! Online-Game: Die Menschheit ausrotten mit der Hilfe von Impfgegnern?
Analoges Gaming ganz ohne Pixel in Klagenfurt. Februar Impressum. See all results. Liebe Streaming Job Reise Zukunft Grün Style Food Musik Kunst Social Media Viral Tech Politik Queer Uni Aktiv Graz Steiermark Klagenfurt Kärnten.
Mannsbuilder: Ein Wiener Projekt zum Thema Männlickeit en. Kindlicher Herzschmerz: von der Mutter und der Heimat getrennt.
Mit Ende zwanzig erkannte ich langsam, warum mir dieser Weg auferlegt wurde, es war meine Berufung anderen Menschen auf ihrem Weg zu helfen, dies konnte ich besonders gut, da ich mich auf Grund meiner eigenen Erfahrung, und meiner Gabe der Empathie besonders gut in Menschen einfühlen konnte.
Hinzu kam, dass ich bereits in frühester Kindheit wusste, dass es da einen besonderen Menschen gibt, nach dem ich lange suchte.
Auch dieser schwierige Lernprozess war nötig um in späteren Jahren andere auf diesem Weg begleiten zu können, seit sind es schon viele Dualseelen die ich zueinander führen durfte.
Durch mein kreatives, künstlerisches Talent entstand sehr schnell etwas neues, einzigartiges und höchst wirksames, die magische Kunst.
Erfahrungen mit Sylvia Suckert. Lee Sallows has pointed out that, due to Subirachs's ignorance of magic square theory, the renowned sculptor made a needless blunder, and supports this assertion by giving several examples of non-trivial 4 x 4 magic squares showing the desired magic constant of Similarly to Dürer's magic square, the Sagrada Familia's magic square can also be extended to a magic cube.
It is also a metaphor for something that is almost right, but is a little off. The constant that is the sum of any row, or column, or diagonal is called the magic constant or magic sum, M.
This can be demonstrated by noting that the sum of 1 , 2 ,. If we think of the numbers in the magic square as masses located in various cells, then the center of mass of a magic square coincides with its geometric center.
The moment of inertia of a magic square has been defined as the sum over all cells of the number in the cell times the squared distance from the center of the cell to the center of the square; here the unit of measurement is the width of one cell.
Then all magic squares of a given order have the same moment of inertia as each other. For the order-3 case the moment of inertia is always 60, while for the order-4 case the moment of inertia is always Dividing each number of the magic square by the magic constant will yield a doubly stochastic matrix , whose row sums and column sums equal to unity.
However, unlike the doubly stochastic matrix, the diagonal sums of such matrices will also equal to unity. Thus, such matrices constitute a subset of doubly stochastic matrix.
This representation may not be unique in general. While the classification of magic squares can be done in many ways, some useful categories are given below.
There is only one trivial magic square of order 1 and no magic square of order 2. As mentioned above, the set of normal squares of order three constitutes a single equivalence class -all equivalent to the Lo Shu square.
Thus there is basically just one normal magic square of order 3. Cross-referenced to the above sequence, a new classification enumerates the magic tori that display these magic squares.
The number of magic tori of order n from 1 to 5, is:. The number of distinct normal magic squares rapidly increases for higher orders. The magic squares of order 4 are displayed on magic tori of order 4 and the ,, squares of order 5 are displayed on ,, magic tori of order 5.
The number of magic tori and distinct normal squares is not yet known for any higher order. Algorithms tend to only generate magic squares of a certain type or classification, making counting all possible magic squares quite difficult.
Traditional counting methods have proven unsuccessful, statistical analysis using the Monte Carlo method has been applied.
The probability that a randomly generated matrix of numbers is a magic square is then used to approximate the number of magic squares.
More intricate versions of the Monte Carlo method, such as the exchange Monte Carlo, and Monte Carlo backtracking have produced even more accurate estimations.
Using these methods it has been shown that the probability of magic squares decreases rapidly as n increases. Using fitting functions give the curves seen to the right.
Over the millennium, many ways to construct magic squares have been discovered. These methods can be classified as general methods and special methods, in the sense that general methods allow us to construct more than a single magic square of a given order, whereas special methods allow us to construct just one magic square of a given order.
Special methods are specific algorithms whereas general methods may require some trial-and-error. Special methods are standard and most simple ways to construct a magic square.
The correctness of these special methods can be proved using one of the general methods given in later sections. After a magic square has been constructed using a special method, the transformations described in the previous section can be applied to yield further magic squares.
Special methods are usually referred to using the name of the author s if known who described the method, for e. De la Loubere's method, Starchey's method, Bachet's method, etc.
Magic squares exist for all values of n , except for order 2. Magic squares can be classified according to their order as odd, doubly even n divisible by four , and singly even n even, but not divisible by four.
This classification is based on the fact that entirely different techniques need to be employed to construct these different species of squares.
Odd and doubly even magic squares are easy to generate; the construction of singly even magic squares is more difficult but several methods exist, including the LUX method for magic squares due to John Horton Conway and the Strachey method for magic squares.
Consider the following table made up of positive integers a , b and c :. The method prescribes starting in the central column of the first row with the number 1.
After that, the fundamental movement for filling the squares is diagonally up and right, one step at a time.
If a filled square is encountered, one moves vertically down one square instead, then continues as before.
When an "up and to the right" move would leave the square, it is wrapped around to the last row or first column, respectively. Starting from other squares rather than the central column of the first row is possible, but then only the row and column sums will be identical and result in a magic sum, whereas the diagonal sums will differ.
The result will thus be a semimagic square and not a true magic square. Moving in directions other than north east can also result in magic squares. Doubly even means that n is an even multiple of an even integer; or 4 p e.
Generic pattern All the numbers are written in order from left to right across each row in turn, starting from the top left hand corner.
Numbers are then either retained in the same place or interchanged with their diametrically opposite numbers in a certain regular pattern.
In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with their diametrically opposite numbers.
A construction of a magic square of order 4 Starting from top left, go left to right through each row of the square, counting each cell from 1 to 16 and filling the cells along the diagonals with its corresponding number.
Once the bottom right cell is reached, continue by going right to left, starting from the bottom right of the table through each row, and fill in the non-diagonal cells counting up from 1 to 16 with its corresponding number.
As shown below:. An extension of the above example for Orders 8 and 12 First generate a pattern table, where a '1' indicates selecting from the square where the numbers are written in order 1 to n 2 left-to-right, top-to-bottom , and a '0' indicates selecting from the square where the numbers are written in reverse order n 2 to 1.
When we shade the unaltered cells cells with '1' , we get a criss-cross pattern. The patterns are a there are equal number of '1's and '0's in each row and column; b each row and each column are "palindromic"; c the left- and right-halves are mirror images; and d the top- and bottom-halves are mirror images c and d imply b.
The pattern table can be denoted using hexadecimals as 9, 6, 6, 9 for simplicity 1-nibble per row, 4 rows. The simplest method of generating the required pattern for higher ordered doubly even squares is to copy the generic pattern for the fourth-order square in each four-by-four sub-squares.
It is possible to count the number of choices one has based on the pattern table, taking rotational symmetries into account. The earliest discovery of the superposition method was made by the Indian mathematician Narayana in the 14th century.
The same method was later re-discovered and studied in early 18th century Europe by de la Loubere, Poignard, de La Hire, and Sauveur; and the method is usually referred to as de la Hire's method.
Although Euler's work on magic square was unoriginal, he famously conjectured the impossibility of constructing the evenly odd ordered mutually orthogonal Graeco-Latin squares.
This conjecture was disproved in the mid 20th century. For clarity of exposition, we have distinguished two important variations of this method.
This method consists in constructing two preliminary squares, which when added together gives the magic square. The numbers 0, 3, and 6 are referred to as the root numbers while the numbers 1, 2, and 3 are referred to as the primary numbers.
An important general constraint here is. The lettered squares are referred to as Greek square or Latin square if they are filled with Greek or Latin letters, respectively.
A magic square can be constructed by ensuring that the Greek and Latin squares are magic squares too.
The converse of this statement is also often, but not always e. Thus the method is useful for both synthesis as well as analysis of a magic square.
Lastly, by examining the pattern in which the numbers are laid out in the finished square, it is often possible to come up with a faster algorithm to construct higher order squares that replicate the given pattern, without the necessity of creating the preliminary Greek and Latin squares.
Satisfaction of these two conditions ensures that the resulting square is a semi-magic square; and such Greek and Latin squares are said to be mutually orthogonal to each other.
For a given order n , there are at most n - 1 squares in a set of mutually orthogonal squares, not counting the variations due to permutation of the symbols.
This upper bound is exact when n is a prime number. In order to construct a magic square, we should also ensure that the diagonals sum to magic constant.
For this, we have a third condition:. The mutually orthogonal Greek and Latin squares that satisfy the first part of the third condition that all letters appear in both the diagonals are said to be mutually orthogonal doubly diagonal Graeco-Latin squares.
Similarly, for the Latin square. The resulting Greek and Latin squares and their combination will be as below. The Latin square is just a 90 degree anti-clockwise rotation of the Greek square or equivalently, flipping about the vertical axis with the corresponding letters interchanged.
For the odd squares, this method explains why the Siamese method method of De la Loubere and its variants work.
This basic method can be used to construct odd ordered magic squares of higher orders. To summarise:. Since there are n - 1! Greek squares this way; same with the Latin squares.
Also, since each Greek square can be paired with n - 1! Dividing by 8 to neglect equivalent squares due to rotation and reflections, we obtain 1, , and , essentially different magic squares, respectively.
Numbers are directly written in place of alphabets. The numbered squares are referred to as primary square or root square if they are filled with primary numbers or root numbers, respectively.
The numbers are placed about the skew diagonal in the root square such that the middle column of the resulting root square has 0, 5, 10, 15, 20 from bottom to top.
The primary square is obtained by rotating the root square counter-clockwise by 90 degrees, and replacing the numbers. The resulting square is an associative magic square, in which every pair of numbers symmetrically opposite to the center sum up to the same value, For e.
In the finished square, 1 is placed at center cell of bottom row, and successive numbers are placed via elongated knight's move two cells right, two cells down , or equivalently, bishop's move two cells diagonally down right.
When a collision occurs, the break move is to move one cell up. All the odd numbers occur inside the central diamond formed by 1, 5, 25 and 21, while the even numbers are placed at the corners.
The occurrence of the even numbers can be deduced by copying the square to the adjacent sides.
The even numbers from four adjacent squares will form a cross. A variation of the above example, where the skew diagonal sequence is taken in different order, is given below.
The resulting magic square is the flipped version of the famous Agrippa's Mars magic square. It is an associative magic square and is the same as that produced by Moschopoulos's method.
Here the resulting square starts with 1 placed in the cell which is to the right of the centre cell, and proceeds as De la Loubere's method, with downwards-right move.
When a collision occurs, the break move is to shift two cells to the right. In the previous examples, for the Greek square, the second row can be obtained from the first row by circularly shifting it to the right by one cell.
Similarly, the third row is a circularly shifted version of the second row by one cell to the right; and so on. Likewise, the rows of the Latin square is circularly shifted to the left by one cell.
The row shifts for the Greek and Latin squares are in mutually opposite direction. It is possible to circularly shift the rows by more than one cell to create the Greek and Latin square.
This essentially re-creates the knight's move. All the letters will appear in both the diagonals, ensuring correct diagonal sum. Since there are n!
Greek squares that can be created by shifting the first row in one direction. Likewise, there are n!
Since a Greek square can be combined with any Latin square with opposite row shifts, there are n! Dividing by 8 to neglect equivalent squares due to rotation and reflections, we obtain 3, and 6,, equivalent squares.
Further dividing by n 2 to neglect equivalent panmagic squares due to cyclic shifting of rows or columns, we obtain and , essentially different panmagic squares.
For order 5 squares, these are the only panmagic square there are. The condition that the square's order not be divisible by 3 means that we cannot construct squares of orders 9, 15, 21, 27, and so on, by this method.
In the example below, the square has been constructed such that 1 is at the center cell. In the finished square, the numbers can be continuously enumerated by the knight's move two cells up, one cell right.
When collision occurs, the break move is to move one cell up, one cell left. The resulting square is a pandiagonal magic square.
This square also has a further diabolical property that any five cells in quincunx pattern formed by any odd sub-square, including wrap around, sum to the magic constant, We can also combine the Greek and Latin squares constructed by different methods.
In the example below, the primary square is made using knight's move. We have re-created the magic square obtained by De la Loubere's method. After dividing by 8 in order to neglect equivalent squares due to rotation and reflection, we get 2, and 3,, squares.
For order 5 squares, these three methods give a complete census of the number of magic squares that can be constructed by the method of superposition.
Even squares: We can also construct even ordered squares in this fashion. Since there is no middle term among the Greek and Latin alphabets for even ordered squares, in addition to the first two constraint, for the diagonal sums to yield the magic constant, all the letters in the alphabet should appear in the main diagonal and in the skew diagonal.
For the given diagonal and skew diagonal in the Greek square, the rest of the cells can be filled using the condition that each letter appear only once in a row and a column.
Dividing by 8 to eliminate equivalent squares due to rotation and reflections, we get essentially different magic squares of order 4.
These are the only magic squares constructible by the Euler method, since there are only two mutually orthogonal doubly diagonal Graeco-Latin squares of order 4.
Euler's method has given rise to the study of Graeco-Latin squares. Euler's method for constructing magic squares is valid for any order except 2 and 6.
Variations : Magic squares constructed from mutually orthogonal doubly diagonal Graeco-Latin squares are interesting in themselves since the magic property emerges from the relative position of the alphabets in the square, and not due to any arithmetic property of the value assigned to them.
This means that we can assign any value to the alphabets of such squares and still obtain a magic square.
This is the basis for constructing squares that display some information e. We will obtain the following non-normal magic square with the magic sum Narayana-De la Hire's method for odd square is the same as that of Euler's.
However, for even squares, we drop the second requirement that each Greek and Latin letter appear only once in a given row or column.
This allows us to take advantage of the fact that the sum of an arithmetic progression with an even number of terms is equal to the sum of two opposite symmetric terms multiplied by half the total number of terms.
Thus, when constructing the Greek or Latin squares,. Thus, we can construct:. The remaining cells are then filled column wise such that the complementary letters appears only once within a row, but twice within a column.
Each Greek letter appears only once along the rows, but twice along the columns. Likewise for the Latin square, which is obtained by flipping the Greek square along the main diagonal and interchanging the corresponding letters.
The above example explains why the "criss-cross" method for doubly even magic square works. Remaining cells are filled column wise such that each letter appears only once within a row.
We proceed similarly until all cells are filled. The Latin square given below has been obtained by flipping the Greek square along the main diagonal and replacing the Greek alphabets with corresponding Latin alphabets.
We can use this approach to construct singly even magic squares as well. However, we have to be more careful in this case since the criteria of pairing the Greek and Latin alphabets uniquely is not automatically satisfied.
Violation of this condition leads to some missing numbers in the final square, while duplicating others. Thus, here is an important proviso:.
The second square is constructed by flipping the first square along the main diagonal. Here in the first column of the root square the 3rd cell is paired with its complement in the 4th cells.
Thus, in the primary square, the numbers in the 1st and 6th cell of the 3rd row are same. Likewise, with other columns and rows. In this example the flipped version of the root square satisfies this proviso.
Here the diagonal entries are arranged differently. The primary square is constructed by flipping the root square about the main diagonal.
In the second square the proviso for singly even square is not satisfied, leading to a non-normal magic square third square where the numbers 3, 13, 24, and 34 are duplicated while missing the numbers 4, 18, 19, and The last condition is a bit arbitrary and may not always need to be invoked, as in this example, where in the root square each cell is vertically paired with its complement:.
Unlike the criss-cross pattern of the earlier section for evenly even square, here we have a checkered pattern for the altered and unaltered cells.
Also, in each quadrant the odd and even numbers appear in alternating columns. That is, a column of a Greek square can be constructed using more than one complementary pair.
This method allows us to imbue the magic square with far richer properties. The idea can also be extended to the diagonals too.
In the finished square each of four quadrants are pan-magic squares as well, each quadrant with same magic constant In this method, the objective is to wrap a border around a smaller magic square which serves as a core.
Subtracting the middle number 5 from each number 1, 2, Harry White, refer to as bone numbers. Putting the middle number 0 in the center cell, we want to construct a border such that the resulting square is magic.
Let the border be given by:. Thus out of eight unknown variables, it is sufficient to specify the value of only four variables.
But how should we choose a , b , u , and v? We have the sum of the top row and the sum of the right column as.
Since 0 is an even number, there are only two ways that the sum of three integers will yield an even number: 1 if all three were even, or 2 if two were odd and one was even.
Hence, it must be the case that the second statement is true: that two of the numbers are odd and one even. The only way that both the above two equations can satisfy this parity condition simultaneously, and still be consistent with the set of numbers we have, is when u and v are odd.
This proves that the odd bone numbers occupy the corners cells. Hence, the finished skeleton square will be as in the left.
Adding 5 to each number, we get the finished magic square. Similar argument can be used to construct larger squares. Consider the fifth-order square.
Disregarding the signs, we have 8 bone numbers, 4 of which are even and 4 of which are odd. In general, for a square of any order n , there will be 4 n - 1 border cells, which are to be filled using 2 n - 1 bone numbers.