A Magic Square of order n is an arrangement of $n^2$ 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. http://en.wikipedia.org/wiki/Magic_square

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Symmetric sums in a magic square

Numbers $1,2,\ldots,16$ are written in a $4 \times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $...
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How can you check if there exists a valid magic square with given initial conditions?

For example, if I have a $4\times4$ magic square that looks like so: \begin{pmatrix} \hspace{0.1ex}2 & 3 & \cdot & \cdot\hspace{1ex} \\ \hspace{0.1ex}4 & \cdot & \cdot & \cdot\...
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How is this a magic square?

I have the following magic square but cannot determine how the square is actually "magic". It's a 3x3 as seen below in red with the green showing a few examples of row sums. Each vertical and ...
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Proof of Rows/Columns/Main Diagonals in a normal Magic Square

Someone else had asked this question but not to enough detail. I understand that you can sum all of the elements from $1$ to $n^2$ that are in the matrix to get $n^2(n^2+1)/2$. Then from there you ...
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On the existence of magic squares of every order different from $2$

I was reading about magic squares and suppose that we speak here only of the magic squares that have in itself numbers from $1$ to $n^2$. It is easy to see that we cannot have $2$x$2$ magic square ...
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Multiplying magic squares like matrices to hopefully arrive at another magic square

Well, the title actually describes what is the problem in question. I was just thinking a bit about magic squares and this question popped-up. It could be that it is not interesting but I do not see a ...
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How do you express a $3\times3$ semi magic square(same sum for each row&col) in form of a set?

By set, I mean like a subspace $W=\{[] \in M_{3\times3} (\mathbb{R})| $something$\}$. Since matrix such as $ \begin{pmatrix} 0 & 0&3 \\ 3 & 0&0 \\ 0 & 3&0 \end{pmatrix}$ $ ...
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Where can I get my super-efficient algorithm for odd order magic squares published? [closed]

It can create any odd size magic square and I believe it can do it faster than any other solution I managed to find. I created this algorithm in 1988 and never told anyone since. It uses no ...
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42 views

How to prove that there are infinite number of magic squares consisted of only consecutive prime numbers?

It is easy to find set of $N^2$ consecutive prime numbers (for small values of N) to build magic square (a square grid, where the numbers in each row, and in each column, and the numbers in the main ...
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36 views

why do odd magic squares have the same rank as their size?

why do odd magic squares have the same rank as their size whats special about odd magic squares? and why do even magic squares alternate The results are below where n is the size and r is the ...
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43 views

Number of Magic squares and their applications

A magic square is a square array of numbers consisting of the distinct positive integers $1, 2, ..., n^2$ arranged such that the sum of the $n$ numbers in any horizontal, vertical, or main diagonal ...
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Least starting fields to full resolve a $5 \times 5$ normal magic sqaure?

The image contains $16$ starting values. The remaining cells can be calculated by arithmetic operations, keeping in mind that the line sum is $65$. Now is there an arrangement where I need less ...
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224 views

Magic square row/colum/diagonal sum proof

When given a magic square of order $N$. How do we know that the sum of integers in the row/column/major diagonal is equal to $N(N^2+1)/2$. I have tested the formula on Durer's "1514" magic square and ...
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Why do siamese magic squares have real eigenvalues, symmetric around zero?

There is a standard method to construct magic squares of odd size, known as the Siamese construction. I'll write $S_m$ for the $m \times m$ Siamese square. For example, here is $S_5$. ...
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Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. (Exercise 705 from Golan, The Linear Algebra a ...
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Magic square with not distinct numbers

There's a 4x4 magic square: 4 0 1 0 3 0 2 0 0 3 0 2 0 4 0 1 Where 0s are different numbers, 1=1, 2=2, 3=3, 4=4. Only the rows and the columns have the same sum, ...
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If $K_n$ is super-magic, what is the sum at each vertex?

If $K_n$ is super-magic, what is the sum at each vertex? A super-magic labeling of a graph is an edge weighting where the edge weights are consecutive integers (that’s the super part), and where if ...
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69 views

What is the sum at each vertex of a super-magic labeling of Kn,n? Explain your answer.

What is the sum at each vertex of a super-magic labeling of Kn,n? Explain your answer. I know that the sum at each vertex of K3,3 equals 15 and K4,4 the sum at each vertex is 34, but I don't know ...
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Exchanging two columns or rows of a regular square yields another regular square

An $n$-by-$n$ square is regular if the two conditions are met: Each of the integers from $0$ to $n^2 − 1$ appears in exactly one cell, and each cell contains only one integer (so that the square is ...
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176 views

Magic Square Diagonals Theorems/Proof

We want to describe via a picture a set of subsets of a square which are something like diagonals, but are not quite the same. We’ll call them steep diagonals. One of them, labelled e, is illustrated ...
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Magic Square given my condition

I am currently studying magic squares and ran into a bit of trouble. The concept I am learning about is a regular square. Below are the conditions of a regular square. We can say that an $n$-by-$n$ ...
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Proof that magic square cannot be transformed

How do I prove that the first magic square below cannot be transformed into the second by a sequence of row and column exchanges. $$(a)\;\pmatrix{15&2&1&12\\4&9&10&7\\8&5&...
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4-by-4 regular square in decimal and base-4

We can say that an $n$-by-$n$ square is regular provided that: Each of the integers from $0$ to $n^2 − 1$ appears in exactly one cell, and each cell contains only one integer (so that the square is ...
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Prove Is a Filled Magic Square

Can someone prove that if $S$ is a filled magic square, and $T$ is obtained from $S$ by switching two rows or two columns, then $T$ is also a filled magic square. So an example that I came up with ...
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Constructing a magic square

I am currently learning about magic squares and I want to construct a magic square. How do I construct a 6-by-6 or a 7-by-7 filled magic square, using the integers 0 to 35 or 0 to 48.
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Prove Magic Square

Suppose that $S$ is a square such that the sum of the entries in each row is some number $R$, and the sum of the entries in each column is some number $C$. Prove that $S$ is in fact a magic square, i....
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Building a 3D matrix of positive integers

I'm trying to build a 3D matrix made up of positive integers that has very specific properties. The matrix dimensions are $N \times N \times (N+1)$ where $N$ is a positive integer. The matrix has two ...
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Magic squares using matrices

Let $M(d)_{d\in \mathbb{R}}\subset M_3(\mathbb{R})$, consisting of all the matrices such that the sum of the elements in each row, column and on the two diagonals is $d$. Such an element $M(d)$ is ...
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173 views

Conditions for magic square.

So I've messing around with magic squares and something occured to me: Assume we have a nxn grid of numbers which respects the sum conditions of a magic square as in it has the appropriate column, ...
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320 views

Magic square solving

I'm trying to solve a 3x3 magic square for 99 which starts at 29. I got the rows, columns and one diagonal but the other diagonal is (way) off. Is there even a magic square which satisfies both ...
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Standard Deviation of a Magic Square

I'm not sure if this is the right stackexchange to post this question to, but I was just wondering if someone had the answer to an interesting observation I've made. I've written a program that ...
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For magic squares prove $id_v+r^2=2c $

I've been presented with the following problem: Let $V$ be the vector space of all $3\times3$ magic squares. Let $r:V\rightarrow V$ be the linear image which rotates a magic square $90^\circ$. Let $c:...
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Sampling from Cartesian product without replacement, but with balanced totals

I am struggling with a combinatorial task that I cannot reduce to any procedure I know: Given two sets $F, G$, I want to sample from $F \times G$ without replacement, but subject to the condition that ...
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Are there associative magic squares of any size except $4k+2$?

An associative magic square is a magic square with the additional property that numbers symmetric to the center sum up to $n^2+1$. For example, the square $\pmatrix{6&9&12&7\\3&16&...
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Are there semimagic knight tours from any starting square?

There are $140$ distinct semimagic knight tours on a normal chessboard ($8\ \times\ 8$). A semimagic knight tour is a knight tour (not necessarily closed) such that a semimagic square appears if the ...
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Are there magic knight tours on a $6\times6$ or $10\times10$ board?

In mathworld, magic tour, it is mentioned that for odd $n$, only semimagic knight tours are possible on a $n\times\ n$ - board. For $n = 8$, it has been verified that there are no magic knight tours,...
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Relaxed magic squares

I found the definition that a relaxed magic square of type $n\times n$ has row and column sums constant, and all numbers from $1$ to $n^2$ appears exactly once. How can one enumerate those, like how ...
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determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order greater ...
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Magic square of a given date

How to create a magic square if we know a date. Eg-22-04-2014 The first column should have 22 2nd-04 3rd-20 and 4th -14. I believe ramanujan created the same thing for his birthday but I don't know ...
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Why does the filling up of odd order magic square with numbers follow the knight movement?

Why does the filling up of odd order magic square with numbers follow the knight movement? I was reading about magic square, where I came up with the knight movement filling up of the magic square ...
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219 views

Strategies for solving Magic Squares

E11/38. By an exhaustive process of elimination I can work this out as 39, but there must be a quicker strategy for solving these kind of questions. Advice please.
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How many nonnegative integer matrices of size $N$ have all row and column sums equal to $D$?

Given the positive integer $N$ and $D$, generate all the non-negative integer matrices which satisfy matrix dimension is $N\times N$; sum of each row elements equals to $D$ sum of each column ...
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I did a proof for magic square of squares and perfect cuboid.

I did the proof for non-existence of magic square of squares. I am looking great guidelines on preparation and submission for publication. the main areas what I want assistance is 1. the format of ...
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Proof of magic square algorithm?

Proof of magic square algorithm? Why does it work? Where the algorithm to create one is to add the next number above diagonally to the right. If you go off the grid, you wrap, as if he grid ...
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Proof that $12$ in a row tic-tac-toe is a tie game?

How can be it proved that tic-tac-toe on an infinite grid (winning with $12$ in a row, a column or a diagonal) can always end in a tie (with optimal strategies of both players)? There is a hint: to ...
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Identity of a Mathematician Mentioned in Euler

I and several others are in the process of translating one of Euler's papers from Latin to English, in particular the one that the Euler Archive lists as E36. In it Euler proves the Chinese Remainder ...
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do you know another Magic Square with this property?

with the repeating digits of $\frac{1}{19} = 0.052631578947368421$ we can construct an exceptional magic square : The number 19 is a cyclic number with a period of 18 before the digits start to ...
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How to prove that a $3\times 3$ Magic Square must have $5$ in its middle cell?

A Magic Square of order $n$ is an arrangement of $n^2$ 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. ...
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1answer
428 views

“Magic matrix” formula inaccuracy

I was reading Martin Gardner's "hexaflexagons and other mathematical diversions" in which he describes a magic matrix - a different type of magic square in which all numbers of different rows and ...
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Mathematical model for magic square

As I spent some time on magic squares, it seems like the magic squares can be formed only with a odd number of rows/columns? Is it that.? If so why? is there a mathematical model that explains magic ...