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Questions tagged [magic-square]

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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Number of weak-sudoku tables

We say that an $n\times n$ table of integers in $\{1,\dots,n\}$ has the weak-sudoku property if each number appears exactly once in each row and each column. The main question is: how many weak-...
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Magic square matrices

A magic square is a square which allow non-negative integers entries in which all row sums and columns sums are equal. Let $H_3(r)$ denotes number of magic squares of size $3*3$ in which each row and ...
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Generate 3x3 magic square without diagonals

The question I recently bought a board game called Novem in which the set up rules indicates that 9 numbered tiles have to be organized in a square with this indication: The sum of the values of ...
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New and hardest question on Magic square matrix

This is about finding the number of magic square matrix. Let $H_3 (r)$ denotes the number of possible magic square matrix of order $3*3$ and sum of line $r$. Prove:- $H_3(r) = \binom{r+4}{4} + \...
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list of “Pythagorean triple” equations for $a^2+b^2=2c^2$

I have come up with an equation recently from a project that I'm doing. it's like the Pythagorean theorem except it's $2c^2$ rather than the familiar $c^2$, then I decided that I wanted a list of all ...
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Proving magic squares determinant is a multiple of 3 when any numbers can be used

I am trying to prove that the determinant of a magic square, where all rows, columns and diagonal add to the same amount, is divisible by 3. I proved it for magic squares which have entries $1,\...
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Magic square with not consecutive numbers

A magic square should be filled with the following numbers: 7,8,9,11,12,13,15,16,17. The numbers: 15, 16 and 17 are already placed as the following: $$\begin{array}{|c|c|c|} \hline &17\\ \...
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Existence of Latin Squares without symmetry around main-diagonal

A Latin square of order $n$ is a matrix $L$ with entries from $[n] \equiv \{0, \dots, n-1\}$ such that each row and column contains every symbol from $[n]$ once. For which orders does there exist a ...
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Algorithm to obtain a magic rectangle of order $3\times 128$

Is there an computer program/ spreadsheet to obtain a magic rectangle of dimensions- $3\times 128.$ Please suggest me an simple algorithm so that i can build it myself if possible.
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variations of a Magic Square Matrix

I am working with magic square matrix, and I would like to know how many variations and how to specify them with n is width(height) of a matrix. For example, this square matrix has n = 3. https://i....
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What are square matrices of this form called?

Is there a name for square matrices whose rows and columns (and only these; indeed, the main diagonal entries always sum to $n \alpha,$ where $n$ is the size of the matrix, and $\alpha$ each entry of ...
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5x5 Magic Square + Even Odd restriction

The Problem Take a $5\times 5$ grid and populate it with the numbers $1$ to $25$ such that the sum of all rows, columns, and both main diagonals are equal. Additionally, only even numbers are ...
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Magic square from $2010$

A multiplicative magic square (MMS) is a square array of positive integers in which the product of each row, column, and long diagonal is the same. The $16$ positive factors of $2010$ can be formed ...
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Reverse engineering a math magic trick involving matrices.

This problem was brought up by my mother from a corporate party along with a question on how that worked. There was a showman who asked to tell him a number from $10$ to $99$ (If i'm not mistaken). ...
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magic square with number theory and sequences& series

A magic square of size N, N ≥ 2, is an N × N matrix with integer entries such that the sums of the entries of each row, each column and the two diagonals are all equal. If the entries of the magic ...
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Magic Bingo Grids

In the realm of video game bingo, it is common to use magic squares to generate cards. If you have $25$ difficulty buckets for goals, then if you lay out those buckets onto a magic square, any bingo ...
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If $G$ is a super edge magic graph with $p$ vertices and $q$ edges then $\displaystyle \sum_{v \in V(G)}f(v)deg(v) $ equals some expression

So I am studying some material on supeer edge magic graphs and the definition is: $G$ (always with $p$ vertices and $q$ edges) is super edge magic iff there is a bijection $f:V(G)\cup E(G) \mapsto \{...
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Magic Square question

Not sure whether this question is correct or not! Please help. Thanks.
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Magic square with multiple math operations

The 3x3 Magic square for 1 thru 9 numbers, and addition only (15) is well known. I am wondering if there are 3x3 Magic squares with answers other than 15 that can be built with multiple math ...
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How do you calculate the determinant of this magic square? [duplicate]

Calculate the determinant of this magic square (which is from Albrecht Dürer's Melancholia) $$\begin{pmatrix} 16 & 3 & 2 & 13 \\ 5 & 10 & 11 & 8 \\ ...
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Looks like a magic square, but can't solve it, please help.

A friend sent me this from her game I tried many combinations and I cannot solve it for the life of me. 9 squares starting from each corner is (top left 61, top right 61, bottom left 69, bottom ...
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a special problem about numbers assigned on polygons

So the problem is stated as follow: We have $a$ numbers of regular $b$-sided polygons. We place them in a fashion such that the sides of polygons are parallel and the vertex of every polygon ...
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What method can be used to solve a magic square problem without just guessing?

The problem is as follows: Fill out the empty boxes of the figure from below by writing an integer number on each of them such in a way that by summing three numbers which are in the same row, ...
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Number Theory - Regular Magic Squares (4x4)

Ok, so I think I found a neat way to solve a $4\times4$ regular magic square (A regular square has one of each number from $0$ to $n^2-1$ and is base $n$. So here's how I did it. We have $16$ ...
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Simultaneous Linear Congruences With Two Variables

(i) Solve the following simultaneous linear congruences: $$17x+23y ≡ 5\mod29$$$$ 10x+4y ≡ 11\mod29$$ (ii) If we were filling a 29 × 29 magic square using the uniform step method $$x_{j} ≡ 16+17j+23 \...
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Squaring an $n$ magic square

Given the following square, where $a,b,c,d \in \mathbb{N}$: $$\left[\begin{matrix} a&b\\ c&d \end{matrix}\right]$$ How do you find the set of $\mathbb{N}$ that satisfies the following ...
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finding sum of entries in magic square directly

On a three by three checkerboard, the $9$ numbers $1, 3, 9, 27, 81, ..., 6561$ are placed so that the product of the $3$ numbers in any row or column is the same. What is that product as an integer? ...
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Magic square variation

I have a rather difficult variation of magic squares: In the below image, all numbers from 1 to 24 must be placed in the 24 closed areas, in such a way that all numbers in areas of each circle must ...
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Can there be a magic square with equal diagonal sums different from equal row and column sums?

I got a task in programming a program that can detect whether a 4x4 square is a magic square or not. At first, I wrote code that met the requirements for all given examples but I noticed one flaw. I ...
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Why is $nM$ equal to $\sum_{i=1}^{n^2}i$?

In this video it is said that $nM$ for a given magic square is equal to $\sum_{i=1}^{n^2}i$, and then the result is also used for magic hexagons. Why does this have to be the case, both for squares ...
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Does there exist an algorithm to find solutions to semi-magic prime squares?

Below is a semi-magic prime square that I have discovered, for which all the entries of the square are prime except $1$, and the sum of all the rows and columns are equal to $37$, also prime. $$\...
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Magic Square - Nigeria Olympiad Second Round 2017 #9

Decide whether the integers $1,2,...,100$ can be arranged in cell $C(i,j)$ of $10×10$ $matrix$ (where $1 \le i,j \le 10$), such that following conditions are satisfied : In every row, the entries ...
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Solving magic square results in incorrect answer

I'm trying to help my daughter with her homework. I'm not looking for an answer, but the process. She has been asked to solve this: ...
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Magic squares in combinatorics

Let $P_{3}(r)$ be the number of 3 x 3 magic squares that are symmetric to their main diagonal. Prove that $P_{3}(r) \leq (r+1)^3$. $r$ in this problem seems to be the sum of each row and column ...
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the dimension of reflective magic squares

An $n \times n$ magic square $A = [ a_{ij}]$ with a magic sum $\mu$ is said to be reflective if $$a_{ij}+a_{n+1-i, n+1-j}=\frac{2\mu}{n}, \forall i,j = 1, \ldots n.$$ If $\mu=0$, we say that it is ...
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Construct $4 \times 4$ magic square with fixed “1”

The method I have found to generate $4\times 4$ magic squares gives me a result in which the number "1" is at of the corners of the square. How can we extend this to a method to generate a magic ...
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Find Hilbert Basis of Magic Square Equations

How to find Hilbert basis of these equations? $x_1 + x_2 + x_3 = x_4 + x_ 5 + x_6 = x_7 + x_8 + x_9 $ $x_1 + x_4 + x_7 = x_2 + x_ 5 + x_8 = x_3 + x_6 + x_9 $
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is there any size $N\cdot N$Sudoku Puzzle where the smaller $\sqrt N\cdot\sqrt N$ squares all form magic squares

Any $N\cdot N$ Sudoku Puzzle has $N$ squares size $\sqrt N\cdot\sqrt N$ that each have the numbers $1$ to $N$ in them.Is there a sudoku puzzle of any size that have magic squares for all of these sub-...
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Arithmetic sequence magic square

If I know that for all $z*z$ magic squares, I can have a magic square consisting of all the terms of the set (1,2,3,...$z^2$), then how do I prove that I can have a magic square consisting of an ...
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Magic square but with multiplication

A 3x3 square is filled out with 9 positive integers such that the product of each row, column, and diagonals are equal. The sum of all 4 corners is less that 10. Find all possible configurations In ...
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Fill out a 3x3 square with 9 different positive integers such that the product of each row, column, and diagonal is equal to each other

I have an idea, which is to put 2 in the middle and have the rest multiply up to an even number, but I can't seem to find an even number with that many factors.
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Showing a set of matrices is a basis of all 3x3 magic squares

I am trying to show that $$\left\{\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix},\begin{bmatrix} 0 & 1 & −1 \\ −1 & 0 & 1 \\ 1 & −1 &...
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Find missing number in grid. [closed]

I'm solving a missing number in square problem. I have tried multiple ways but haven't been able to find any common rule. My attempts are- $((5+7+6)+1) \times 6=114$ $4 \times ((3+5)+1) = 36$ ...
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How many combination has a magic square

Let's assume that I have magic square 3x3 and I know middle number. How many combinations can I create if I try to solve that? Is there a way how to find them all?
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Is there a solution for parker's square?

From wiki: Mathematician Matt Parker attempted to create a 3x3 magic square using square numbers in a YouTube video on the Numberphile channel. His failed attempt is known as the Parker Square. I ...
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What are the possible sums of an $n \times n$ magic square?

An $n\times n$ magic square summing to $S$ is an assignment of distinct integers to the $n^2$ entries of an $n \times n$ grid such that each row, column, and main diagonal sums to $S$. It is well ...
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Ramanujan's method for solving a Magic Square

Ramanujan came up with a formula for solving a magic square. Theorem 1: Let $m_1,m_2$ denote the sums of the middle row and middle column respectively for a $3\times 3$ square array of numbers. Let ...
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Find basis vectors of the vector space of all $4 \times 4$ magic squares

I'm taking a course in linear algebra and I need to solve this problem: Let's define a magic square as a matrix whose sums of all the numbers on a line, a column and on both the main diagonal and ...
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$3\times3$ Magic Square and hidden formula [Solved]

I was doing the $3 \times3$ Magic Square of Squares problem -- found here: http://www.multimagie.com/English/SquaresOfSquaresSearch.htm -- and I figured out that if such Magic square exists with $9$ ...
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Steep Diagonals and Magic Squares - Prove and State a Theorem

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 ...