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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.

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.

For example, using $1\dots9$, this magic square sums to $15$: $$ \begin{matrix}2&7&6\\9&5&1\\4&3&8\end{matrix} $$

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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. How to prove that a normal $3\times 3$ magic…
Archisman Panigrahi
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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). The number $83$ was named after which he took a…
roman
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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 $S_5$. 17 24 1 8 15 23 5 7 14 16 4 6 …
David E Speyer
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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 than $3$? First of all, the determinant of a…
Peter
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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 Theorem, but my question is this: In Section 1,…
Benjamin
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Fewest required values in magic square?

A magic square of order $n$ is an $n \times n$ grid containing each of the numbers $1,2,\dots,n^2$, so that the numbers in each row, column, and diagonal sum to the same number $n(n^2+1)/2$. This question follows on from Is half-filled magic square…
András Salamon
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$3 \times 3 $ Magic Square of Squares

Below is $3 \times 3$ magic square in which seven of the entries are squared integers, found by Andrew Bremner of Arizona State University (and independently by Lee Sallows of the University of Nijmegen): $$ \boxed{ \begin{array} {ccc} 373^2 & 289^2…
Pedja
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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 the main anti-diagonal are the same. Prove that $4…
Juice
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A game of magic Egyptian tilings

Background I've recently been formulating a game that incorporates elements from Egyptian fractions, magic squares, and tilings. It is a single-player game in which the objective is to tessellate a square with sides of length $1$ with tiles that…
Max Lonysa Muller
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Possible solution for the magic square of squares problem

I was fiddling around with this problem for 3x3 magic squares after seeing another Numberphile video and I got to a point where I'm not sure where the error in proving no such magic squares exists is, so I would appreciate someone pointing it…
milin
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Eigenvector of magic square

I'm trying to show: A "magic square" $A$ is a matrix $n\times n$ with slots $1,2,\cdots, n^2$ such that the sum of the elements of each row (and column) is the same . Prove that $\frac{n(n^2+1)}{2}$ is a eigenvalue of the matrix $A$. I was trying…
Hiperion
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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 sum to 80. Each number must be placed only once.…
Pradeep Suny
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Magic square 9, Amazons, and the 2-(81,9,1) design

Consider the following order-7 magic square. The rows, columns, and diagonals all add up to the same sum: 175. Also, all the broken diagonals add up to the same sum, making this a pandiagonal magic square. The square is like a torus -- by shifting…
Ed Pegg
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$n$ by $n$ Primally Magic Squares

(Again copied verbatim from a September 2009 thread I made.) A Primally Magic Square (PMS) is exactly like a traditional magic square with a change of criteria. Where a traditional magic square is one where the sum of each row, column, and both…
El'endia Starman
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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 column have sum equals $r$. Prove that $$H_3(r)…
Parker
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