Mathematics

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 13 20 22 10 12 19 21 3 11 18 25 2…
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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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 to…
Hiperion
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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 out.…
milin
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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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