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.
175 questions
1 vote
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Smallest possible "generalized additive-multiplicative magic square" whose entries are distinct nonzero complex numbers?
An additive-multiplicative magic square is a square array of distinct positive integers such that every row, every column, and each of two diagonals all give the same sum $S$, every row, every column,...
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1 vote
2 answers
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Is there a better lowerbound and upperbound for the minimum value and maximum value of a magic square given the target sum?
I have been playing a bit with magic squares, we will consider this definition: Take any $9$ distinct positive integers and put them into a $3\times3$ square. If the sum of the columns, rows, and ...
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0 votes
1 answer
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Dimensions of rational magic squares
The exercises are from Kostrikin's Introduction to Algebra, 3rd edition, Volume 2, Section 1.2, Exercises 10 and 11. Definition Let $A = (a_{ij}) \in M_n(\mathbb{Q})$ be an $n \times n$ matrix. If $A$ ...
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2 votes
2 answers
214 views
Magic Squares + Graphs Problem: Lower and Upper Bounds?
Let $n$ be an integer and consider an $n \times n$ grid. Each cell is a graph on the same number of vertices, which does not have to be $n$. The graph vertices all have the same labels. The first rule ...
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1 vote
1 answer
137 views
Prime Number Pattern in Magic Sum of Pseudo / Semi Magic Squares of Squares
Question: Why does this pattern of prime numbers appear in the magic sum of pseudo-magic squares of squares, and does this represent an infinite "family" of solutions? Consider a magic ...
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0 votes
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Is there a general form of magic squares?
I am a student interested in numbers. One day I was playing with magic squares (the sum of the numbers in each row, each column and each diagonal is the same) when I tried to find a general form of a $...
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1 vote
3 answers
168 views
Parameterization of solutions for $a^2+b^2=c^2+d^2=2f^2$ with integer solutions
I have little to no experience with Diophantine equations, and I have run across this problem while experimenting with magic squares. $a^2+b^2=c^2+d^2=2f^2$ a parameterization for $x^2+y^2=z^2+w^2$, ...
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0 votes
2 answers
163 views
Is it known that the center of a magic square of squares is one third the total?
I was messing around with magic squares and proved that the center value in a magic square of squares must be one-third of what each row, column, and diagonal sum to. I was wondering if this was an ...
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1 vote
2 answers
137 views
The smallest array $N \times N$ in which every row and every column is an emirp.
What is the smallest $N$ for which an $N \times N$ square array of digits exists such that every row, every column, and both diagonals of this array are primes when read in both forward and backward ...
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3 votes
2 answers
139 views
Magic square of square order containing a nice subsquare?
The Benjamin Franklin's magic square (shown at the very end) is a $16 \times 16$ matrix filled with the numbers $\left\{1,2,\dots, 256 \right\}$ with the following nice properties the sum of numbers ...
1 vote
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Applicability of W.S. Andrews' 'Magic Cubes and Squares' for Modern Combinatorics Course
I'm currently enrolled in a combinatorics course and am considering using 'Magic Cubes and Squares' by W.S. Andrews as a supplementary resource. Given that the book is quite old, I would like to know ...
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3 votes
1 answer
209 views
Are incomplete magic squares with some integral entries necessarily purely integral
A magic square is an $n \times n$ table with rational entries (including negative values and $0$) such that the sum of entries in every row, column and both diagonals is some unknown value $k$. Magic ...
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2 votes
1 answer
184 views
Formula for number of $(3,k)$ magic squares
Let $n,k\in\mathbb{N}.$ By a $(n,k)$ magic square, we mean a $n×n$ matrix containing non-negative integer entries such that the sum of entries of any given row or column is $k.$ Note that we don't ...
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9 votes
1 answer
216 views
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 ...
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2 votes
0 answers
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Dimension of $n\times n$ magic squares space is a polynomial in $n$
A classical exercise in basic linear algebra is finding the dimension of the space of $n\times n$ magic squares. The solution usually goes via looking at the defining equation, namely, by noticing ...
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