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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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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Some conjectures based on surprising observations on magic squares

Let us restrict ourselves to odd magic squares constructed using the Uniform Step Method of Lehmer (See [Apostol51] for the construction technique). A magic square constructed using Lehmer’s Uniform ...
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Antimagic square induction problem

I'm currently trying to solve all the problems in Jay Cumming's book of proofs, and on the induction section, I'm having trouble solving a certain problem involving antimagic squares, the exercise is ...
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Finding or parametrizing integer solutions to $pq(p^2-q^2)=rs(r^2-s^2)$

Background: The order-3 magic square of squares problem (MSS3) is a well-known open problem that involves finding eight separate arithmetic progressions of three squares (APSs). In particular, two ...
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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, ...
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Magic square $29\times29$: Linear Congruences and Uniform Step Method

This linear congruency I was given is part 1 to a 2 part question, I was able to get this. [ \begin{split} 14\cdot27(x+y)\equiv14\cdot16&\pmod{29}\Longrightarrow\\ ...
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Show that $det(A)$ is divisible with the sum of all elements in $A$.

We have $A(3×3)$ matrix (the sum of the elements on each row, column and diagonal are the same) with non-zero natural entries. Show that $det(A)$ is divisible with the sum of all elements in $A$. I ...
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Finding a rational (or integral) parametrization for a system of (homogeneous?) quadratic forms?

The solution of the $3 \times 3$ magic square of squares involves finding $8$ arithmetic progressions of squares (APS) using $9$ integers. I'm looking at a much smaller portion of the problem: $3$ ...
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Pythagorean "like" quadruples, help with general solutions.

a long time ago I posted a question to find a general solution to a modified Pythagorean equation, mainly $a^2+b^2=2c^2$ that question was eventually answered. But now I need more help. I now have 3 ...
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Finding all valid 5x5 sudoku/bingo boards where diagonals must also be unique?

I'm working on a personal project to create a bingo card for a video game. The bingo card contains items the player can use during normal play, and my goal is to be able to generate a bingo board ...
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Magic Squares of Squares

Let $A =$ $\{a_1, a_2, ..., a_n\}$ and $B =$ $\{b_1, b_2, ..., b_m\}$ be two sets of integers. If $a + b$ is a square for all $ a ∈ A $ and $ b ∈ B $. $A$ and $ B $ are then said to be Square Additive ...
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Understanding specific explanation of centre value of 3x3 magic square

EDIT. Put another way, we know from the summation that each line must equal 15. 4 lines should equal 60, but there is some over-counting. How can we know that this over-counting is by 15, without ...
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Magic squares for everybody: for statesmen and pedestrians

The book [1] is a book focused mainly in Franklin's magic squares but it has very interesting and suggestive sections and paragraphs that accompany this topic (summarizing is a jewel). I refer that ...
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Minimum cost of forming a magic square

Hello i have a question regarding finding the minimum cost of converting a 3 x 3 matrix in to a magic square. So i have a simple question, why can't we solve it by finding the sum of the each row then ...
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Euler's 1779 Conjecture magic squares with 6x6 grid

I must be missing something, because it seems this question about Euler's 1779 Conjecture from Quanta Magazine is trivial: "Six army regiments each have six officers of six different ranks. Can ...
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Magic square of square

I am trying to solve magic square of square First let me explain my approach when will be the magic square form if we have $$\begin{array}{|c|c|c|} \hline A² &B²&C² \\ \hline D²&E²&F²\\...
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Is this magic square solvable with no more information? [closed]

This magic square question was given to my brothers sixth grader: ...
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How far have people checked if there is a 3×3 magic square of squares?

Basically the titles says it all. I wrote a small Python program to search for 3×3 magic squares with square number entries (using the property of magic squares that Edouard Lucas has discovered -->...
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Why does the below process for a 4x4 grid with consecutive numbers 1-16 yield the magic constant 34?

I have encountered a procedure that produces the magic constant 34 for a 4x4 grid with the numbers 1-16 arranged in consecutive order, and I can't figure out why it works. It is as follows: Arrange ...
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Sum in Magic star puzzle

I have the following problem: Place the first 11 natural numbers in the circles so that the sum of the four numbers at the tops of each of the five sectors-beams of the star equals 25. I came up with ...
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Is it possible for a nonnormal magic square to have the same magic constant as a normal magic square of the same order?

I've been programming some quick and fast solutions to 4x4 magic square problems, and I've come across an assumption I've been using: The only squares with the magic constant 34 are the normal 4x4 ...
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Can we approximate the occurrence of an arbitrary structure in a subset of $\mathbb{N}$ from the growth rate of that set?

I'm wondering if it makes sense to use the growth rate of a subset of $\mathbb{N}$ to approximate when an arbitrary structure or construction that is defined independently from the set is likely to ...
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Question on The multiplicative magic square

Consider this magic square: $$\begin{array} {|r|r|}\hline a & b & c \\ \hline d & e & f \\ \hline j & h & i \\ \hline \end{array} $$ Where $a,b,c,d,e,f,j,h,i\in \mathbb N^*$ ...
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about magic square

Let us define a magic square as a matrix whose entries are rational numbers and have the same sum on columns, rows and both main diagonals. I am trying to prove the exercise : For every $n\ge4$, there ...
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Magic squares and matrices

A $3 × 3$ magic square with entries in $\mathbb{R}$ is a $3 × 3$ matrix $A = (a_{ij} )$ $\in$ $\mathbb{R}^{3× 3}$such that the sums of the numbers in each row, each column, and both main diagonals are ...
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Find the matrix of a Transformation $\Bbb R^{3×3} \to\Bbb R^8$

Denote by $MQ$ the set of 3×3 magic squares and let $MQ_a$ be the set of magic squared with magic sum $a$. The sets $MQ$ and $MQ_0$ are vector spaces over $\Bbb R$ with matrix addition and scalar ...
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Is there a link between two magic squares with the same constant?

For instance we consider the magic squares of order $3$ with the constant $15$. We can find : \begin{array}{ | l | c | r | } \hline 8 & 3 & 4 \\ \hline 1 & 5 & 9 \\ \...
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The set of all semi-magic squares is a subspace of the vector space of 3 × 3 matrices.

I am interested in proving the following statement and would appreciate some guidance or help: ...
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Sets of integers that can be placed in a magic square

Given $X$, a set of $n^2$ integers what are necessary and sufficient conditions for forming a magic square with them? For eg: if $X$ has members from an arithmetic sequence, we can create a magic ...
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Solving a 5x5 Magic Square

Is there any strategy involved in solving a 5x5 magic square like this? The array above the square are all the missing values.
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Solving a $3\times 3$ magic square with exponentiated entries.

In one of my previous questions, I sought the solution to the most elementary magic square. This time, I seek an answer to a much more complicated case. I seek solutions for the following magic square(...
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Given a 4*4 square where all rows columns and diagonals must sum to a given value, what's the min number of squares needed to make the solution unique

If I have a $4\times 4$ square where all the rows columns and $2$ main diagonals must sum to a specific given value (same in each case), what's the minimum number of squares that are required to be ...
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Why multiple of nine with special order produce regular shape with permutations?

Few years ago, by making some kind of introspection into basic numbers, i've found some interesting things in multiple of nine. So far I have not found an answer to my questions, so I allow myself to ...
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What type magic square did I just create?

I was just putting numbers on the paper and creating some patterns and while doing so, I noticed something. 1 2 3 4 2 4 1 3 3 1 4 2 4 3 2 1 This gives the value ...
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How can a compact complete $4\times4$ magic square be constructed, with 4 given values in the top row?

An $n\times n$ magic square with magic total $T$ is termed compact if every $2\times2$ block has the sum $4T/n$. By a block is meant those elements where two neighbouring rows (or the top and bottom ...
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Variations on Ramanujan's magic square

There is a well known Ramanujan's magic square of the 4th order (see Figure 1.). The underlying structure of Ramanujan's magic square is given in Figure 2. $$\begin{array}{ |c|c|c|c| } \hline 22 &...
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Find $y+z$ in the magic square, understanding the solution

I am having trouble understanding the alternate solution in my algebra book on system of equations for the problem below. In the magic square shown, the sums of the numbers in each row, column, and ...
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How many $4\times4$ normal magic squares have all subsquares sum to the magic constant?

How many essentially distinct normal $4\times4$ magic squares have all $2\times2$ subsquares sum to the magic constant? Relevant information I apologize if this basic question has already been asked ...
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3 x 3 Grid Square

For a question that I need to answer, we have to show that there is only one arrangement for which, in a $3\times 3$ grid in which the numbers from $1$ to $9$ are placed and $5$ is the centre number, ...
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Proof of Magic Constant Formula

A magic square is a NxN square grid filled with distinct positive integers in the range 1,2...$N^2$. Each cell contains a different integer. The sum of the integers in each row, column and diagonal is ...
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Find matrix for linear transformation L(M) = transpose(M) in the given basis B of 3x3 magic-squares

I already found a basis B for 3x3 magic-squares, but I am unsure of where to start on part b. Finding the transformation matrix for $L(M) = \text{transpose}(M)$ in said basis.
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More conditions for 3x3 magic squares

so, there are the base rules for 3x3 magic squares, rows, columns, and diagonals add to the same number known as the "magic number", all numbers are distinct, and only natural numbers are used. But I ...
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Problem of values in a $3\times3$ magic square

We give the following integers : $$\left[\begin{matrix} -2 & x_2 & x_1 \\ 17 & -8 & 9 \\ 3 & x_3 & x_4 \end{matrix}\right]$$ The magic constant is equal to $18$ then I found ...
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Is it possible complete a $3\times3$ magic square such that

Is it possible complete a $3\times3$ magic square such that the constant is equal to $(-22)$ and the initial form is : $$ \left[ \begin{array} {ccc} b& (+2) & a \\ d & c & (-34) \\ ...
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I don’t get how this card trick works

So my friend made a square like this with this amount of cards in each pile 4 3 2 1 3 2 2 3 1 2 3 4 He then said that each line, horizontal and vertical, ...
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How to show there is no magic cube of order 2?

I am reading Richard A. Brualdi's Introductory Combinatorics (Ch1.2). Three-dimensional analogs of magic squares have been considered. A magic cube of order n is an n-by-n-by-n cubical array ...
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Proof for Ramanujan's Oblong magic square

In his first notebook, Ramanujan discusses a $3 X 4$ magic square which he calls oblongs. In this he suggests that following would be the elements of a magic square: $$ \begin{array} {|r|r|r|} \hline ...
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Is it possible to generalize Magic Squares to infinite dimensional matrices?

I know that magic squares exist: Summing over every row or column and diagonal one gets the same sum. My question is whether it is possible to generalize magic squares in such a way that the numbers ...
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packing uniform cuboids into regular cube

I have a pile of uniform cuboids, of side a,b,c. I would like to make a regular cube. The sides are not in harmonic ratio ( cf. de Bruijn) but are in fact white sugar lumps. The minimum regular cube ...
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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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