# 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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### 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 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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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-...