Question

Below is a magic square, meaning that the sum of the numbers in each row, in each column, and in each of the $2$ main diagonals are equal. What is the value of $n$? [asy]size(125); for(int i = 0; i<4; ++i) { draw((0,i)--(3,i),linewidth(1)); } for(int j = 0; j<4; ++j) { draw((j,0)--(j,3),linewidth(1)); } label("$n-2$",(.5,.5)); label("3",(.5,1.5)); label("$n+2$",(.5,2.5)); label("$n+3$",(1.5,.5)); label("$n-1$",(1.5,1.5)); label("$1$",(1.5,2.5)); label("$2$",(2.5,.5)); label("$2n-5$",(2.5,1.5)); label("$n$",(2.5,2.5)); [/asy]

Answer 1

The property  of equality of the rows, columns, and diagonals in the magic

square can be used to find the value of n.

The value of n is 6

Reasons:

The magic square can be presented as follows;

$\begin{array}{|c|c|c|}n-2&3&n + 2\\n + 3&n - 1&1\\2&2 \cdot n - 5&n\end{array}\right]$

Given that the sum of the numbers in each row and in each column and in

each of the two diagonals are equal, to find the value of n, two rows having

different number of the variable n can be equated as follows;

The top row = n - 2 + 3 + n + 2 = The bottom row = 2 + 2·n - 5 + n

n - 2 + 3 + n + 2 = 2 + 2·n - 5 + n

2·n + 3 =  3·n - 3

3·n - 3 = 2·n + 3 (symmetric property)

3·n - 2·n = 3 + 3

n = 6

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