Question

Fill this square using all the numbers from $21$ to $29$.
Rule: The total of each side is $75$.

$\text{ }$	$\text{ }$	$\text{ }$
$\text{ }$	$25$	$\;\; \;\;$
$\text{ }$	$\;\; $	$\text{ }$

Answer 1

Hint: To fill the square by using the numbers from$21$ to $29$, we will use start from top of the middle column because it is an odd square of $3\times 3$. We will fill the lowest number that is $21$ in that box and then we will use the pattern of up-one and right one to fill the other boxes of squares.

Complete step-by-step solution:
Since, we have given $9$ numbers from $21$ to $29$ for filling a $3\times 3$square. Since, the square is an odd square. So, we will start to fill the number from the top of the centre column with given smallest number that is $21$ as:

$\;\;\;\;\; $	$21$	$\;\;\;\;$
$\;\; $	$25$	$\;\; $
$\;\;\;\;\;\;\ $	$\;\; \;\;\;$	$\;\;\;\;\;\ $

Now, we will fill the next number that is $22$ in the last box of the last row or column by using the method of up-one and right-one as:

$\;\; $	$21$
	$25$
$\;\; $		$22$

Here, we will go further with same rule and will fill the next number in the second box of first column as:

	\[\text{21}\]
$23$	$25$
		$22$

Again we will use the pattern of up-one and right-one to fill the next number but that place is already filled with the number $21$. So, we will fill $24$in the box below the number $23$ as:

	\[\text{21}\]
$23$	$25$
$24$		$22$

Since, the number \[\mathbf{25}\] is already filled in the right place. So, we will go further for number $26$ and it will be filled in the box right next to the number $21$as:

	\[\text{21}\]	$26$
$23$	$25$
$24$		$22$

Here, the place for number $27$ will be last box of the centre row because of same situation as number $24$ as:

	\[\text{21}\]	$26$
$23$	$25$	$27$
$24$		$22$

Now, by using up-one and right-one pattern we will fill the number $28$ in the first box of the square as:

$28$	\[\text{21}\]	$26$
$23$	$25$	$27$
$24$		$22$

Since, we have only one box left and only one number is remaining to fill. So, we will place that number, $29$ in the remaining box as:

$28$	\[\text{21}\]	$26$
$23$	$25$	$27$
$24$	$29$	$22$

This is the filled magic square with numbers from $21$ to $29$.

Note: Here, we will discuss the up-one and right-one pattern. This pattern tells us to do the next step to fill the next number in the right box. In this pattern we need to go up and then right for the next number to fill. If there is any box, we will fill that box with number and if there is not any box and we are the right to the any box, we will go to the box of furthest left of same row to fill number and if we are top of the any box, we will go to the bottom of that column to fill number. But we get the place already occupied by any number; we will come back to the previous box and will fill the number below that number.