Secret to Solving the Square Puzzle

During my childhood days, I was very much interested in math tricks though I didn't learn much of them.  One of the puzzle that I got to know is square puzzle where numbers from certain range are arranged in such a way that the sum of every rows and columns would be same.  Also, the sum of numbers in both the diagonals would also be the same.  I learned one method to solve such puzzle(probably from my father) but what bugged me was another solution done by XYZ which was different from what I had learned.  I tried to learn how XYZ did that solution but could not.  I even asked XYZ how s/he would do it, but s/he never taught the trick.  Failed after my few attempts, I didn't try and eventually forgot that thing.

This puzzle came up again when I was browsing Quora.  I was looking a thread of Mathematics trick and happened to see the trick.  With all excitement, I crammed that technique and was delighted.  That very technique is today's recipe.  However, it only applies to odd-numbered-boxes squares.



The algorithm to solve the puzzle is given stepwise as:

1. First place the first number(say 1) in the middle box of the first row.
2. Then, place the next number in the neighboring northeast box.  If the northeast box happens to be outside the whole square , place the number in the opposite row/column to where first number was.  Note that the number should be placed in the same row/column.
3. If there already is a number in the northeast box, then place the number just below the previous number.  If the number goes outside the box, do the same as said in (2).
4. Continue this process until all the boxes are filled.

This may seem obscure at the first sight, but explanation with examples and figures will do the work.

Take a cube of 9 boxes, 3 boxes in each row and column.

Now, place the first number(in this case, 1) in the middle box of 1st row.

Now, place the next(consecutive) number in the box northeast to previous one.  The number would go out of the square, so we place it in the opposite row in the same horizontal position.  It is shown diagrammatically in the picture below.

Similarly, third number(i.e 3) is also placed.

Now, comes turn of forth number.  Here, northeast position has been occupied by number 1.  So, we place number '4' just below the previous number(i.e 3).

In this way whole square is filled.

We can work this way for any odd-numbered-rows squares.  

There is another technique too to solve this puzzle which we may discuss in our next article if I get requests to.

Well, Mathematics is not just interesting to Mathematics guys, it can be interesting to anyone.  It is not that people need to have Mathematics in their gene to understand it.  Practice and hard work are secret to Mathematics like they are for all the fields.  

People get discouraged of Mathematics because of boring teachers, not-motivating instructors and wrong methods of study.  Let's be those teachers, instructors, motivators who encourage others to learn and understand Mathematics, who show others how Mathematics is beautiful and applicable.  Sharing is what makes us happy.  Let's share our knowledge and be happier.  Mathematics is not a golden egg to hide in your treasure-store.