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Example: 3 x 3 Magic Square starting with 1

Step 1: Put 1 in the middle of the first row.
Step 2: Your first move takes you out at the top of the right column, so you drop down to the bottom of that column and put 2 in that cell.
 
Step 3: Your second move takes u out at the right of the square, shift 3 to the extreme left of the row and put in that cell.
Step 4: Your next move leads to difficulty, it takes you to a cell already filled. So you drop down one row and put 4 below 3.
Step 5: Your next moves are unobstructed and hence move upto top of the last column.
Step 6: Your are now at the upper right hand corner of the square. You drop down one row and put 7 beneath 6.
 
Step 7: Your next move takes you at the right again, so you shift left and put 8 at the first cell of the first row.
Step 8: Your final move takes you to the top of the second column, so you drop to the bottom.