Peter H. answered • 05/27/14
Tutor
New to Wyzant
Tutoring in Math, Science, and Computer Engineering
Hi,
If we look up "magic square", we will find that it is a square arrangement of numbers, such that the numbers in each row, and in each column, and in the diagonals, all add up to the same number. We can also find that the magic square for 3x3 is --
2 7 6
9 5 1
4 3 8
For the first row, the numbers 2, 7, and 6 add up to 15. We see that each row, column, and diagonal adds up to 15, so this is a "magic square". One property of the magic square is that if we do the same operation to each of the elements, we still have a magic square. For example, if we multiply each element by 2, we get --
4 14 12
18 10 2
8 6 16
and now all rows, columns, and diagonals add up to 15 * 2, or 30.
You were given the job to make a magic square with -16,-12,-8,-4,0,4,8,12,16. For our magic square at the beginning, we used 1, 2, 3, 4, 5, 6, 7, 8, 9. It is interesting to see that if we take the sequence {1,...,9} and multiply each by 4 and subtract 20, we get the {-16,...,16} sequence. For example,
4 * 1 - 20 = -16
...
4 * 9 - 20 = 16
This tells us that if we take our magic square that added to 15, and take each element * 4, then subtract 20, we should still have a magic square, and solve our problem.
Give it a try !
