This was the question:

Can you make a 4-cell x 4-cell magic square, in which each of the numbers is a multiple of 5, using the numbers 5 - 80 (inclusive). The sum of each of the columns, rows and diagonals must be 170.

Professor Homunculus' answer:

Here is how we might start to solve this one:

• Ask yourself, "Do I know how any 4-cell x 4-cell magic squares?" If you do, great, if you don't, you first have to learn a basic 4-cell x 4-cell magic square. Fortunately, you can use the one shown here. You should still learn the basics of 4 x 4 squares, though.

Here is a basic 4 x 4 magic square:

• If you know your basic 4 x 4 magic square, you know that the rows, columns and diagonals each add up to 34.
• What is the relationship of 34 to the required 170? Well, if you divide 170 by 34, you will get 5. Hmmm.
• Therefore, every number in every cell could be multiplied by 5, and they would fulfill the condition of being multiples of 5,
  using the numbers 5 - 80. The columns, rows and diagonals would also add to 170.

If you multiplied each of the numbers in the square above by 5, you would get this magic square:

and it would fulfill all the criteria of the question.

This solution was an example of purely rational thought. You could have done it empirically if you didn't know any 4 x 4 magic squares. It would have probably taken you a long time, but you would have learned even more than you learned here.

What are some interesting things you notice about this square? Here are some of the things I noticed:

• There are two odd, and two even numbers in each row, column, and diagonal
• That makes the total of the units column of any row, column or diagonal add up to 10
• Therefore, the total of the tens columns of any row, column or diagonal must add up to 160, (and it does) because you are going to carry the 1 from the units to the tens column, each time.

If you noticed other interesting things, please let me know by sending me an e-mail to mathmojo@dmcom.net.

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