Magic Squares

By de Mestre, Neville | Australian Mathematics Teacher, Fall 2013 | Go to article overview

Magic Squares


de Mestre, Neville, Australian Mathematics Teacher


The following is the basis of an extension topic that I recently gave to mathematically-advanced Year 7 students at a local primary school. It could also be used for higher classes.

Consider the following 3 x 3 square array of numbers.

The total for each row, each column and each of the two diagonals is 30. Such square arrays are termed magic squares. Here is an easy one for your students to complete.

The next one is a little harder.

The problem here is that the total is not known. Some of your students may get to the answer by trial and error. At least they will find out that not all 3 x 3 square arrays are magic. Others may like to use algebra by letting one of the unknown corner numbers be z, say.

Next ask your students to make up a magic square using the numbers 1, 1, 1, 2, 2, 2, 3, 3, 3. After a short time ask them what total are they aiming for?

Some will guess correctly, but you really want someone to tell you that the required total can be determined mathematically as the sum of all the numbers divided by 3. Now they should try to find the magic square, and many will get the row and column totals correct, but not both the diagonals. The correct answer must have the three 2s along a diagonal. The reason for this is that since 2 + 2 + 2 = 6 (the required total), we cannot form two other triplets from the remaining six numbers which each add to 6. Therefore 2. 2, 2 cannot be in a row or column. The solution can be obtained by pencil and paper, but it is more productive to use a hands-on approach. Write the numbers on nine slips of paper, and then arrange them in three groups each adding to six. The simplest are three lots of 1 + 2 + 3. Place them in three rows to form the 3 x 3 array. Now move them within their rows so that the columns each add to 6. Finally check the sums of both diagonals and, if they do not come to 6, swap whole rows or swap whole columns until they do.

Next, ask them to step back a little and make up a 2 x 2 magic square. They should soon discover that the only ones possible have the same number in each cell.

Here are some other groups of nine numbers that your students could try to place in 3 x 3 magic squares.

1. 1, 2, 3, 4, 5, 6, 7, 8, 9

2 1/24, 1/12, 1/8, 1/6, 5/24, 1/4, 7/24, 1/3, 3/8

3. 7, 37, 43, 67, 73, 79, 103, 109, 139

4. x-4, x-3, x-2, x-1, x, x+1, x+2, x+3, x+4

The hands-on approach is suggested for these. …

The rest of this article is only available to active members of Questia

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Note: primary sources have slightly different requirements for citation. Please see these guidelines for more information.

Cited article

Magic Squares
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen
Items saved from this article
  • Highlights & Notes
  • Citations
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Search by... Author
    Show... All Results Primary Sources Peer-reviewed

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.