hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su

The Math Fun Facts App!

List All : List Recent : List Popular
About Math Fun Facts / How to Use
Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

# Magic Squares, indeed!

Perhaps you've seen the magic square

8 1 6
3 5 7
4 9 2
which has the property that all rows, columns and diagonals sum to 15. Well, it has another "magic" and "square" property! If you read the rows as NUMBERS, forwards and backwards, and square them, then

8162 + 3572 + 4922 = 6182 + 7532 + 2942.

Magic?

Presentation Suggestions:
If they like this fun fact, ask them to take a minute to see what happens with the columns when you read them forwards and backwards and take their sums of squares. Then try the "diagonals" which wrap around the square... they also share a similar property!

The Math Behind the Fact:
This holds for ANY 3x3 magic square (though if the entries contain more than one digit, you will have to carry the extra places) using techniques of linear algebra. For instance, for this magic square:

13 6 11
8 10 12
9 14 7
you can check that:

(1300+60+11)2 +(800+100+12)2 +(900+140+7)2 = (1100+60+13)2 +(1200+100+8)2 + (700+140+9)2.

The Gardner references makes this observation for this specific 3x3 magic square, and the Benjamin-Yasuda reference proves the generalization for all 3x3 magic squares.

Su, Francis E., et al. "Magic Squares, indeed!." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Keywords:    matrix
Subjects:    algebra, number theory
Level:    Easy
Fun Fact suggested by:   Arthur Benjamin
Suggestions? Use this form.
3.81
current
rating
Click to rate this Fun Fact...
*   Awesome! I totally dig it!
*   Fun enough to tell a friend!
*   Mildly interesting
*   Not really noteworthy
and see the most popular Facts!
Get the Math Fun Facts
iPhone App!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!