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
816^{2} + 357^{2} + 492^{2} =
618^{2} + 753^{2} + 294^{2}.
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 BenjaminYasuda reference proves the generalization for all 3x3 magic squares.
How to Cite this Page:
Su, Francis E., et al. "Magic Squares, indeed!."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.

