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 3×3 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 3×3 magic square, and the Benjamin-Yasuda reference proves the generalization for all 3×3 magic squares.

**How to Cite this Page:**

Su, Francis E., et al. “Magic Squares, indeed!.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**References:**

A. Benjamin and K. Yasuda, “Magic Squares Indeed!”, Amer. Math. Monthly, Feb. 1999.

Martin Gardner, *Penrose Tiles to Trapdoor Ciphers*.

**Fun Fact suggested by: **

Arthur Benjamin