A magic square is a square of numbers in which all rows and columns and both diagonals add to the same number, or the magic total. The 3×3 magic square, also known as the Lo Shu square (“scroll of the River Lo” square), uses the numbers 1 to 9 and has a magic total of 15. I haven’t seen it explicitly stated anywhere on the net, perhaps because it’s trivially obvious to proper mathematicians, but in this and other 3×3 magic squares, the magic total must be three times the central number. Here is the proof:

4 | 9 | 2 |

3 | 5 | 7 |

8 | 1 | 6 |

a | b | c |

d | e | f |

g | h | i |

1. a + b + c = a + e + i = b + e + h = c + e + g

2. 3(a + b + c) = (a + e + i) + (b + e + h) + (c + e + g)

3. 3a + 3b + 3c = 3e + a + i + b + h + c + g

4. 2a + 2b + 2c = 3e + g + h + i

5. 2a + 2b + 2c – (g + h + i) = 3e

6. 3e = a + b + c = magic total

Update: In fact, this fact about 3×3 squares is mentioned a lot on the web. See, for example, Negative Magic Squares, which describes a proof discovered by Māori mathematicians in 736 B.C.E.

Some 3×3 magic squares using entirely prime numbers (except for 1 in the first square):

00043 00001 00067

00061 00037 00013

00007 00073 00031 mt = 111 = 37 x 3

00071 00005 00101

00089 00059 00029

00017 00113 00047 mt = 177 = 59 x 3

00083 00029 00101

00089 00071 00053

00041 00113 00059 mt = 213 = 71 x 3

00103 00007 00109

00079 00073 00067

00037 00139 00043 mt = 219 = 73 x 3

00107 00011 00149

00131 00089 00047

00029 00167 00071 mt = 267 = 89 x 3

00139 00007 00163

00127 00103 00079

00043 00199 00067 mt = 309 = 103 x 3

12841 09769 15013

14713 12541 10369

10069 15313 12241 mt = 37623 = 12541 x 3

12721 07753 17167

16993 12547 08101

07927 17341 12373 mt = 37641 = 12547 x 3

13183 08059 16417

15787 12553 09319

08689 17047 11923 mt = 37659 = 12553 x 3