# Behavioral Sciences Research Core

## Math Tricks for All Ages

## Humongous Magic Square

Here is one of the biggest magic squares around. It is 15-by-15 squares in all. It was originally published by a German mathematician in 1604. (I got it from *Mathematics From the Birth of Numbers*. This is not only a nifty magic square, but the explanation that follows it below gives you a general method for creating odd-sided magic squares.

8 | 121 | 24 | 137 | 40 | 153 | 56 | 169 | 72 | 185 | 88 | 201 | 104 | 217 | 120 |

135 | 23 | 136 | 39 | 152 | 55 | 168 | 71 | 184 | 87 | 200 | 103 | 216 | 119 | 7 |

22 | 150 | 38 | 70 | 54 | 167 | 70 | 183 | 86 | 199 | 102 | 6 | 118 | 6 | 134 |

149 | 37 | 165 | 53 | 166 | 69 | 182 | 214 | 198 | 101 | 214 | 21 | 5 | 133 | 21 |

36 | 164 | 52 | 180 | 68 | 181 | 84 | 197 | 100 | 213 | 116 | 4 | 132 | 20 | 148 |

179 | 51 | 179 | 67 | 195 | 83 | 196 | 99 | 212 | 115 | 3 | 35 | 19 | 147 | 35 |

50 | 194 | 66 | 194 | 82 | 210 | 98 | 211 | 114 | 2 | 130 | 18 | 146 | 34 | 162 |

193 | 65 | 193 | 225 | 209 | 97 | 225 | 17 | 1 | 129 | 17 | 49 | 33 | 161 | 49 |

64 | 192 | 80 | 208 | 96 | 224 | 112 | 15 | 128 | 16 | 144 | 32 | 160 | 48 | 176 |

191 | 79 | 207 | 14 | 223 | 111 | 14 | 31 | 30 | 143 | 31 | 63 | 47 | 175 | 63 |

78 | 206 | 94 | 222 | 110 | 13 | 126 | 29 | 142 | 45 | 158 | 46 | 174 | 62 | 190 |

221 | 93 | 221 | 109 | 12 | 125 | 28 | 141 | 44 | 157 | 60 | 77 | 61 | 189 | 77 |

92 | 220 | 108 | 11 | 124 | 27 | 140 | 43 | 156 | 59 | 172 | 75 | 188 | 76 | 204 |

10 | 107 | 10 | 42 | 26 | 139 | 42 | 74 | 58 | 171 | 74 | 91 | 90 | 203 | 91 |

106 | 9 | 122 | 25 | 138 | 41 | 154 | 57 | 170 | 73 | 186 | 89 | 202 | 105 | 218 |

Did you find the pattern? Sort of? That's okay, it isn't completely clear until you're given the secret. So how does it work?

You begin by putting a **1** in the square that is just to the right of the center square (which is here the 8th row and 9th column). Then you enter increasing number, moving diagonally upward to the right. In this particular square you go from **1** to **7**. Any time you reach the right side of the magic square is continued on the lefthand side of the magic square, one row up. So, in this case, since the **7** hits the last column in the second row, we put the **8** on row one starting in the far left column.

When you come to the top, or first, row, then you continue in on the bottom row of the next column. In this case, **8** happens to be in the top row in the first column, so we have put **9** in the bottom, or fifteenth, row of the next column (which is the second column in this case). Then you continue increasing your numbers diagonally up toward the right like before.

When you come to a square that already has a number in it, then you skip over that number and restart on the same row but two columns over. This happens when **15** runs into **1** in this magic square. Look and see where **16** in comparison to **15** in the magic square.

One special case in this magic square happens when **120** hits the upper right hand corner. Normally it would go in the bottom square of the far left column (the bottom left corner). But it is blocked by **106**. So the two-step change gets made by putting **121** in the top square of the next column. Then, because the numbering has now reached the top row, it has to be continued on the bottom row of the next column, which is why **122** is in the third column of the bottom row.

You continue in this way until you get to the last space, and enter **225** just to the left of **113**, which is in the center of the whole magic square.

Oh, by the way, did you figure out what every row, column, and diagonal adds up to? You can add one or two of them up if you wish, or you can do the following: Figure out the total number of squares (15 squared, actually, which = 15 X 15 = 225). Then multiply the total number of squares (225) by the total number of squares plus one (226) and divide the answer by 2. That equals (225 X 226) / 2 = 26,425. That's what the total of all of the squares is. Since there are 15 squares in each of the columns or rows and diagonals, you divide 26,425 by 15 and that equals 1,695, which is exactly what each and every row, column, and diagonal adds up to.

**Now that's magic!**